Hydropower plants and power systems

Transcription

1 Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology 1494 Hydropower plants and power systems Dynamic processes and control for stable and efficient operation WEIJIA YANG ACTA UNIVERSITATIS UPSALIENSIS UPPSALA 2017 ISSN ISBN urn:nbn:se:uu:diva

2 Dissertation presented at Uppsala University to be publicly examined in Polhemsalen, Ångtröm 10134, Lägerhyddsvägen 1, Ångströmlaboratoriet, Uppsala, Friday, 19 May 2017 at 13:15 for the degree of Doctor of Philosophy. The examination will be conducted in English. Faculty examiner: Doctor Christophe Nicolet (Swiss Federal Institute of Technology in Lausanne (EPFL)). Abstract Yang, W Hydropower plants and power systems. Dynamic processes and control for stable and efficient operation. Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Science and Technology pp. Uppsala: Acta Universitatis Upsaliensis. ISBN As the largest global renewable source, hydropower shoulders a large portion of the regulation duty in many power systems. New challenges are emerging from variable renewable energy (VRE) sources, the increasing scale and complexity of hydropower plants (HPPs) and power grid. Stable and efficient operation of HPPs and their interaction with power systems are of great importance. Theoretical analysis, numerical simulation and on-site measurement are adopted as main study methods in this thesis. Various numerical models of HPPs are established, with different degrees of complexity for different purposes. The majority of the analysis and results are based on eight HPPs in Sweden and China. Stable operation (frequency stability and rotor angle stability) and efficient operation are two important goals. Regarding the stable operation, various operating conditions are analysed; the response time of primary frequency control (PFC) and the system stability of isolated operation are investigated. A fundamental study on hydraulic-mechanical-electrical coupling mechanisms for small signal stability of HPPs is conducted. A methodology is proposed to quantify the contribution to the damping of low frequency oscillations from hydraulic turbines. The oscillations, with periods ranging from less than one up to hundreds of seconds, are analysed. Regarding the efficient operation, a description and an initial analysis of wear and tear of turbines are presented; a controller filter is proposed as a solution for wear reduction of turbines and maintaining the frequency quality of power systems; then the study is further extended by proposing a framework that combines technical plant operation with economic indicators, to obtain relative values of regulation burden and performance of PFC. Weijia Yang, Department of Engineering Sciences, Electricity, Box 534, Uppsala University, SE Uppsala, Sweden. Weijia Yang 2017 ISSN ISBN urn:nbn:se:uu:diva (

3 There is no elevator to success. You have to take stairs. 一步一个脚印 Dedicated to my parents and my love

4 List of Papers This thesis is based on the following papers, which are referred to in the text by their Roman numerals. Stable operation regarding frequency stability: I II III IV V VI Weijia Yang, Jiandong Yang, Wencheng Guo, Wei Zeng, Chao Wang, Linn Saarinen, Per Norrlund. A mathematical model and its application for hydro power units under different operating conditions, Energies, 2015, 8(9), DOI: /en Weijia Yang, Jiandong Yang, Wencheng Guo, Per Norrlund. Response time for primary frequency control of hydroelectric generating unit, International Journal of Electrical Power and Energy Systems, 74(2016): DOI: /j.ijepes Weijia Yang, Jiandong Yang, Wencheng Guo, Per Norrlund. Frequency stability of isolated hydropower plant with surge tank under different turbine control modes, Electric Power Components and Systems, 43(15): DOI: / Wencheng Guo, Jiandong Yang, Weijia Yang, Jieping Chen, Yi Teng. Regulation quality for frequency response of turbine regulating system of isolated hydroelectric power plant with surge tank. International Journal of Electrical Power & Energy Systems, 2015, 73: DOI: /j.ijepes Wencheng Guo, Jiandong Yang, Jieping Chen, Weijia Yang, Yi Teng, Wei Zeng. Time response of the frequency of hydroelectric generator unit with surge tank under isolated operation based on turbine regulating modes. Electric Power Components and Systems, 2015, 43(20), DOI: / Wei Zeng, Jiandong Yang, Weijia Yang. Instability analysis of pumped-storage stations at no-load conditions using a parametervarying model. Renewable Energy, 90 (2016): DOI: /j.renene

5 VII Wei Zeng, Jiandong Yang, Renbo Tang, Weijia Yang. Extreme water-hammer pressure during one-after-another load shedding in pumped-storage stations. Renewable Energy, 99 (2016): DOI: /j.renene VIII Jiandong Yang, Huang Wang, Wencheng Guo, Weijia Yang, Wei Zeng. Simulation of wind speed in the ventilation tunnel for surge tank in transient process. Energies, 9.2 (2016): 95. DOI: /en Stable operation regarding rotor angle stability: IX X Weijia Yang, Per Norrlund, Chi Yung Chung, Jiandong Yang, Urban Lundin. Eigen-analysis of hydraulic-mechanical-electrical coupling mechanism for small signal stability of hydropower plant, Submitted to: Renewable Energy, under review, Weijia Yang, Per Norrlund, Johan Bladh, Jiandong Yang, Urban Lundin. Hydraulic damping on rotor angle oscillations: quantification using a numerical hydropower plant model, Submitted to: IEEE Transactions on Energy Conversion, under review, Efficient operation and balancing renewable power systems: XI XII Weijia Yang, Per Norrlund, Linn Saarinen, Jiandong Yang, Wencheng Guo, Wei Zeng. Wear and tear on hydro power turbines influence from primary frequency control, Renewable Energy, 87(2015) DOI: /j.renene Weijia Yang, Per Norrlund, Linn Saarinen, Jiandong Yang, Wei Zeng, Urban Lundin. Wear reduction for hydro power turbines considering frequency quality of power systems: a study on controller filters, IEEE Transactions on Power Systems, DOI: /TPWRS XIII Weijia Yang, Per Norrlund, Jiandong Yang. Analysis on regulation strategies for extending service life of hydro power turbines, IOP Conference Series: Earth and Environmental Science. Vol. 49. No. 5. IOP Publishing, DOI: / /49/5/ XIV Weijia Yang, Per Norrlund, Linn Saarinen, Adam Witt, Brennan Smith, Jiandong Yang, Urban Lundin. Burden on hydropower units for balancing renewable power systems. Submitted to: Nature Energy, under formal peer review, XV Linn Saarinen, Per Norrlund, Weijia Yang, Urban Lundin. Allocation of frequency control reserves and its impact on wear on a hydropower fleet, Revised and resubmitted to: IEEE Transactions on Power Systems,

6 XVI Linn Saarinen, Per Norrlund, Weijia Yang, Urban Lundin. Linear synthetic inertia for improved frequency quality and reduced hydropower wear and tear, Submitted to: IEEE Transactions on Power Systems, under review, Reprints were made with permission from the respective publishers. The author has further contributed to the following papers, not included in this thesis: 1 Weijia Yang, Jiandong Yang, Wencheng Guo, Per Norrlund. Time-domain modeling and a case study on regulation and operation of hydropower plants, Modeling and Dynamic Behavior of Hydro Power Plants, Institution of Engineering and Technology (IET), (Book chapter) ISBN: Wencheng Guo, Jiandong Yang, Weijia Yang. Modeling and stability analysis of turbine governing system of hydro power plant, Modeling and Dynamic Behavior of Hydro Power Plants, Institution of Engineering and Technology (IET), (Book chapter) ISBN: Jiandong Yang, Wei Zeng, Weijia Yang, Shangwu Yao, Wencheng Guo, Runaway stabilities of pump-turbines and its correlations with s-shaped characteristic curves (In Chinese), Transactions of the Chinese Society for Agricultural Machinery, 46(2015): DOI: /j.issn

7 Contents 1 Introduction Power system stability Features of hydropower generating systems Hydraulic mechanical electrical coupling system Problems of oscillations Previous research Dynamic processes and modelling of hydropower plants Regulation quality and operating stability Efficient operation: wear, efficiency and financial impacts Brief summary Hydropower research at Uppsala University Scope of this thesis Outline of this thesis Methods and theory Principles of methods Numerical simulation On-site measurement Theoretical derivation Engineering cases: HPPs in Sweden and China Various hydropower plant models Numerical models in TOPSYS Model Model 4 and 4-S Numerical models in MATLAB Model 2-L (in Simulink) Model 5 and 5-S (in SPS) Models for theoretical derivation Model 3-F Model 3-L Model Numerical models in MATLAB for HPPs with Kaplan turbines (Model 2-K) System components Turbine characteristic from measurements

8 4 Stable operation regarding frequency stability Case studies on different operating conditions Comparison of simulations and measurements Discussion Response time for primary frequency control Specifications of response of PFC Formula and simulation of response time Frequency stability of isolated operation Theoretical derivation with the Hurwitz criterion Numerical simulation Stable operation regarding rotor angle stability Hydraulic mechanical electrical coupling mechanism: eigenanalysis Influence of water column elasticity (T e ) Influence of mechanical components of governor (T y ) Influence of water inertia (T w ) Influence on tuning of PSS Quantification of hydraulic damping: numerical simulation Method and model Quantification of the damping coefficient Influence and significance of the damping coefficient Discussion on quick response of hydraulic mechanical subsystem Efficient operation and balancing renewable power systems Wear and tear due to frequency control Description and definition Cause Analysis on influencing factors Controller filters for wear reduction considering frequency quality of power systems Method and model On-site measurements Time domain simulation Frequency domain analysis: stability of the system Concluding comparison between different filters Framework for evaluating the regulation of hydropower units The framework Methods Burden quantification Regulation performance Regulation payment Summary of results

9 8 Conclusion and discussion Future work Summary of papers Acknowledgements Svensk sammanfattning 中文概要 (Summary in Chinese) Appendices Appendix A Appendix B References

10 Abbreviations and Symbols Abbreviation Description AVR automatic voltage regulator GV guide vane GVO guide vane opening HPP hydropower plant OF opening feedback PF power feedback PFC primary frequency control PID proportional-integral-derivative PI proportional-integral PJM PJM Interconnection LLC PSAT Power System Analysis Toolbox PSS power system stabilizer RB runner blade RBA runner blade angle SPS SimPowerSystems SISO single-input and single-output SvK Svenska Kraftnät TSO transmission system operator VRE variable renewable energy 0-D zero dimensional 1-D one dimensional 2-D two dimensional 3-D three dimensional 10

11 Latin symbols: Symbol Unit Description a [pu] runner blade angle a w [m/s] velocity of pressure wave A [m 2 ] cross section area of pipeline A P [m 2 ] cross section area of turbine inlet A S [m 2 ] cross section area of turbine outlet a tk [pu] runner blade angle at time step tk BL a [pu] runner backlash BL gv [pu] guide vane backlash B M, B P, [m 2 /s] intermediate variables of method of characteristic C M, C P [m 3 /s] intermediate variables of method of characteristic b p [pu] governor droop b p2 [pu] governor droop of the rest of the units in the grid b p3 [pu] governor droop in Model 3 c [m/s] pressure propagation speed in penstock D [pu] common damping coefficient D 1 [m] diameter of runner D p [m] inner diameter of the pipe D t [pu] equivalent hydraulic turbine damping coefficient ( the damping coefficient ) [pu] d-axis component of the sub-transient internal emf E fd [pu] excitation EMF e g [pu] coefficient of load damping [pu] q-axis component of the transient internal emf [pu] q-axis component of the sub-transient internal emf e qy, e qω, partial derivative of turbine discharge with respect to guide [pu] e qh vane opening, speed and head e y,e ω,e h [pu] partial derivative of turbine power output with respect to guide vane opening, speed and head f [pu] frequency or turbine rotational speed f D [pu] Darcy-Weisbach coefficient of friction resistance f 0 [Hz] rated frequency of power system (50 Hz in this thesis) f c [Hz] given frequency f g [Hz] generator frequency f i [Hz] frequency of oscillation corresponding to an eigenvalue f p [pu] frictional coefficient of penstock f t [pu] frictional coefficient of tunnel G [pu] comprehensive gate opening g [m/s 2 ] gravitational acceleration 11

12 G 1 [pu] gain from frequency deviation to power deviation for the Kaplan unit G 2 [pu] gain from frequency deviation to power deviation for the lumped hydropower plant G F [pu] fitting function of the comprehensive gate opening G g [pu] transfer function describing the grid G P [pu] transfer function describing the head variation due to the discharge deviation in the penstock G PI [pu] gain from GVO deviation to frequency deviation for the PI controller G S [pu] transfer function describing the head variation due to the discharge deviation in the surge tank G t [pu] transfer function describing the Francis turbine and waterway system h [pu] water head H [m] water head in the pipeline h 0 [pu] initial water head H 0 [m] net head of turbine h 1 [s -1 ] derivative of water head with respect to time H p [m] water head at turbine inlet H s [m] water head at turbine outlet h y0 [pu] head loss of draw water tunnel I d, I q [pu] d- and q-axis component of the armature current J [kg m 2 ] moment of inertia K 1 [pu] scaling factor in Model 1 K 2 [pu] scaling factor of the lumped HPP in Model 2-K-2 K 3 [pu] scaling factor of the lumped HPP in Model 2-K-3 K a [pu] gain of exitation system (automatic voltage regulator) K d [s] governor parameter for the proportional term K i [s -1 ] governor parameter for the integral term K p [pu] governor parameter for the derivative term K s [pu] gain of power system stabilizer K ω, K Pe [pu] gain of power system stabilizer for selecting different input L [m] length of penstock M [s] system inertia M 11 [N/m 2.5 ] unit mechanical torque M g [N m] resistance torque of generator m g [pu] relative resistance torque of generator M R [MW] regulation mileage M R-base [MW] base value of regulation mileage M t [N m] mechanical torque m t [pu] relative mechanical torque n [rpm] rotational speed 12

13 n 11 [m 0.5 /s] unit rotational speed n c [rpm] given rotational speed n r [rpm] rated rotational speed P A,i, P B,i [pu] absolute value of a local maximum or local minimum of speed deviation Pay mile [pu] amount of mileage payment Pay strength [pu] amount of strength payment p c [MW] given power P e, P m [pu] electromagnetic active power and mechanical power p g [pu] generator power p l [pu] load p m [pu] active power p m, k [pu] active power at time step k p m0 [pu] initial active power p m2 [pu] active power of the lumped HPP P m-rated [MW] rated power of generating unit p r [MW] rated power of generating unit P RMSE [pu] a root mean square error used for quantifying Dt P step [MW] increase in output power caused by a frequency step change from 50 Hz to 49.9 Hz q [pu] discharge q 0 [pu] initial discharge Q 11 [m 0.5 /s] unit discharge [m 0.5 /s] Q e, Q g [pu] reactive power of generator Q p [m 3 /s] discharge of turbine inlet Q s [m 3 /s] discharge of turbine outlet q t [pu] discharge of turbine q y [pu] discharge of draw water tunnel s [s -1 ] complex variable in Laplace transform S R [MW/Hz] regulation strength S R1, S R1-pu [pu] regulation strength of the Kaplan unit S R2, S R2-pu [pu] regulation strength of the lumped hydropower plant S R-base [MW/Hz] base value of regulation strength S RT [pu] regulation strength of all the units in the grid t [s] time T 0, T 1, T 2 [s] parameters of power system stabilizer, [s] open-circuit d-axis transient and sub-transient time constants T del-a [s] delay time in runner control T del-gv [s] delay time in guide vane control T e [s] time constant of water column elasticity, Te = L/c T f [s] period of frequency oscillation T F [s] surge tank time constant T j [s] mechanical time constant 13

14 T j [s] mechanical time constant t k / number of time step t p [s] time constant in grid inverse model [s] open-circuit d-axis sub-transient time constants T r [s] time constant in exitation system (automatic voltage regulator) T s [s] time constant of surge T w [s] water starting time constant T wp [s] water starting time constant of penstock T wt [s] water starting time constant of tunnel T wy [s] water starting time constant of draw water tunnel T y [s] time constant of guide vane servo T ya [s] time constant of runner blade servo V [m/s] average flow velocity of pipeline section V 1 [pu] signal between washout and phase compensation block in power system stabilizer V g [pu] voltage at the generator terminal V gd, V gq [pu] d- and q-axis component of the voltage at the generator terminal V PSS [pu] output signal of power system stabilizer V s [pu] infinite bus voltage V sd, V sq [pu] d- and q-axis component of the infinite bus voltage x [m] position X d,, [pu] d-axis synchronous, transient and sub-transient reactance of generator, [pu] ; x f [pu] relative value of speed (frequency) deviation, xf = (fg fc)/fc X q, [pu] q-axis synchronous and sub-transient reactance of generator X s [pu] total reactance of transmission line (between generator and infinite bus) y [pu] guide vane opening y c [pu] given value of guide vane opening Y GV, dist [pu] movement distance of guide vane y PI [pu] guide vane opening signal between PI terms and servo y PID [pu] guide vane opening signal after PID terms Y RB, dist [pu] movement distance of runner blade y servo [pu] guide vane opening signal after the servo z [pu] relative change value of water level in surge tank 14

15 Greek symbols: Symbol Unit Description α, α p [pu] elasticity coefficient of penstock α HP [m -2 ] correlation coefficient of kinetic energy at turbine inlet α HS [m -2 ] correlation coefficient of kinetic energy at turbine outlet δ [rad] power (or rotor) angle Δ / stands for a deviation from a steady state value Δf [pu] frequency deviation from set-point value Δh [pu] water head deviation from initial value ΔH [m] HP HS 2 H - Q 2 2 P 2gAP 2gAS Δh p [pu] water head deviation from initial value due to hydraulic dynamics in penstock Δh s [pu] water head deviation from initial value due to hydraulic dynamics in surge tank Δn [pu] speed deviation Δq [pu] discharge deviation from initial value Δt [s] time step in simulation Δy [pu] guide vane opening deviation from set-point value Z [m] absolute change value of water level in surge tank Δη [pu] efficiency change η [pu] turbine efficiency η I [pu] interpolation function of the turbine efficiency η Sj [pu] average value of the instantaneous efficiency during the operation period under a specific strategy (Sj) η st [pu] on-cam steady state efficiency θ [rad] angle between axis of pipeline and horizontal plane ξ / damping ratio of an oscillation φ [rad] power factor angle at the generator terminal ω [pu] angular velocity of the generator ω 0 [rad/s] synchronous angular velocity in electrical radians (equals to 2πf0) Note that the symbols in subsection of mathematical variables for theory introduction are explained within the text, and they are not listed here. 15

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17 1 Introduction Hydropower has played an important role in the safe, stable and efficient operation of electric power systems for a long time. Hydropower not only generates electricity as the largest global renewable source, but also shoulders a large portion of the regulation and balancing duty in many power systems all over the world. Hydropower technology is relatively mature, but new challenges are still emerging. First, with current trends toward de-carbonization in the electricity sector [1], the amount of electricity generated by variable renewable energy (VRE) sources has been constantly growing [2, 3]. Dealing with generation intermittency of VRE in an effective and efficient manner is a growing research field [3-6]. High VRE integration [7] and fewer heavy synchronously connected generators, which imply less inertia [8], lead to crucial consequences for power system stability. Second, a hydropower generation system is a complex nonlinear power system including hydraulic, mechanical and electrical subsystems (details in section 1.2). The generator size and the complexity of waterway systems in hydropower plants (HPPs) have been increasing. Especially in China [9, 10] dozens of HPPs with at least 1000 MW capacity are being planned, designed, constructed or operated. Third, many large HPPs are located far away from load centres, forming many hydro-dominant power systems, such as the cases in Sweden [11] and China [12]. In recent years, there has been a tendency that the new turbines experience fatigue to a greater extent than what seem to be the case for new runners decades ago [13], and the maintenance needs at HPPs are affected [14], due to more regulation movements caused by increasingly more integration of VRE. In some countries, as in Sweden, primary frequency control (PFC) is a service that the transmission system operator (TSO) buys from the power producers. In other countries, as in Norway and China, there is also an obligation for the producers to deliver this service, free of charge. However, there are costs related to this, e.g. due to design constraints and auxiliary equipment when purchasing a new unit or system, due to wear and tear that affects the expected life time and maintenance intervals, and due to efficiency loss when a unit operates in a condition that deviates from the best efficiency point, etc. Based on the aforementioned aspects, the demand on the quality of regulation emanating from hydropower units has been increasing. Stable and efficient operation of HPPs and their interaction with power systems is of great importance. 17

18 1.1 Power system stability Power system stability is generally defined as a property of a power system, and it enables the system to remain in a stable operating state under normal operating conditions and to restore an equilibrium after a disturbance [15]. Three forms of power system stability are defined as follows [16]. (1) Frequency stability refers to the ability of a power system to maintain steady frequency after a severe system disturbance leading to a significant imbalance between generation and load. It depends on the ability to maintain and restore equilibrium between system generation and load, with minimum unintentional loss of load [16]. (2) Rotor angle stability refers to the ability of synchronous machines in a power system to remain synchronized after a disturbance. It depends on the ability to maintain and regain stability between electromagnetic torque and mechanical torque of each synchronous machine in the system [16]. (3) Voltage stability means the capability of a power system to maintain steady voltages at all buses in the system after a disturbance from a given initial operating condition. It is determined by the ability to maintain and restore equilibrium between load demand and load supply from the power system [16]. In this thesis, the main focus is on the frequency stability and the rotor angle stability of power systems. In order to maintain frequency stability, generating units change their power output automatically according to the change of grid frequency, to make the active power balanced again. This is the PFC. PFC of electrical power grids is commonly performed by units in HPPs, because of the great rapidity and amplitude of the power regulation. PFC supplied by hydropower units is a core content of this thesis. It is worth noting that the term stability is also used with respect to control theory, which is introduced in subsection Features of hydropower generating systems In this section, important features of hydropower generating systems are highlighted, as main analysis objects of the works throughout the thesis Hydraulic mechanical electrical coupling system A HPP is a complex nonlinear system integrating hydraulic mechanical electrical subsystems, as shown in Figure

19 Figure Simple illustration of a HPP: a hydraulic mechanical electrical coupling system. Figure 1.2. Common ranges of some standard time constants in the hydropower system, indicating the interactions among the multiple variables. The definitions of the symbols are in the section Abbreviations and symbols. Time constants regarding the electrical subsystem (generator) are in red, and time constants regarding hydraulic and mechanical subsystems are in blue. A core scientific challenge is to reveal the coupling mechanisms and oscillation characteristics of diverse physical variables within multiple subsystems. For describing transient processes in HPPs, there are several common time constants, such as: transient and sub-transient time constants of generator 1 (accessed on March 14th, 2017) 19

20 (,, ), mechanical time constant (T j ) regarding the electrical subsystem; water starting time constant (T w ), time constants of water column elasticity (T e ), servo (T y ), and surge (T s ) in surge tank or gate shaft, etc. regarding the hydraulic and mechanical subsystems. The common ranges of these time constants are presented in Figure 1.2, indicating the interactions among the multiple variables Problems of oscillations There are different oscillation issues with various periods existing in hydropower generating systems, as illustrated by the measured data in Figure 1.3 (350 s oscillation), Figure 1.4 (60 s oscillation) and Figure 1.5 (1 s oscillation). In terms of different categories of stability studied in this thesis, oscillation periods regarding frequency stability (> 20 s) is normally larger than the ones regarding rotor angle stability (< 5 s). It is worth noting that in the power system field, a rotor angle oscillation with a frequency between Hz is defined as a low frequency oscillation [17]; however in this thesis, this frequency range is relatively high. Hence, the oscillations regarding frequency stability is called very low frequency oscillation, and the term has already been used in [18]. In this thesis, the very low frequency oscillation regarding frequency stability is investigated in chapter 4 and chapter 6; then the low frequency oscillation regarding rotor angle stability is studied in chapter 5. Figure 1.3. Oscillations with a period around 350 s: measurements of guide vane opening (GVO) and power output under a load step disturbance in a Chinese HPP with a surge tank. 20

21 Figure 1.4. Oscillation with a period around 60 s: measured frequency of the Nordic power system Figure 1.5. Oscillation with a period around 1 s: measured electromagnetic power under disturbances of excitation voltage in a Swedish HPP. 1.3 Previous research Dynamic processes and modelling of hydropower plants Previous studies have obtained many meaningful achievements regarding the dynamic processes and modelling of HPPs, and these works could be roughly divided into the following three categories. (1) The first category mainly works on the electrical perspective (generators and power grids) by simplifying the hydraulic and mechanical systems. This is a standard approach for small signal stability analysis of power systems in the classical books, such as in [15] and [19]. Power system stabilization was investigated based on hydropower generators through excitation control in several representative early works [20-22]. New controllers of reactive power were proposed for grid-connected operation [23, 24] and the emergency start- 21

22 up process [25]. Besides, in studies on inter-area mode oscillations, the hydro turbine model is normally very simple, such as in [26-29]. (2) The second category is to simplify the electrical sub-system by applying the first-order swing equation of generating units, focusing on the hydraulicmechanical subsystem. Accuracy of second order hydraulic turbine models was studied [30, 31]. Hydraulic-turbine and turbine control-models for system dynamic studies were proposed [32, 33]. Field tests were conducted to validate a hydro turbine-governor model structure and parameters [34], and modelling and controller tuning of a HPP with units sharing a common penstock section were presented [35]. A nonlinear model of penstock and a hydraulic turbine model were proposed for simulation of hydraulic transients [36]. A basic simulation tool for analysis of hydraulic transients in HPPs was established [37]. Modelling of the dynamic response and control of Francis turbines were conducted [38] [39]. A one-dimensional and three-dimensional (1D-3D) coupling method was proposed for hydraulic system transient simulations [40]. A physics based hydraulic turbine model was built for system dynamics studies [41]. For HPPs with Kaplan turbines, a nonlinear digital simulation model was proposed [42], and differential evolution-based identification of a nonlinear model was conducted [43]. HPP models and control were reviewed in [44], and modelling and dynamic behaviour of HPPs was introduced in the two books [45] and [46]. (3) In recent years, the works on the coupling of the hydraulic-mechanicalelectrical subsystems with relatively detailed models on all three subsystems have been emerging. Based on the simulation software SIMSEN, high-order modelling of HPPs in islanded power networks was conducted [47], and dynamical behaviour of variable speed and synchronous machines with power system stabilizer (PSS) were compared [48]. Based on the simulation software TOPSYS, transient processes for hydraulic, mechanical and electrical coupling system were studied [49, 50]. Hydro turbine and governor models in a free and open source software package (Power System Analysis Toolbox, PSAT) were developed and implemented [51]. There are other software packages for analysis of dynamic processes of HPPs, e.g. LVTrans [52], the Modelica Hydro Power Library 2, and Alab 3, etc. An object-oriented framework was applied to the study of electromechanical oscillations at a HPP [53]. An advanced-model of synchronous generators for HPP numerical simulations was built [54]. Nonlinear dynamic analysis and robust controller design were conducted for turbine regulating system with a straight-tube surge tank [55]. 2 (accessed on March 22nd, 2017) 3 (accessed on March 22nd, 2017) 22

23 1.3.2 Regulation quality and operating stability Previous research on the regulation and control of hydropower units is introduced here, in terms of frequency stability and rotor angle stability. (1) On frequency stability The hydraulic turbine regulation was comprehensively introduced in a book [56]. Oscillatory behaviour of a British HPP was reproduced by numerical simulation [57]. Hydro turbine-governor model validation in Pacific Northwest was conducted [58]. The effects of governor settings on the stability of the Turkish power system frequency were investigated [59]. Analysis of very low frequency oscillations in hydro-dominant power systems (the Colombian power system) and design of robust control to damp oscillatory modes were conducted [18, 60]. Speed and active power control of HPPs was studied [61], and impact from hydraulic transients [62] and a dynamic model of a Kaplan turbine regulating system [63] on power system dynamic stability were discussed. Load frequency regulation of HPPs in isolated systems was studied [64-66]. Influence of the hydraulic system layout [67], the elastic property of penstock [68] and the effect of surge tank throttling [69] on the system stability were analysed. Stability analysis of governor-turbine-hydraulic system was conducted by state space method and graph theory [70]. A fuzzy sliding mode controller was designed via input state feedback linearization method [71]. A series of nonlinear modelling and stability analysis of hydro-turbine governing system were conducted [72-76]. Comprehensive theoretical analysis on response and stability of HPPs based on various forms of surge tanks and tunnels were carried on [77-80]. (2) On rotor angle stability Various controllers were proposed or optimized for enhancing rotor angle stability of hydropower generating systems. An application of a multivariable feedback linearization scheme was presented [81]. An approach was proposed for the damping of local modes of oscillations resulting from large hydraulic transients [82]. Nonlinear decentralized robust governor control [83], nonlinear coordinated control of excitation and governor [84] and dynamic extending nonlinear H control [85] for hydropower units were studied. Damping low frequency oscillations with hydro governors was investigated [86], and the effect of governor settings on low frequency inter area oscillations was assessed [87]. Eigenvalue analysis on the stability of HPPs was performed based on phase variables a, b, c instead of d, q-components [88, 89]. An eigenanalysis was conducted for the oscillatory instability of a HPP, and the influence of the water conduit dynamics was studied [90]. The second-order oscillation mode of hydropower systems was studied in [91] based on a linear elastic model and modal analysis. 23

24 1.3.3 Efficient operation: wear, efficiency and financial impacts Previous studies have obtained a considerable amount of illuminating achievements regarding the wear and tear of hydropower turbines. From the point of view of hydraulics, the hydro turbine failure mechanisms [92] and the fatigue damage mechanism [93] were comprehensively reviewed; the fatigue design and life of Francis turbine runners were investigated [94-96]; the pressure at discharge control and frequency control was analysed for the life time of high head runners and low head runners [14]; consequences of PFC to the residual service life of Kaplan runners were investigated [97]; dynamic loads on Francis runners and their impact on fatigue life were examined [13, 98]. Within the tribology research field, the wear on bearing materials of GVs in hydro turbines is a main topic [99-101]. Wear and tear is also mentioned in studies on the design and tuning of controllers of hydropower units [ ], but it is not a main objective. In terms of reducing efficiency loss, investigations were conducted on turbine design [105, 106] and strategies of operation and power dispatch [ ]. Kaplan turbine efficiency improvement was proposed, e.g. through draft tube design [114, 115] and control methods optimizations [116, 117]. Operating performance enhancements of Kaplan turbines were comprehensively described [118]. From the aspect of research regarding scheduling and financial feasibility, the hydropower unit start-up cost [ ] was investigated. The costs and financial impacts of operation, production and maintenance of HPP were studied [ ] Brief summary Many meaningful studies have been carried out on the topics above, however there are still limitations in these works. In terms of stable operation, most of the previous works only conducted numerical simulations and did not theoretically explain the oscillation mechanism. While in case of works that include theoretical analysis, the hydraulic-mechanical-electrical subsystems is still relatively simple, and only limited factors are discussed. Hence, a fundamental and comprehensive study on the stability of the multi-variable hydropower systems is very important. Regarding the efficient operation, to the author s knowledge, there have been few publications on wear and tear issue in the field of hydropower regulation and control. It is necessary to study the wear and tear from aspects of regulation strategies. Besides, in front of producers and TSOs, there are new challenges that are to minimize regulation burden, maintain good regulation performance, and to agree on reasonable compensation structures. Therefore, a systematic study quantifying and evaluating the trade-off between burden and performance of hydropower regulation is needed. 24

25 1.4 Hydropower research at Uppsala University The hydropower research at the Division of Electricity in the Department of Engineering Sciences at Uppsala University was initiated in Four PhD theses have been published until now. Ranlöf conducted electromagnetic analysis of hydroelectric generators [126]; Bladh studied the interaction between hydropower generators and power systems [127]; Wallin focused on measurement and modelling of unbalanced magnetic pull in hydropower generators [128]; Saarinen investigated hydropower and grid frequency control [129]. Presently, the main topics of the hydropower research at the department are axial magnetic leakage flux in hydropower generators, actively controlled magnetic bearings, brushless exciter systems, frequency control and dynamic processes in HPPs. 1.5 Scope of this thesis In the title of this thesis, stable operation and efficient operation are highlighted as two important goals. Regarding the stable operation, the frequency stability and the rotor angle stability are studied. For the frequency stability, the main focus is on PFC; secondary frequency control is also involved, such as in Paper XV, but it is not explicitly presented in the following context. For the rotor angle stability, the main objective is the local modes [15]. Regarding the efficient operation, here it is a general term that refers to a good operating condition or status with respect to overall economic performance; more exactly, it relates to wear and tear of units, generating efficiency, etc. Here, the term economic operation is not adopted, since the study is from a physical perspective by applying some indicators, and no practical profit or economic value is explicitly demonstrated. In terms of common mathematical methods in the hydropower study field, they can be divided into the following three categories, from an aspect of modelling sophistication of hydropower units: 3-D (three-dimensional), 1-D (onedimensional), and 0-D (zero-dimensional) methods. Here, 0-D means that a generating unit is simplified as a node in a model, and it is normally applied in analyses of complex systems. In this thesis, all the works are conducted from the 1-D (one-dimensional) perspective, as demonstrated in Figure 1.6. A goal of this thesis is starting to build a bridge between studies of detailed components (e.g. turbines, generators, etc.) with 3-D methods and studies of large systems (e.g. power systems, cascade HPPs, etc.) with 0-D methods. 25

26 Figure Scope of mathematical method in this thesis: 1-D method 1.6 Outline of this thesis In chapter 2, the methods and theory of the research in this thesis are briefly introduced. In chapter 3, various HPP models for different study purposes are developed and presented. In chapter 4, the stable operation of HPPs regarding frequency stability of power systems is analysed, mainly based on Papers I III. In chapter 5, the stable operation of HPPs regarding rotor angle stability of power systems is studied, mainly based on Paper IX and Paper X. In chapter 6, the efficient operation of HPPs during balancing actions for renewable power systems is investigated, mainly based on Papers XI XIV. In chapter 7, the results of the thesis are summarized. In chapter 8, the conclusion and discussion are condensed, and future work is suggested in chapter 9. 4 Parts of Figure 1.6 are from figures on the following websites (accessed on March 17th, 2017):

27 A brief summary of the papers, acknowledgements and summaries in Swedish and in Chinese are included in the thesis. At last, appendices are supplied for detailed information. Categorization and relation of the main works and papers in this thesis are illustrated in Figure 1.7. Figure 1.7. Categorization and relation of the main works in this thesis. The Roman numerals indicate the corresponding paper numbers. 27

28 2 Methods and theory In this chapter, the methods and theory of this thesis are briefly displayed. In section 2.1, principles of three main methods used in this thesis are presented. In section 2.2, practical engineering cases adopted in this thesis are introduced. 2.1 Principles of methods In this section, principles of numerical simulation, on-site measurement, and theoretical derivation applied in the thesis are concisely introduced Numerical simulation Time-domain numerical simulation is a core method of this thesis. Various numerical models of HPPs are established, with different degrees of complexity for different purposes. The implementations are based on VC++, MATLAB / Simulink and MATLAB / SimPowerSystems (SPS); the PSAT is also applied to conduct basic power system simulations. Details of the numerical models are introduced in section On-site measurement On-site measurement in HPPs in Sweden and China is an important approach of this thesis, for observing practical engineering problems and validating the numerical simulations. The majority of measurement data can be directly obtained from the measurement system installed in HPPs. Additional measurements were also conducted, and part of the measurement devices in Swedish HPPs are illustrated in Figure 2.1 and Figure 2.2. Detailed information regarding settings and results of measurements are included in the following sections. 28

29 (a) (b) Figure 2.1. Measuring device for turbine actuator movement for a Francis unit. (a) A distance transducer of wire type, which measures position of servo link; (b) An angular transducer which measures GVO. (a) (b) Figure 2.2. Measurement devices for a Kaplan unit. (a) Guide vane (GV) servo with feedback transducer, connected with orange cables, on its right. (b) Feedback device on the top of the generator shaft, for runner blade angle (RBA). The red device on top is the transducer. The rod is close to 14 m long and ends in the runner hub. 29

30 2.1.3 Theoretical derivation First, the basic theory of the Laplace transform and transfer functions is presented, and they are applied in the governor equation analysis in the studies of the response time (section 4.2) and wear of the turbine (section 6.1). Second, the Routh Hurwitz criterion is introduced, and it is adopted to analyse the frequency stability in section 4.3. Third, the Nyquist stability criterion and the describing function method are introduced, and they are utilized for discussing the frequency stability in section 6.2. The content in these three parts are based on two books [130, 131]. Lastly, the state matrix and eigen-analysis are briefly presented based on [15], and they are applied for analysing the small signal stability in section Laplace transform and transfer function This part is based on [131], for introducing the basic theory of the Laplace transform and the transfer function. Laplace transform and inverse Laplace transform: Solving linear ordinary differential equations is a main purpose of the Laplace transform, through the following three steps: (1) transforming the differential equations into an algebraic equation in s-domain; (2) dealing with the simple algebraic equation; (3) taking the inverse Laplace transform to obtain the solution. The mathematical definition of the Laplace transform and the inverse Laplace transform are found in [131]. Transfer function for a single-input and single-output (SISO) system: For a linear time-invariant SISO system, the definition of a transfer function of it is Gs L gt, (2.1) and all the initial conditions set to zero. Here, L is the Laplace transform, and g(t) means the impulse response of the system with the scalar functions input u(t) and output y(t); s is a complex variable in the Laplace transform. The relationship between the Laplace transforms is Ys Gs, (2.2) Us with all the initial conditions set to zero. Here, Y(s) and U(s) are the Laplace transforms of y(t) and u(t), respectively. For an nth-order differential equation with constant real coefficients, as presented in n n1 d yt d yt dyt a 1... n n a n 1 1 a0y t dt dt dt, (2.3) m m1 d u t d u t du t bm bm 1... b m m 1 1 b0u t dt dt dt 30

31 by simply taking the Laplace transform of both sides of the equation and assuming zero initial conditions, the transfer function of the linear time-invariant system is obtained, as shown in n n1 s an 1s... a1sa0y s. (2.4) m m1 bs b s... bsb U s m m1 1 0 The transfer function from u(t) to y(t) is m m1 Ys bs m bm s... bsb Gs n n1 U s s a s... asa n (2.5) Routh Hurwitz stability criterion This section is based on [131], for presenting the basic theory of the Routh Hurwitz stability criterion. In order to determine the stability of linear timeinvariant SISO systems, a simple way is to study the location of the roots of the characteristic equation of the system. The stable and unstable regions in the s-plane are shown in Figure 2.3 [131]. Figure 2.3. Stable and unstable regions in the s-plane (s=σ+jω) For a linear time-invariant system that has a characteristic equation with constant coefficients, the Routh-Hurwitz criterion can be used to investigate the stability by simply manipulating arithmetic operations. For a linear time-variant SISO system of which the characteristic equation is n n1 Fs as n an 1s... as 1 a0 0, (2.6) the analysis process of Routh-Hurwitz criterion is presented in the following steps. (1) Step one is forming the coefficients of the equation in (2.6) into the following tabulation: an an2 an4 an6... a a a a... n1 n3 n5 n 7 31

32 (2) Step two is to obtain the Routh's tabulation (or Routh's array); which is illustrated in Figure 2.4 for a sixth-order equation: 32 as as... asa 0. (2.7) s a6 a4 a2 a0 5 s a5 a3 a1 4 aa 5 4 aa 6 3 aa 5 2 aa 6 1 aa 5 0 a6 s 0 A B a a a a Aa3 a5b Aa1a5a 0 A0a50 s C D 0 A A A 2 BC AD Ca0 A0 C 0A0 s E a0 0 C C C ED Ca0 s F E Fa0 E 0 s a0 F Figure 2.4. The Routh's tabulation for a sixth-order equation (3) Step three is to determine the location of the roots by studying the signs of the coefficients in the first column of the tabulation. If the signs of all the elements of the first column are the same, the roots of the equation are all in the left half of the s-plane that is the stable region. Besides, the number of roots in the right half of the s-plane (unstable region) equals the number of changes of signs in the elements of the first column Nyquist stability criterion This part is based on [131], for introducing the basic theory of the Nyquist stability criterion. For a SISO system with the closed-loop transfer function that is Gs M s, (2.8) 1 GsHs the characteristic equation roots must satisfy s 1Gs Hs 1Ls 0. (2.9) Here, L(s) is the loop transfer function. The stability of a closed-loop system can be determined by the Nyquist criterion, through analysing the frequency-domain plots (the Nyquist plot) of the loop transfer function L(s). Absolute stability: There are two types of absolute stability as follows. (1) Open-loop stability: A system is open-loop stable, if all the poles of L(s) are in the left-half s-plane. (2) Closed-loop stability: A system is closed-loop stable, if all the zeros of the 1+ L(s) or all the poles of the closed-loop transfer function are in the left-half s-plane

33 Relative stability: The relative stability indicates how stable the system is, and it is measured by how close the Nyquist plot (or Nyquist curve) of L(s) is to the (-1, j0) point, in the frequency domain. The gain margin and phase margin are frequently used criteria to reflect relative stability of control systems. The detailed definitions of the two types of margin can be found in [131] Describing function method for non-linear systems This part is based on [130], to introduce the basic theory of the stability analysis for non-linear systems and the describing function method. The describing function method is commonly used together with the Nyquist criterion to analyse the stability of non-linear systems. Here, a system without external signals is taken as an example, the block diagram is shown in Figure 2.5 [130]. Figure 2.5. Block diagram for the describing function method In the system, G(s) is the transfer function of a linear part, the function f describes a static nonlinearity, and the input and the output signal of f are assumed to be et Csin t. (2.10) w t f Csin t By expanding this function into a Fourier series, the function w can be written in the form w t f Csin t f0 C ACsin t C (2.11) + A C sin 2t C A C sin 3 t C For the fundamental component (with the angular frequency ω), the gain is argg i. Gi and the change in phase is Then, a complex number is defined as i C ACe Yf C, (2.12) C which is the describing function of the nonlinearity. By introducing the Fourier coefficients 33

34 1 2 ac f Csincosd ACsinC 0, (2.13) 1 2 bc f Csinsind ACcosC 0 the complex number can be described as bc iac Yf C. (2.14) C The equations of the system can be presented in an equation with the two unknowns C and ω f 1 Y C G i. (2.15) The solutions of the equation are the amplitude and frequency of a corresponding oscillation. No solution of the equation indicates that the system likely does not oscillate. By rewriting (2.15) as 1 Gi Yf C, (2.16) it is clearer that the solution can be interpreted as an intersection of the following two curves in the complex plane: the Nyquist curve G(iω) plotted as a function of ω and the -1/Y f (C) curve plotted as a function of C. An intersection of these two curves indicates the possible presence of an oscillation in the system [130], and time domain simulations might be needed to obtain clearer results State matrix and eigen-analysis This part is based on [15], to introduce the basic theory of the state matrix and eigen-analysis. The eigenvalues of a matrix are the values of the scalar parameter λ for which the following equation has non-trivial solutions (i.e., other than Ф = 0) A. (2.17) Here, A is an n n matrix (e.g., for a power system), and Ф is an n 1 vector. The n solutions of λ (i.e., λ 1, λ 2,, λ n ) are eigenvalues of A. The free motion of a dynamic system can be described by a linearized form x Ax, (2.18) where the column vector x means the state vector, and the derivative of the state variable with respect to time is ; A is the state matrix of the system. The eigenvalues are applied to determine the stability of the system as follows. Real eigenvalue: A real eigenvalue indicates a non-oscillatory mode. A positive real eigenvalue corresponds to an aperiodic instability, and a negative real eigenvalue represents a decaying mode. A larger magnitude of a negative eigenvalue means a faster decay. 34

35 Complex eigenvalue: Complex eigenvalues appear in conjugate pairs (for a state matrix A of which all the elements are real), and each pair indicates an oscillatory mode. More exactly, for a pair of complex eigenvalues j, (2.19) the frequency (in Hz) of the corresponding oscillation is f, (2.20) 2 which stands for the actual or damped frequency. The damping ratio is described as, (2.21) 2 2 which determines the rate of decay of the oscillation amplitude. More detailed introduction of eigen-analysis and its application to power system analysis can be found in [15]. 2.2 Engineering cases: HPPs in Sweden and China In this thesis, the majority of the analysis and results are based on real engineering cases. Eight HPPs in Sweden and China are applied as study cases in different parts of the thesis for various purposes. Table 2.1 presents the basic information of these HPPs, and more details are given in the following sections and Appendices. The Swedish HPPs are owned by Vattenfall, the largest hydropower owner and operator in Sweden. Table 2.1. Basic information of the engineering cases of this thesis. The values of each HPP are rated values for a single unit. HPP Location Type of turbine Active power [MW] Water head [m] Discharge [m 3 /s] Rotational speed [r/min] 1 Sweden Francis China Francis China Francis China Francis Sweden Francis Sweden Kaplan Sweden Kaplan Sweden Kaplan

36 3 Various hydropower plant models In this thesis, various HPP models for different study purposes are developed. For a clearer demonstration and classification of models, they are distinguished from two aspects: (1) the implementation and (2) the generator modelling approach, as shown in Figure 3.1. There are three categories regarding the implementations: TOPSYS (a software for transient processes in HPPs) [50, 132], MATLAB and theoretical derivation; models in TOPSYS and MATLAB are for numerical simulation. In terms of the generator modelling approach, the first-order model is used in HPP models for studies regarding frequency stability, and high-order generator models are built in HPP models to investigate rotor angle stability. Figure 3.1. Various HPP models in this thesis, and the models are distinguished from two aspects: implementation and generator modelling approach. F, K, L and S mean Francis, Kaplan, Lumped and Simplified respectively. There are totally ten models in six categories. The arrows pointing to Sophisticated generally indicates the degree of complexity of among these categories. As shown in Figure 3.1, there are totally ten models in six categories. F, K, L and S mean Francis, Kaplan, lumped and simplified respectively. Generally speaking, the degree of complexity in TOPSYS models 36

37 is higher than in the MATLAB models, and the theoretical models are further simplified, as indicated by the arrows in Figure 3.1. Table 3.1 shows the main differences and characteristics of the ten models. Table 3.1. Ten models built in this thesis. The descriptions of abbreviations are as follows: Char. Characteristics, F Francis, K Kaplan, L lumped, S simplified. P m is the turbine mechanical power. The implementations in MATLAB are divided into Simulink and SPS. The column Chapter/Section shows the corresponding parts in the thesis for main applications of the models. Model Implementation Generator model Turbine type Main corresponding papers Chapter/ Section 1 TOPSYS 1st-order Francis I, II, III, XI 4, K Simulink 1st-order Kaplan XIV L Simulink 1st-order Francis XII F Theoretical 1st-order Francis III L Theoretical 1st-order Francis XII TOPSYS 5th-order Francis X S TOPSYS 5th-order None X SPS 7th-order Francis IX S SPS 7th-order None X Theoretical 5th-order Francis IX 5.1 Model Turbine characteristic Water column Surge tank Grid Swing equation 1 Char. curves Elastic Yes Various General 2-K Specific model Elastic Yes Nordic grid General 2-L Ideal Rigid No Nordic grid General 3-F Linear Rigid Yes Isolated General 3-L Ideal Rigid No Nordic grid General 4 Char. curves Elastic Yes SMIB General 4-S None None No SMIB Constant Pm 5 Linear Elastic No SMIB General 5-S None None No SMIB Constant Pm 6 Linear Elastic No SMIB General Details of these ten HPP models are introduced in the following sections. It is worth noting that there are several other models extended or slightly transformed from these ten models for various purposes, and they are presented in the corresponding parts in section 4 through section 6. Additionally, some 37

38 power system models are also applied in the thesis but not presented here, such as a model implemented in the PSAT in section Numerical models in TOPSYS In this section, Model 1, Model 4 and Model 4-S are presented. These models are based on the software TOPSYS [50, 132] that is developed by applying Visual C++ for scientific studies and consultant analyses of transient processes in HPPs. These models are now implemented in TOPSYS as an extension version. The graphical user interface of TOPSYS is shown in Figure 3.2. In the basic version of TOPSYS, the model of waterway systems and hydraulic turbines has the following merits. (1) Equations for compressible flow are utilized in the draw water tunnel and penstock, considering the elasticity of water and pipe wall. (2) Various types of surge tanks and tunnels are included in the model library. (3) Turbine characteristic curves are used, instead of applying a linearized model with transmission coefficients. These characteristics lay a solid foundation for this work to achieve efficient and accurate simulation results. Figure 3.2. Graphical user interface of TOPSYS and a model of a Swedish HPP (HPP 1 in this thesis) Model 1 Based on Paper I, a mathematical HPP model, especially a governor system model for different operating conditions, is presented in this subsection. Model 1 is applied for case studies on different operating conditions of HPPs 38

39 and the main results are presented in section 4, including comparisons with on-site measurements. Model 1 is also utilized for investigating the influence of PFC on wear and tear in section Generator and power grid As shown in Figure 3.1, the main difference of Model 1 and Model 4 is the generator modelling approach. Model 1 adopts the first-order swing equation to describe the whole generator. Equations for three grid-connection conditions are presented as follows. (1) For the single-machine isolated operation, the equation has the general form, as shown in dn 30egpr J M t Mg n 30 2 dt nr. (3.1) (2) For the Single-Machine Infinite Bus (SMIB) operation, it is assumed that the rotational speed is constant at the rated value or other given values, yielding n nc,( fg fc). (3.2) (3) Under the off-grid operation, the values of M g and e g are 0, and the corresponding equation is dn J M, (3.3) t 30 dt which can be considered as a special case of (3.1). The generator frequency, f g, is transferred from the speed, n Waterway system Models of waterway systems in HPPs are only shortly presented, more details can be found in [50]. Considering the elasticity of water and pipe wall, equations for one-dimensional compressible flow in draw water tunnel and penstock are described by the continuity equation and the momentum equation. The continuity equation is 2 2 H H aw V aw V A V sin θ V 0. (3.4) x t g x ga x The momentum equation is H V V VV g V fd 0. (3.5) x x t 2Dp The details of all the symbols in this thesis can be found in Abbreviations and Symbols. The set of hyperbolic partial differential equations are solved by a common and widely used approach, the method of characteristics [133]. Moreover, different forms of pipelines, channels and surge tanks are included in the model, and more information can be found in Paper 1. 39

40 Turbine The Francis turbine model is discussed in this part. Figure 3.3 shows the model and illustrates some of the notations. The equations of the model are presented below. C P C Figure 3.3. Illustration of the Francis turbine model in TOPSYS The continuity equation is Q S Q. P (3.6) The equations of the method of characteristics are + C: QP CP BPHP C: QS CM BMHS. (3.7) The turbine flow equation is 2 QP Q11 D1 H P H S H. (3.8) The equations of unitary parameters are n nd H H H, (3.9) / P S Mt M11 D1 HPHS H. (3.10) The equations of turbine characteristic curve are (3.11) M f n Y. (3.12) , Here, functions f 1 and f 2 mean the interpolation of the turbine characteristic curves. The equation pg M n (3.13) t 30 shows the transform from the torque to the power output Governor system The governor system is a feature of Model 1, with various equations for different operating conditions and control modes. Figure 3.4 demonstrates the complete control block diagram of the proportional-integral-derivative (PID) governor system. The main non-linear factors (dead-zone, saturation, rate limiting and backlash) are included. All the variables in the governor system are per unit values. The S 1, S 2 and S 3 blocks are selectors between different signals, and the zero input to the selector means no input signal. S 40

41 Figure 3.4. Block diagram of the governor system in TOPSYS For the normal operation, which means the isolated and the grid-connected operation with load, there are three control modes: frequency control, opening control and power control. This study establishes a governor model with a switchover function of control mode. (1) Frequency control mode (PFC) In the frequency control mode, as shown in Figure 3.4, the feedback signal contains not only the frequency value, but also the opening or power, which forms the frequency control under opening feedback (OF) and power feedback (PF), as respectively described by 2 d ypid dypid bk p d (1 bk ) 2 p p bk p iypid yc dt dt, (3.14) 2 d xf dx f Kd K 2 p Kix f dt dt 2 d pg dpg dypid ek p d ek ( ) 2 p p ek p i pg pc dt dt dt. (3.15) 2 dxf dx f Kd K 2 p Kix f dt dt The symbols in governor equations are illustrated in Figure 3.4. (2) Opening control mode (secondary frequency control) In the opening control mode, the governor controls the opening according to the given value (y c ). As demonstrated by Figure 3.4, the opening control is equivalent to the frequency control under the OF without the frequency deviation input (x f ). The equation of the opening control is 41

42 2 d ypid yc d ypid yc p d bk p p bk (1 ) 2 dt dt. (3.16) bk y y 0 p i PID c Moreover, the modelling of the opening control process can be simplified by ignoring the engagement of the PID controller, i.e. setting the opening directly equal to the given value, as shown in ypid y. (3.17) c (3) Power control mode (secondary frequency control). In the power control mode, the governor controls the opening according to power signals, leading the power output to achieve the given value. The equation is 2 d ( pg pc) d( pg pc) ek p d ek 2 p p dt dt. (3.18) dpc dypid ek p i( pg pc) 0 dt dt It is worth noting that a simpler controller, without the proportional (P) and derivative (D) terms, is applied in many real HPPs, as shown in dpc dypid ek p i( p g pc) 0 dt dt. (3.19) In the governor equations (3.14) (3.19), only the value of y PID is solved for. For the servo part, the output opening (y servo ) is obtained by solving dyservo ypid Ty y. (3.20) servo dt Then, the value of final opening (y) is the value after the non-linear functions i.e. saturation, rate limiting and backlash. Table 3.2. States of selectors in different control modes Control mode Equation S 1 state S 2 state S 3 state Frequency control (3.14) (3.15) Opening control (3.16) (3.17) Power control (3.18) (3.19) The selectors (S 1, S 2 and S 3 ) in the governor system are related to each other. Table 3.2 shows various states of selectors in different control modes. Table 42

43 3.3 concludes the equation set, in this study, of hydropower units under different operating conditions. Discussion of equations for start-up, no-load operation, emergency stop and load rejection can be found in Paper I. Table 3.3. Equation set of the hydropower unit under different operating conditions Operating condition Equation set Governor Generator Turbine Frequency control (3.14) or (3.15) (3.1) or (3.2) Normal Opening control (3.16) or (3.17) (3.1) or (3.2) operation Power control (3.18) or (3.19) (3.1) or (3.2) Start-up Open-loop (3.18) (3.3) (3.6)- Closed-loop (3.15) (3.3) (3.13) No-load operation (3.15) (3.3) Emergency stop (3.18) (3.3) Load rejection (3.15) (3.3) Model 4 and 4-S This subsection is based on Paper X. Comparing to Model 1 with a first-order generator model, Model 4 and 4-S applies a fifth-order model for investigating the rotor angle stability, as demonstrated in Table 3.1 and Figure 3.1. The models of the whole hydraulic-mechanical subsystem (i.e. the waterway system, the turbine and the governor system) in Model 4 are the same as the ones in Model 1. Hence, only the model of electrical subsystem, including the automatic voltage regulator (AVR) and the PSS, is presented here in a per-unit system. Model 4-S ( S is short for simplified ), a simplified version of Model 4, is designed specifically for studying the hydraulic turbine damping of rotor angle oscillations. Hence Model 4-S is introduced in section Generator and power grid The generator is described by a classical fifth-order model [19] that consists of five differential equations, as shown in 43

44 d 10 dt d Tj Pm Pe dt de. (3.21) q Td 0 Efd Eq Id( Xd Xd ) dt de q Td 0 Eq Eq Id( Xd Xd ) dt de d Tq 0 Ed Iq( Xq Xq ) dt The d- and q-axis components of the voltage at the generator terminal are V gd E 0 - = d X I q - d V E gq q X d 0 I. (3.22) q The electric active power and the reactive power are described in Pe Vgd Vgq Id = Q Vgq -V e gd I. (3.23) q The network model for the SMIB system is shown in Vsd Vs sin Vgd Iq = = + X s V sq Vs cos. (3.24) Vgq -Id All the resistances in the system are ignored, and the transformer and transmission line are simplified as a reactance (X s ) AVR and PSS The AVR is described by a standard first-order model [15], as presented in Figure 3.5 and the following equation defd Tr Efd KaVg VPSS. (3.25) dt Figure 3.5. Block diagram of the AVR in Model 4 and 4-S The PSS is described by a second-order model [15], as shown in Figure 3.6 and 44

45 dv1 d V1 Ks dt dt T0. (3.26) d VPSS T1 d V1 V1 VPSS dt T2 dt T2 T2 The output limit (saturation) of the AVR and the PSS are included. Figure 3.6. Block diagram of the PSS in Model 4 and 4-S 3.2 Numerical models in MATLAB In this section, models in MATLAB/Simulink and MATLAB/SPS are presented Model 2-L (in Simulink) The main purpose of Model 2-L ( L is short for lumped ) is to study controller filters for the efficient operation and the grid frequency quality in section 6.2, which is based on Paper XII. Model 2-L is for simulating the Nordic power system [129, 134], and it is shown in Figure 3.7. In the model, all the power plants are lumped into one. Figure 3.7. Mode 2-L: a Simulink model of the Nordic power system, for computing the power system frequency. The governor system is shown in Figure 3.8. The governor system is the key part, as shown in Figure 3.8. The dead zone [135, 136], floating dead zone [137], and first-order linear filter [138] are included, as the frequency filter and guide vane opening (GVO) filter. In the 45

46 numerical model, the floating dead zone is equivalent to the backlash [135], which has an impact during every GVO direction change. Figure 3.8. The Simulink model of a general turbine governor with different filters, for computing GV movements. The blocks of the filters are highlighted in green. The transfer functions of the first-order linear filters and the proportionalintegral (PI) controller are described by: 1 Linear frequency filter: Ff 1() s Tf 1 s1 (3.27) Linear GVO filter: 1 Ff 2() s Tf 2 s1 (3.28) PI Controller: K ps Ki C( s) 1 bpk ps bpki (3.29) The equations of the plant and the grid are as follows. Plant: Ts w 1 Ps () K 0.5Ts w 1 (3.30) Grid: 1 Gs () Ms D (3.31) The parameters of the plant and the grid are shown in Table 14.4 in Appendix B Model 5 and 5-S (in SPS) The aim of Model 5 and 5-S is to conduct simulations for for investigating the rotor angle stability in section 5, as shown in Table 3.1 and Figure 3.1. More exactly, both Model 5 and 5-S are developed for verifications, by applying the widely-used software SPS. Model 5 is built to verify the theoretical eigenanalysis (Model 6) in section 5.1; Model 5-S, a simplified version of Model 5, is utilized specifically for verifying the electrical subsystem in TOPSYS (Model 4) in section 5.1. Model 5 is introduced in this section based on Paper IX, and Model 5-S is presented in section 5.2 together with Model 4-S. The overall model that contains several subsystems is in per-unit system, as illustrated in Figure 3.9, it is a SMIB system with the extended HPP model. 46

47 Figure 3.9. Block diagram of Model 5: the SMIB system with the extended HPP model. The inputs of reference values are not shown. The signal of rotational speed (ω) is in blue, and the signal of electromagnetic power (P e ) is in red Generator, transformer, AVR and PSS The generator and the transformer adopt the existing blocks Synchronous Machine 5 and Three-Phase Transformer (two windings) 6 from the SPS library. Compared to a standard example Synchronous Machine 7 existing in SPS, Model 5 is an extended version with the PSS added and with more detailed governor, turbine and waterway models. It is worth noting that the generator model here is seventh-order: the stator transients are included, while they are ignored in the fifth-order generator model in Model 4 and 4-S. The numerical model of the AVR has the same structure as the one in Model 4 and 4-S, as shown in Figure 3.5. In terms of the PSS, in Model 4 and 4-S, only the input of speed deviation (Δω) is considered; while in Model 5 and 5-S, the input of the deviation of the electromagnetic power (ΔP e ) is also included, for the practical case in Swedish HPPs, e.g. HPP 5 in Table 2.1. The PSS is shown in Figure 3.10 and the following equation dv1 dipss V1 Ks, ( IPSS K KPePe ) dt dt T0. (3.32) d VPSS T1 d V1 V1 VPSS dt T2 dt T2 T2 5 (accessed on March 14th, 2017) 6 (accessed on March 14th, 2017) 7 (accessed on March 14th, 2017) 47

48 Figure Block diagram of the PSS in Model 5 and 5-S Governor system, turbine and waterway system The numerical model of the governor is presented in Figure 3.11, including the nonlinear components that are important due to their decent influence on the response of GVO. It is worth noting that the PF signal here is the electromagnetic power, not the mechanical power. The turbine and waterway system is described by a linearized model, through applying the standard method with six coefficients [15], as shown in Figure The elasticity of the water column is considered and the frictional loss is ignored here. Figure Block diagram of the governor system in Model 5. The nonlinear components are with dashed outline. Figure Block diagram of the linear model of turbine and waterway system in Model Models for theoretical derivation In this section, various models derived for theoretical analysis are presented. 48

49 3.3.1 Model 3-F Model 3-F ( F is short for Francis turbine ) is mainly for analysing the frequency stability, and the results are comparted with simulations by Model 4 in section 4.3, based on Paper III. Comparing to Model 4, Model 3-F is linearized on the basis of the following assumptions. (1) Rigid water column equations are adopted in the draw water tunnel, neglecting the elasticity of water and pipe walls. (2) The penstock modelling is ignored. (3) Head loss at the bottom of the surge tank is not considered. (4) Steady-state turbine characteristic is linearized by using transmission coefficients. (5) Nonlinear characteristics of the governor (e.g. saturation, rate limiting and dead zone) are ignored Details of the model The basic equations are described in the schematic diagram shown in Figure Figure Schematic diagram for Model 3-F: a HPP with a surge tank The equation describing the draw water tunnel is 2h y 0 dq y z qy Twy H 0 dt. (3.33) The continuity equation of the surge tank is dz qy qt T. (3.34) F dt The equations of the turbine torque and the turbine discharge are mt ehzexxeyy qt eqhz eqxx eqyy. (3.35) The first-order generator equation is dx Ta mt ( mg egx) dt. (3.36) The governor equation for the frequency control is 49

50 2 d y dy ( Ty bkt p P y) (1 bk ) 2 p P bkt p i y dt dt. (3.37) dx bky p i ( KP Kx i ) dt Comparing with Equation (3.14), the servo (T y ) is included and K d is ignored. The governor equation for the power control is 2 d y dy dpc Ty e ( ) 2 pki pc p. (3.38) g dt dt dt Comparing with Equation (3.19), the servo (T y ) is included Characteristic equations for Routh-Hurwitz stability criterion The characteristic equations, introduced in section 2, of the system for Routh- Hurwitz stability criterion is presented here. Two characteristic equations are deduced for the frequency control and the power control respectively. (1) Frequency control By applying the Laplace transform to Equations (3.33) through (3.37), a fifth-order characteristic equation for frequency control is derived as: d x d x d x d x dx a a a a a a x. (3.39) dt dt dt dt dt The coefficients a 0 through a 5 are explained in Appendix A. (2) Power control By applying the similar treatment as above on Equations (3.33) through (3.36) and Equation (3.38), a fifth-order characteristic equation for power control is obtained: dx dx dx dx dx a 5 a 5 4 a 4 3 a 3 2 a 2 1 a 0 x 0. (3.40) dt dt dt dt dt The coefficients through are explained in Appendix A Model 3-L Model 3-L ( L is short for lumped ) is developed for investigating the stability of the Nordic power system in frequency domain, in section 6.2 based on Paper XII. The corresponding model for time domain simulation is Model 2-L Details of the model The block diagram of the power system, for frequency domain analysis, is shown in Figure Comparing to the time domain simulation Model 2-L, the transfer functions of the PI controller, the plant and the grid in Model 3-L are the same, as shown in Equations (3.29), (3.30) and (3.31). However, the non-linear components of the actuator are ignored here. 50

51 Figure Block diagram of the power system model for frequency domain analysis The term F(s) in Figure 3.14 is a general transfer function for the filter. It is replaced by specific transfer functions of different filters according to the corresponding case. The transfer functions of the linear filters are the same as the ones in Model 2-L, as shown in Equations (3.27) and (3.28). The describing functions, as introduced in section , are applied to analyse the nonlinear filters in frequency domain. The describing function of a dead zone of size E dz [130] is 2 1 Edz Edz Edz Ndz ( A) 2arcsin 2 1 A Edz. (3.41) A A A The describing function of a floating dead zone of size E fdz (backlash) [130] is 2 1 Efdz Efdz 2Efdz Efdz N fdz ( A) arcsin A A A A (3.42) 2 1 2Efdz Efdz j A E fdz A A Here, the parameter A means the amplitude of the periodical input signal Transfer functions for Nyquist stability criterion In order to investigate the system stability, the open-loop systems with different filters are examined by the Nyquist stability criterion, as introduced in section For analysing the system with the dead zone and floating dead zone, the transfer function of the open-loop system is 1(s) Cs ( ) P( sgs ) ( ) 3 2 b01s b11s b21sb. (3.43) a01s a11s a21s a31sa41 For the open-loop system with the linear filter, the transfer function is 2(s) Ff 1() s C()P() s s G() s 3 2 b02s b12s b22sb. (3.44) a02s a12s a22s a32sa42 All the transfer function coefficients here are shown in Appendix A. 51

52 3.3.3 Model 6 Model 6 is built for eigen-analysis on the rotor angle stability in section 5.1 based on Paper IX. The corresponding model for time domain simulation is Model Details of the model The overall structure of Model 6 is the same with Model 5, as shown in Figure 3.9. It is a SMIB system with an extended model of a HPP described by a state matrix, ignoring all the nonlinear factors. The generator and the power grid are modelled with the same approach as Model 4, as shown in Equations (3.21) through (3.24). The AVR and the PSS (with the speed input and the power input) are the same as the ones in Model 5 and 5-S, with the output saturation removed, as described in Equations (3.25) and (3.32). The turbine and waterway system is described by a linear model, the same as the one in Model 5. The corresponding equation of the turbine is q e qy y e q e qh h. (3.45) Pm ey y e eh h The corresponding equation of the elastic water column is h Ts w 2 2. (3.46) q 1T s From Equations (3.45) and (3.46), two differential equations are deduced: dh h1 dt (3.47) d h1 1 d y d d h ht 2 w eqy eq eqh dt T r dt dt dt The governor system is a linearized version of the one in Model 5, as illustrated in Figure 3.11, excluding the nonlinear components with dashed-outline. The equation of frequency control with the OF is dypi d 1 bk p p bk p iypi K p Ki. (3.48) dt dt The equation of the PF is dypi dpe d bk p p bk p ipe K p Ki. (3.49) dt dt dt Note that the PF signal here is the electromagnetic power, not the mechanical power. The servo is described by dy Ty ypi y. (3.50) dt e 52

53 State matrix for eigen-analysis As introduced in section , the small signal stability of the system can be analysed by investigating the eigenvalues of the state matrix. For the whole system of Model 6, there are twelve differential equations, i.e. five equations for the generator, one equation for the AVR, two equations for the PSS, two equations for the turbine with the waterway system, one equation for the servo and one equation for the PI controller. Hence, a state matrix with twelve corresponding state variables is derived, as shown in 0 a1, a a 0 a a 0 0 a 0 a 0 0 E q E q a a a E d a5, a5, E d E fd a6,1 0 0 a6,4 a6,5 a6, a6,12 E fd y a PI 7,1 a7,2 a7,3 a7,4 a7,5 0 a7,7 a7,8 0 a7, ypi y a8,7 a8, y h a9,1 a9,2 0 a9,4 a9,5 0 a9,7 a9,8 a9,9 a9, h1 h a10, h a V 11,1 a11,2 a11,3 a11,4 a11,5 0 0 a11,8 0 a11,10 a11,11 0 V1 1 a12,1 a12,2 a12,3 a12,4 a12,5 0 0 a12,8 0 a12,10 a12,11 a 12,12 V VPSS PSS 2,1 2,2 2,4 2,5 2,8 2,10 a3,1 0 a3,3 a3,4 0 a3, E q 4,1 0 4,3 4, E q (3.51) All the non-zero elements,,, of the state matrix are given in Appendix A. Analyses are conducted based on damping ratios corresponding to different eigenvalues in the system for each case. The smallest damping ratio is selected as the main indicator of the system stability. 3.4 Numerical models in MATLAB for HPPs with Kaplan turbines (Model 2-K) Model 2-K ( K is short for Kaplan turbine ) is a numerical HPP model with a Kaplan turbine implemented in applying Simulink, calibrated with measurements from two Swedish HPPs (HPP 6 and HPP 7). It is established for quantifying relative values of regulation burden and performance of PFC in section 6.3 based on Paper XIV. Model 2-K is divided into several sub-models, i.e. Model 2-K-1, Model 2- K-2 and Model 2-K-3 for different purposes that are introduced in section 6.3. In this section, Model 2-K-1 and Model 2-K-2 are presented, and Model 2-K- 3 is introduced additionally in section 6.3. The per-unit (pu) system is adopted for describing all the models. The overall structure of the model is shown in Figure The open loop hydropower plant model with the red dashed outline is Model 2-K-1, for 53

54 simulating the transient processes within a HPP. In this thesis, it is mainly applied for computing the efficiency (η), power output (p m ), GVO and RBA. The closed loop model with the blue dashed outline is Model 2-K-2, for simulating the frequency quality of the whole power system. Figure Model 6: Overall model structure of a hydropower system with a Kaplan turbine. Some detailed set points and feedback signals are omitted here, but included in the more detailed block scheme shown in the following content System components Kaplan turbine and waterway system The Kaplan turbine and waterway system model is illustrated in Figure The active power from the turbine is described by the classical simplified nonlinear model [15, 32]: q q qg h G h h, (3.52) 0 0 3/2 pm qh Gh. (3.53) The above two equations are for single-regulated turbines, e.g. Francis turbines. While for the double-regulated turbine, G is a comprehensive gate opening that is identified from the values of GVO (y) and RBA (a), as shown in a fitting function G GF y, a. (3.54) The efficiency value is from an interpolation function I ya,. (3.55) These two functions are achieved from on-site measurement data that is presented separately in section

55 Figure Block diagram of a model of Kaplan turbine and waterway system in Model 2-K. The signal of RBA is presented in blue, for distinguishing it from the GVO signal. The head is affected by hydraulic dynamics from the elastic penstock, draw water tunnel and surge tank [32], as shown in hh0 h. (3.56) h hp hs In terms of frequency domain, the transfer function describing the head variations due to discharge deviations in the penstock is 2 hp Twp pte fps Tes fp Gp s. (3.57) 2 2 q Te 1 pte s For the head variations due to discharge deviations in the surge tank, the transfer function is hs Twts ft Gs s 2 q TwtTs s Ts fts 1. (3.58) When the turbine damping (D) [32] is included, the equation of the active power becomes 3/2 p Gh DG f. (3.59) m Governor system with filters The model of a governor system for a Kaplan turbine is demonstrated in Figure

56 Figure Block diagram of governor system of the Kaplan turbine in Model 2-K. A standard PID (proportional integral derivative) controller with droop, common mechanical components and a 2-D (two-dimensional) lookup table, and the artificial filter for the signal of RBA are included. The filter for RBA is a floating dead zone (or floating dead band) that is the same as the one presented in section 3.2.1, which is equivalent to the backlash in the Simulink model Power grid For investigating the frequency quality under different operation strategies, a simplified model [129, 139] representing the Nordic power grid is applied, as shown in Figure The transfer function is the same as the one in Model 2-L, as described in Equation (3.31). Figure Block diagram of a model of lumped HPP in Model 2-K. The model contains a governor and a simplified representation of Francis turbine Lumped HPP The lumped HPP represents the rest of regulating units in the power grid, by assuming that all the regulation in the grid is provided by hydropower. The principle of modelling the lumped HPP is the same as it is for Model 2-L, expect for the disposition of the corresponding scaling factor. As shown in Figure 3.18, the model contains a governor and a simplified representation of a Francis turbine and waterway system, as described in 56

57 Ts w 1 Gt () s 0.5Ts 1. (3.60) w Turbine characteristic from measurements Model development regarding the turbine characteristics is a key point here. As shown in Table 3.1, a specific model for describing turbine characteristic is developed for Model 2-K, comparing to the common approach by applying turbine characteristics curves. More exactly, the comprehensive gate opening and efficiency, as shown in Equations (3.54) and (3.55), are modelled from specific on-site index tests [140, 141] data from HPP 6 and HPP 7. Here, the principle of the approach is introduced and more details can be found in Paper XIV. The measured scatter data of comprehensive gate opening and efficiency can be calculated for a limited operating region and applied for the fitting or the interpolation. By applying the surface fittings, the turbine characteristics for a larger operation range can be obtained. A relatively higher efficiency accuracy is demanded by the economic analysis, and it can be achieved by interpolation of the efficiency data. However the measurement data are only available in a certain range, hence extra data for larger operating points are added from the fitting data for the extrapolation Fittings The comprehensive gate opening and efficiency for each HPP are fitted to a quadratic polynomial surface using GG y, a p p ap yp a p ay p y, (3.61) F ya, p p a p y p a p ay p y F G G G G G G. (3.62) where, p Gij and p ηij are the coefficients of the fitting. The fittings are demonstrated in Figure 3.19, taking HPP 7 as an example. The fitting of an operating point far from the on-cam range might not be accurate, however, this will not affect the results of this study because the fitting is only applied for small disturbance simulations Interpolations and extrapolation of efficiency Piecewise cubic interpolation is applied to obtain the final efficiency data, as shown in Figure By adopting the added points (in red) from the fitting data, the small operation range covered by the index tests is extended, then the efficiency data can support all the small disturbance simulations (in section 6.3). 57

58 (a) (b) Figure Fitting of data of HPP 7: (a) comprehensive gate opening G F and (b) turbine efficiency data. The efficiency value is normalized with respect to the maximum efficiency value. Figure Interpolation of turbine efficiency data of HPP 7. The red scatters are extracted from the fitting for extrapolation. 58

59 4 Stable operation regarding frequency stability The stable operation of HPPs regarding frequency stability of power systems is analysed in this chapter. In section 4.1 (based on Paper I), case studies on different operating conditions of HPPs are conducted, and the simulation performance of Model 1 based on TOPSYS is presented. In section 4.2 (based on Paper II), response time of PFC in HPPs is investigated under grid-connected operation. In section 4.3 (based on Paper III), frequency stability of HPPs in isolated operation is studied. All the content in these three sections mainly focus on active power control, of which the ultimate goal is achieving a better frequency stability of power systems. 4.1 Case studies on different operating conditions The application of Model 1 based on TOPSYS is presented in this section by comparing simulations with on-site measurements, based on four engineering cases: a Swedish HPP (HPP 1 shown in Figure 3.2) and three Chinese HPPs (HPP 2 HPP 4 shown in Figure 4.1). Figure 4.1. TOPSYS models of three Chinese HPPs: HPP 2 HPP 4 in Table 2.1. The aim of Model 1 is to achieve accurate simulation and analysis of different operation cases, e.g. small disturbance, large disturbance, start-up and no-load operation, etc. A good simulation in Model 1 is a crucial basis of the thesis, 59

60 such as for the studies in section 4.2 (grid-connected operation), section 4.3 (isolated operation) and section 6.1 (grid-connected operation) Comparison of simulations and measurements For normal operation (small disturbances in grid-connected operation and isolated operation), the comparison between simulations with Model 1 and measurements are shown in Figure 4.2 through Figure 4.4. Figure 4.2. Grid-connected operation: power output and opening from simulation and measurement under sinusoidal frequency input (HPP 1). In the figures of this thesis, the M refers to measurements and the S means simulation. Figure 4.3. Grid-connected operation: power from simulation ( S- ) and measurement ( M- ) under step frequency input (HPP 2). Overall, the simulation has a good agreement with the measurements. As shown in Figure 4.2, the effect of backlash is reflected: the GVO keeps stable for a short period during the direction change process (e.g. around t = 28 s). In Figure 4.3, after the frequency step change, the phenomenon of power reverse regulation caused by water inertia is simulated accurately, as well as the 60

61 gradual power increase or decrease due to the surge (after 20 seconds). However, the simulation of the power decrease has a lower value than the measurement. This deviation could be ascribed to the characteristic curve, to some extent, the on-site measurements inevitably deviate from the simulation that is based on the data from the model tests. The oscillation after a load step change is examined by simulation and compared with the measurements, as shown in Figure 4.4. The simulation reflects the real operating condition well: under the power control mode in HPP 3, the power oscillates with the surge oscillation under certain governor parameter settings due to a relatively small cross section of the surge tank. Figure 4.4. Isolated operation: Simulation ( S- ) and measurement ( M- ) of the power oscillation under power control mode (HPP 3). Figure 4.5. Frequency and opening from simulation and measurement during a startup process (HPP 2). The S means the simulation with the original characteristic curve of turbine, and the S2 means the simulation with the modified characteristic curve. For the start-up process, a case study of HPP 2 is simulated and compared with measurements, as shown in Figure 4.5. In the simulation with the original characteristic curve of the turbine (S), the simulated frequency increase process is approximately 30% shorter than the measured one, hence the opening 61

62 from simulation decrease to the no-load opening slightly earlier than the measured opening. Therefore, the curve was modified by decreasing the efficiency. With this revised characteristic curve, the new simulation (S2) fits the measurement well. It demonstrates that the inaccurate simulation mainly hinges on errors in the characteristic curve, which is especially error-prone in the smallopening operation range. Due to that for the small-opening operation range, the original input data achieved from the characteristic curve is not accurate enough and very sparse, it is hard to obtain a good predictive simulation. Figure 4.6. Simulation and measurement of the (a) GVO and pressure in the volute; and (b) pressure in the draft tube, during a load rejection process (HPP 4). For the load rejection process, the pressures at the inlet of the volute and in the draft tube are simulated and compared with measurements in HPP 4, as demonstrated in Figure 4.6. For the simulated pressure, there is a small static deviation from the measurement after load rejection. It might be due to the water head error caused by the characteristic curve and imprecise parameters of the waterway system. Moreover, the pulsating pressure at volute and draft tube in the measurement cannot be reproduced by the simulations because of the limitation of the one-dimensional characteristic method. The pressure measurement in the draft tube might also be difficult to compare to modelled values, due the swirl not being modelled in the 1-D modelling approach, making the actual water velocity past the pressure transducer deviate from the mean velocity in an unknown way Discussion The results above show that Model 1 can yield trustworthy simulation results for different physical quantities of the unit under various operating conditions. The main error sources of the simulation are the characteristic curves of the turbine, provided by manufacturers, which directly causes small deviations of power output and affects the rotation speed and pressure values. The reason 62

63 might be that the characteristic curves do not really describe the on-site dynamic process accurately, and the error is especially obvious in the smallopening operation range. Furthermore, waterway system parameters might also have errors that impact the simulation. 4.2 Response time for primary frequency control For evaluating the regulation quality of hydro units in PFC, a key is the power response time. How do the regulation and water way system affect the response time? How should governor parameters be set to control the power response time? These problems are the focuses of this section. Figure 4.7. Illustration of different times under frequency step disturbance. The opening means GVO. The aim of this section is to investigate general rules for controlling the power response time of PFC. Firstly, specifications of the response of PFC in different regions are introduced. Then, from the analytical aspect, a time domain solution for GVO response and a response time formula are deduced. Case studies of HPP 2 are conducted by simulations based on Model 1, to investigate various influencing factors. The response time (deployment time) T response and the delay time T delay of power response process are the key indicators in this section, as shown in Figure 4.7. The difference between the power response and the GVO response is the focus. 63

64 4.2.1 Specifications of response of PFC Strictly speaking, the parameters need to be tuned and tested for PFC in every HPP, for meeting the requirement of specifications that varies in different regions. (1) China Electricity Council Based on specifications of China Electricity Council [142], if the units are operating on 80 % of the rated load, the power response for a frequency step should meet a series of requirements. The most crucial requirements are: the power adjustment quantity should reach 90 % of the static characteristic value within 15 seconds. If the rated head of the unit is larger than 50 m, the power delay time should be less than 4 seconds. (2) ENTSO-E According to the specifications of ENTSO-E [143], the time for starting the action of primary control is a few seconds starting from the incident, the deployment time of 50 % of the total primary control reserve is at most 15 seconds, and the maximum deployment time rises linearly to 30 seconds for the reserve from 50 % to 100 %. (3) The Nordic power grid Currently, the Norwegian TSO Statnett has no specific requirements on the response time, but prescribes limits on certain quantities, such as on the delay between frequency deviation and incipient GV motion, on the resolution in frequency measurement, on the permanent droop, and on how to measure these parameters [144]. In Sweden, the TSO Svenska Kraftnät (SvK) has demands on response time, but no requirements on details [145]. The requirements depend on the magnitude of the frequency deviation, and if it exceeds 0.1 Hz, 50 % should be delivered within 5 s, and 100 % within 30 s Formula and simulation of response time Based on the theory in section , a formula for the GVO response time of PFC under opening control is deduced, for a PI controller with droop and servo. The main variables of the formula are governor parameters, as described in 1 bk p p T1 ln (1 bpkp bpkt i y) 1 bk. (4.1) p i Here, Δ is the target value, for example, it is set to 90% according to specifications of China Electricity Council [142]. Simulations based on HPP 2 under different conditions are conducted to analyse the sensitivity of response time with respect to the main parameters. The default settings of the simulation are given in Appendix B. In order to 64

65 investigate the influence of surge in upstream surge tank, a HPP model without surge tank is built, still by applying Model 1. More exactly, in the simplified model, the surge tank and the upstream pipeline before the tank are replaced by a reservoir. The simulation results are shown in Table 4.1. The power response time, T 4, can be expressed as T4 T1 T T1 T1 T2 T3. (4.2) The time difference ΔT (as shown in Figure 4.7), between the power response time (T 4 ) and the analytical response time of GVO (T 1 ), is mainly affected by the rate limiting and numerical algorithm (ΔT 1 ), the water inertia (ΔT 2 ) and the surge (ΔT 3 ). More detailed results can be found in Paper II. Table 4.1. The response time of frequency step under different conditions. T 1 and T 2 are calculated by Equation (4.1) and simulation respectively, and T 3 is simulated with the simplified model; Response time of opening or power means the time when the opening or power reaches the target value Δ. All the simulations are conducted with rate limiting which is 12.5%/s. The green bar in each cell indicates the relative magnitude of the values in the corresponding cells, except for the cells highlighted by yellow (the values in the yellow cells are much larger). No. Parameters K p K i b p T y Δ Response time of opening (s) Formula Simulation T1 Response time of power (s) Time difference ΔT (s) ΔT3= T4 -T % % % % % % % % % % % T2 Without surge tank, T3 With surge tank, T4 ΔT1= T2 -T1 ΔT2= T3-T2 4.3 Frequency stability of isolated operation For HPPs with surge tank, the Thoma criterion [146, 147] is often violated to diminish the cross section of surge tank with the scale getting larger nowadays. Therefore, the surge fluctuation is aggravated and frequency stability becomes more deteriorative [148]. Recently, some huge Chinese HPPs encountered this instability problem during the commissioning, measurements 65

66 under a load step disturbance are shown in Figure 1.3. Hence, the focus of this section is on stabilizing the very low frequency oscillation (see section 1.2.2) of an isolated HPP caused by surge fluctuation. In this section, by means of theoretical derivation based on Model 3-F, stability conditions under two control modes are contrasted through adopting the Hurwitz criterion. Then, the frequency oscillations are simulated and investigated with different governor parameters and operation cases, by applying Model 1. The engineering case here is HPP Theoretical derivation with the Hurwitz criterion Based on Model 3-F in section and the theory in section , a stability condition of frequency oscillation under frequency control and power control is obtained. The stability region is the region which satisfies the stability condition in K i -n coordinates by substituting the system parameters of different states into the stability condition. Here, n (n = F/F th ) stands for the coefficient of cross section area of the surge tank, where F and F th are the real area and Thoma critical area, respectively. Figure 4.8. Stability region in K i -n coordinates of two control modes A set of curves of stability region boundaries is achieved under two control modes based on the stability condition, as shown in Figure 4.8. The stability region of power control is much larger than which of frequency control. There is no proportional gain (K p can be regarded as 0) in power control, and it is conducive to the stability. However under frequency control, when K p is set to near 0, the stability region is still smaller than for power control. 66

67 4.3.2 Numerical simulation Based on Model 1 in TOPSYS, numerical simulations are conducted to validate the result of the theoretical derivation, as shown in Figure 4.9 and Figure Through the simulation, the conclusion drawn in the theoretical derivation is verified: the power control produces a better effect on stability than the frequency control. More exactly, under the frequency control, it is hard to stabilize the frequency by adopting any of the three sets of parameters. Even when K p is set to nearly 0, to compare with the power controller that is without proportional component (K p = 0), the frequency instability still occurs. While under the power control, frequency stability is well ensured, and the contradiction between rapidity and stability is also indicated. Figure 4.9. Frequency oscillation under frequency control with different governor parameters Figure Frequency oscillation under power control with different governor parameters Besides, it is necessary to have an additional discussion on the power control. The applying of the power control in the isolated operation condition is an ideal case, which cannot be implemented in the practical HPP operation. It is because that the load is unknown in reality, therefore the given power cannot be set properly. However, the conclusion based on the idealized case can supply the understanding and guidance for the stability in an islanded operation, which means the operation of a generating unit that is interconnected with a relatively small number of other generating units [136]. In the islanded system, some units operate in the frequency control mode to balance the changing peak load, and other units adopt the power control to maintain the stability. This issue is also a suggested topic for future work. More results and discussions are in Paper III. 67

68 5 Stable operation regarding rotor angle stability The stable operation of HPPs regarding rotor angle stability of power systems is studied in this chapter. In section 5.1 (based on Paper IX), a fundamental study on hydraulic-mechanical-electrical coupling mechanism for small signal stability of HPPs is conducted by eigen-analysis. Considerable influence from hydraulic-mechanical factors is shown, and it is further quantified in section 5.2 (based on Paper X): An equivalent hydraulic turbine damping coefficient and the corresponding methodology are proposed to quantify the contribution on damping of rotor angle oscillations from hydraulic turbines based on refined simulations. In section 5.3, the quick hydraulic mechanical response is discussed to support the results in this chapter. The engineering case of all sections is HPP Hydraulic mechanical electrical coupling mechanism: eigen-analysis This section aims to conduct a fundamental study on hydraulic-mechanicalelectrical coupling mechanism for small signal stability of HPPs, focusing on the influence from hydraulic-mechanical factors. For the local mode oscillation [15] in a SMIB system, the theoretical eigenanalysis (section ) is the core approach based on Model 6 (section 3.3.3) that is a twelfth-order state matrix. Numerical simulation by applying Model 5 (section 3.2.2) is also conducted for validation. As shown in Figure 1.2, three principal time constants for water column elasticity (T e ), water inertia (T w ), and servo (T y ) in the hydraulic-mechanical subsystem are the main study objects. They are analysed under two modes of frequency control (OF and PF) without the PSS. Then, the influence from the hydraulic-mechanical subsystem on tuning of the PSS is investigated. Detailed parameter values and operating settings are given in Appendix B Influence of water column elasticity (Te) Through Model 6, for each case (one combination of the value of K p and T e ), the smallest damping ratio (ξ) of all oscillation modes is plotted. As shown in 68

69 Figure 5.1 (a), when the value of T e is small (short penstock), the increased response rapidity of the frequency control (indicated by an increase of the K p value) with OF leads to a smaller damping ratio of the system. On the contrary, when the value of T e is larger than a certain value, the system becomes more stable with stronger frequency control. Moreover, the trend is inverted when the governor applies PF, as shown in Figure 5.1 (b); the increased strength of the frequency control stabilizes the system with the small value of T e. Also, PF generally leads to higher damping ratios than OF. (a) (b) Figure 5.1. The smallest damping ratio (ξ) of all oscillation modes under different values of governor parameters (K p ) and time constant of water column elasticity (T e ). (a) The feedback mode is OF; (b) The feedback mode is PF. The observations above are validated by time domain simulations. As presented in Figure 5.2, the black line, the blue line and the red line correspond to cases with a high, medium and low damping ratio respectively. The simulation results of these three sets of parameters fit the damping ratio well. In short, the impact of water column elasticity is important and it differs from the feedback mode of the frequency control. 69

70 Figure 5.2. Time domain simulation of the process after the three phase fault: Rotational speed. The result validates the cases in Figure Influence of mechanical components of governor (Ty) The rapidity of the GVO response is highly affected by the mechanical components in the governor system, e.g. servo, backlash, rate limiter, etc. In the state matrix, these components are simplified and represented by the servo time constant (T y ). As shown in Figure 5.3, a small value of T y leads to a quicker response of GVO, and brings clearer influence on system stability. The influence of T y is more obvious when K p is larger. A time domain simulation in Figure 5.4 (a) illustrates the influence and it corresponds to the result in Figure 5.3 (b). Figure 5.3. The smallest damping ratio (ξ) of all the oscillation modes under different values of servo time constant (T y ) and governor parameters (K p ). (a) The feedback mode is OF, T e = 1.0 s; (b) The feedback mode is OF, T e = 0.01 s; (c) The feedback mode is PF, T e = 1.0 s; (d) The feedback mode is PF, T e = 0.01 s. 70

71 (a) (b) Figure 5.4. Simulation of rotational speed after the three phase fault. The governor adopts the OF; (a) cases under different values of T y : the result validates the cases in Figure 5.3 (b); (b) cases with and without rate limiter Moreover, the linear theoretical model can result in negative damping ratios (Figure 5.1 and Figure 5.3). However the simulated oscillations are not divergent, because of the added damping by the nonlinear components (mainly from the rate limiter) in the numerical model, as shown in Figure 5.4 (b) Influence of water inertia (Tw) The water inertia, represented by the water starting time constant (T w ), is normally regarded as adverse to system stability, especially in islanded operating conditions, as presented in section 4.3. By contrast, for the SMIB system, the influence of water inertia is not monotonic, as demonstrated in Figure 5.5 and Figure 5.6. It is shown that the effect of water inertia differs from that of water column elasticity. Figure 5.5 (a) and Figure 5.6 present that a larger value of T w leads to smaller damping ratio when the value of water column elasticity (T e ) is around 0.4. However, the increase of T w results in slightly more stable cases when the value of T e is large or small, as demonstrated in Figure 5.6 and Figure 5.5 (b). When the governor adopts the PF, the system is more stable under 71

72 larger water inertia in this case, and this is validated by time domain simulations (Figure 5.7). Figure 5.5. The smallest damping ratio (ξ) of all oscillation modes under different values of water starting time constant (T w ) and governor parameters (K p ). (a) The governor adopts the OF, T e = 0.4 s; (b) The governor adopts the OF, T e = 0.01 s; (c) The governor adopts the PF, T e = 0.4 s; (d) The governor adopts the PF, T e = 0.01 s. Figure 5.6. The smallest damping ratio (ξ) of all the oscillation modes under different values of T w and T e. The governor adopts OF. 72

73 Figure 5.7. Simulation of rotational speed after the three phase fault under different values of T w. The governor adopts PF. The result validates the cases in Figure 5.5 (d) Influence on tuning of PSS Here, the influence of hydraulic-mechanical factors on tuning of the PSS is investigated. The damping effects under implementation of the PSS with different settings of gain K s under various conditions are shown in Figure 5.8. Figure 5.8. The smallest damping ratio (ξ) of all the oscillation modes under different gains of PSS (K s ): (a) OF in governor and speed input in PSS; (b) OF in governor and power input in PSS; (c) PF in governor and speed input in PSS; (d) PF in governor and power input in PSS. Two main insights are obtained here. (1) The optimal parameters values vary with different types of feedback. More exactly, for achieving the largest damping ratio, the K s value differs in four cases, shown in Figure 5.8 (a) - (d). The optimal value of K s changes with different types of PSS, meanwhile, the tuning is also affected by the water column elasticity. (2) The stability margin changes considerably under various conditions. The frequency control with 73

74 the PF generally results in a higher damping ratio, and this is validated by the time domain simulations in Figure 5.9. In short, it can be observed that there is still room for optimizing the parameters and performance of the PSS by considering the effect of the hydraulic-mechanical factors. Figure 5.9. Simulation of rotational speed after the three phase fault under different modes (OF and PF) of governor. The PSS adopts speed input and the gain (K s ) is set to 4.0. The result validates the cases in Figure 5.8 (a) and (c). 5.2 Quantification of hydraulic damping: numerical simulation Damping coefficient is a common term (D) used in power system stability analysis, and its general form is described in a linearization of the swing equation T jpmped (5.1) However, the variation range of D in the hydropower field is still unclear; it is normally assumed to be positive and often set to zero to obtain a conservative result. Therefore the swing equation is rewritten as T j PmPe. (5.2) The aim of this section is to quantify the contribution from a hydraulic turbine to the damping of local mode electromechanical oscillations [15]. An equivalent hydraulic turbine damping coefficient (D t, simplified as the damping coefficient in the following context) is introduced here, as described in T j Pm Pe Pm, const Dt Pe. (5.3) In this study, the focus is on the mechanical power (P m ), instead of the electromagnetic power (P e ) that is the main analysis object in previous studies. The purpose of introducing the damping coefficient is as follows. (1) Quantifying the value of D t can clarify an long-standing issue: how large is the damping contribution from the hydraulic system? (2) For analysis of large 74

75 power systems, the mechanical power simulation in HPPs is inevitably simplified and less accurate, misleading the analysis of power system oscillations. The quantified damping coefficient can be easily implemented in models of complex multiple-machine systems, hence the mechanical power in the system can be set to constant without losing the influence from the hydraulic system on the system stability (shown in section 5.2.3). (3) Considering the damping coefficient can affect the system parameter tuning, including the PSS tuning (shown in section 5.2.3). In this section, firstly, the corresponding methodology is introduced. Then, the quantitative results of the damping coefficient are presented in different cases with and without the application of PSS. Lastly, the influence and significance of the damping coefficient are demonstrated in case studies Method and model Method of quantifying the damping coefficient The method of quantifying the damping coefficient is based on simulations by Model 4 and Model 4-S ( S is short for simplified ), as shown in Table 5.1. The only difference between these two models is that the swing equation in Model 4 and Model 4-S is (5.2) and (5.3) respectively; it means that the mechanical power is simplified as constant and the whole model of the hydraulic subsystem is ignored in Model 4-S. Table 5.1. Different numerical models in this section Model Description Purpose 4 TOPSYS model with refined hydraulic-mechanical subsystem introduced in section Refined simulation 4-S 5-S Simplified version of Model 4, with the swing equation in (5.3) A MATLAB/SPS model mentioned in section Quantifying the damping coefficient by the comparison with Model 4 Verifying the TOPSYS model (Model 4 and 4-S) The detailed steps are as follows: (1) Step 1: Apply Model 4 to simulate transient processes of a HPP after a three phase fault; (2) Step 2: Adopt Model 4-S to simulate transient processes with different values of the damping coefficient D t. Other conditions remain the same as the case in Step 1. (3) Step 3: Compare the damping performance (reflected by the rotational speed) from the two simulation models; among results from Model 4-S with 75

76 different values of D t, one of the curves has the best agreement with the simulation from Model 4; thus the corresponding value of D t is the quantified damping coefficient for this case. The detailed method for determining the agreement between results from two models in Step 3 is by applying the root mean square error (RMSE), as described in 1 N RMSE i S i N i1 2 4, 4, P P P (5.4) Here, P i means a value of a local maximum or local minimum (peaks of a curve) of speed deviation (Δω); the subscript 4 and 4-S stand for Model 4 and Model 4-S respectively; N is the total number of local maxima and minima during a certain time period (10 seconds in this paper). The minimum value of the P RMSE indicates the best agreement of two curves of rotational speed simulated by Model 4 and Model 4-S. The engineering case of this section is HPP 5, as shown in Figure Detailed parameter values and operating settings are given in Appendix B. Figure TOPSYS model of HPP 5, of which a single unit is connected to an infinite bus Model verification Here, the verification of the TOPSYS Model is presented. The hydraulic-mechanical subsystem of the model has been verified by measurements in different cases, as shown in section 4.1. Therefore, the main focus here is verifying the electrical subsystem, by comparing the electrical transients simulated from Model 4-S (TOPSYS) and Model 5-S (SPS) in Table 5.1. Model 5-S is a standard SPS model, as shown in Figure 5.11, and the swing equation of it is described by (5.3). 76

77 Figure Model 5-S in SPS. The block with dashed outline shows the implementation of the simplified mechanical power by applying the damping coefficient (D t ). (a) (b) (c) (d) Figure Comparison between Model 4 (TOPSYS) and Model 4-S (SPS), without the application of PSS. In both models, the mechanical power P m is constant. (a) Rotational speed; (b) Excitation voltage; (c) Generator terminal voltage; (d) Electromagnetic active power. Comparisons between simulations by Model 4-S and Model 5-S are shown in Figure In both models, the mechanical power is constant. Generally, the simulations by Model 4-S have a good agreement with the results from Model 77

78 5-S. The main difference occurs in the damping performance, which is expected. The reason is the stator transients, which contribute to the damping [15], are included in the SPS model but ignored in the equivalent circuit model in TOPSYS. Therefore, the TOPSYS model leads to slightly more conservative results; however it does not affect much the quantification of the damping coefficient, since both Step 1 and 2 are conducted by the TOPSYS model Quantification of the damping coefficient Here, the quantification results of the damping coefficient are presented for different cases with and without the application of the PSS. The simulation cases are listed in Table 5.2. Table 5.2. Different simulation cases under the three phase fault. Other settings of all the cases are the same, apart from the descriptions. The delay means the delay time in the turbine governor. Case Description Purpose 1 Delay = 0.30 s; No PSS Demonstrating a positive and a negative damping coefficient under cases without 2 Delay = 0.50 s; No PSS the PSS 3 Delay = 0.25 s; with PSS Demonstrating a positive and a negative damping coefficient under cases with the 4 Delay = 0.50 s; with PSS PSS Without the application of the PSS, two examples of quantifying the damping coefficients are shown in Figure The values of the damping coefficient are quantified as 2.0 and -1.1 respectively. The main reason for the difference in the damping performance is the phase shift in the mechanical power response with respect to the rotational speed deviation, and the delay time is the most influential fact affecting the phase shift. A crucial point here is that the damping coefficient can vary over a considerable range and can even be negative, while previously the contribution is unclear and normally assumed to be positive. The change processes of the GVO and mechanical power under case 1 and case 2 are shown in Figure The phase shift between the mechanical power and GVO is approximately 180º, clearly demonstrating the non-minimum-phase response of the mechanical power. Also, it is shown clearly that the phase shift between the power and the speed is changed due to different delay times. 78

79 (a) (b) Figure Quantification of the damping coefficients for cases without the PSS (a) (b) Figure Simulation of GVO, mechanical power and speed. (a) Case 1; (b) Case 2. The GVO and mechanical power are deviations from initial values. The curves of speed are exactly the same as the ones in Figure For the cases with application of the PSS, the influence from the hydraulic turbine is still obvious. As shown in Figure 5.15, the damping coefficient is quantified as 1.5 and -2.1 respectively for case 3 and case 4. The quantifying method is basically the same as above, the only difference is that the PSS is activated in both Model 4 and Model 4-S. Considering the influence from mechanical power can contribute to a better tuning of PSS, as further discussed below. The quantified damping coefficient 79

80 is convenient to add in cases of PSS tuning for complex multiple-machine systems in which the detailed hydraulic modelling needs to be ignored. (a) (b) Figure Quantification of the damping coefficients under the application of the PSS Figure Simulations of three cases with the implementation of PSS Influence and significance of the damping coefficient In this part, firstly, the influence of the damping coefficient on the PSS tuning is presented for a SMIB system. Secondly, the effect of the damping coefficient on multi-machine system stability is shown, based on the WSCC 3-machine 9-bus system [149] without the implementation of PSS Influence on the PSS tuning Three cases after the three phase fault are simulated by Model 4-S, as shown in Figure In order to neutralize the effect of a negative damping (-2.0), the gain of PSS (K s ) needs to be increased from 2.0 to

81 Influence on multi-machine system stability For showing the influence of the damping coefficient on multi-machine system stability, a simple case study is conducted based on the WSCC 3-machine 9-bus system [149], by applying the PSAT 8. Figure Model of the WSCC 3-machine 9-bus system in PSAT Figure Rotor angle difference (δ 21 ) between machine 1 and machine 2 under two conditions The system model is shown in Figure 5.17, a fault occurs on bus 7 at 1.0 s and the clearing time is s. The PSS is not applied and the AVR type is the same as the one above. The setting of the AVR is: K a = 400 pu, T r = 0.01 s. Two cases are compared: the first is the original system in which the damping coefficients of all the machines are 0. For the second, a positive damping coefficient (D t = 2.0 pu) is applied in machine 2. 8 Power System Analysis Toolbox (PSAT): (accessed on March 14th, 2017) 81

82 The results are shown in Figure The rotor angle difference (δ 21 ) between machine 1 and machine 2 is taken as an indicator of the system stability [149]. The original system (D t = 0) is unstable (red dashed line); while under the second condition, the contribution from the damping (D t = 2.0 pu) in machine 2 leads to a stable system. 5.3 Discussion on quick response of hydraulic mechanical subsystem A key point of this chapter is whether the responses of GVO and mechanical power of turbines are quick enough to trigger an obvious coupling effect between the hydraulic-mechanical subsystem and the electrical subsystem. Previously the effect of turbine governor has often been ignored in the small signal stability analysis [15]; however in recent years, the rapidity of PFC has been demanded in order to ensure quick response, as shown in section 4.2. Figure 5.19 shows the on-site measurements in a Swedish HPP, supporting the simulations in this chapter in the following aspects. (1) The fast GVO response is demonstrated clearly, and the largest rate can reach the rate limit (0.1pu/s). (2) A delay in the governor system, around 0.25 s between two GVO signals, is shown. (3) In terms of the measured active power, a non-minimum phase response is clearly presented. Furthermore, the rapid GVO response after a three phase fault is also shown in simulations in previous studies [81, 82, 85]. Also, in this thesis, practical nonlinear components (servo, backlash, rate limit) are included and all these factors tend to slow the governor response. In short, the concern on the response rapidity of hydraulic mechanical subsystem is fully considered in this study, and the cases are realistic. Figure Measurements in a Swedish HPP after a step change of the GVO setpoint: active power and deviation of GVO control signal and feedback signal 82

83 6 Efficient operation and balancing renewable power systems In this chapter, the efficient operation of HPPs during balancing actions for renewable power systems is studied, focusing on PFC that acts on a time scale from seconds to minutes. In section 6.1 (based on Paper XI, Paper XII and Paper XIII), the problem description, cause and initial analysis of wear and tear of turbines are presented. Based on the analysis results, a controller filter is proposed in section 6.2 (based on Paper XII) as a solution for reducing the wear of turbines and maintaining the regulation performance, reflected by the frequency quality of power systems. Then in section 6.3 (based on Paper XIV), the study is further extended by proposing a framework that combines technical plant operation with economic indicators, to obtain relative values of regulation burden and performance of PFC. 6.1 Wear and tear due to frequency control Description and definition In terms of wear and tear of hydropower turbines, there are different views and corresponding indicators to evaluate. From a point of view of control, this study focuses on the movements of the GVs in Francis turbines. The GV movements are expressed by the variations of GVO. Two core indicators are discussed, as shown in Figure 6.1: (1) The first is the movement distance which is the accumulated distance of GV movements; (2) The second is the movement amount which means the total number of movement direction changes. One movement corresponds to one direction change. Some distance and amount of GV movement for a regulating unit are always expected. Hence, blindly decreasing the movement is obviously not advisable. However, excessive values of these two indicators bring three types of wear and tear as follows. (1) In the perspective of tribology, there is a linear positive correlation between movement distance and material deterioration on the bearing [100]. (2) From the standpoint of hydraulics, direction changes of the actuator leads to dynamic loads on the turbine runner [13, 96]. 83

84 (3) A huge amount of actuator movement implies a multitude of load cycles, which might increase the structure fatigue. Figure 6.1. Illustration of two very important indices of wear and tear: distance and amount of GV movement. Figure 6.2. Histogram of simulated GVO movements for real frequency record of a week in March, is the number of movements with the distance from 0 to 0.2%. In [97], the measurements show that when Kaplan turbines operate in frequency control mode instead of in discharge control mode, the movement distance of blade angle range will be increased up to ten times, and the amount of load cycles increases from to for the same period of time. It is worth noting that there is a great amount of GV movements with small amplitudes. Reference [150] found that during 4 months of observations in HPPs, between 75 and 90% of all GVO movements are less than 0.2% of full stroke. In this thesis, a simulation is conducted and it demonstrates a similar result: 93.9% of all GVO movements are less than 0.2% of full stroke, as shown in Figure 6.2. More importantly, from the engineering experience, the wear and tear on the materials from small movements is believed to be more serious than from large movements. On the other hand, the regulation value from the small movements is not very obvious. Therefore, decreasing the number of small movements should be a priority. 84

85 6.1.2 Cause Here, a crucial reason for small GV movements is revealed: fluctuations of power system frequency. In order to exemplify the characteristics, measured frequency data of the Nordic power grid frequency is discussed, as shown in Figure 6.3 and Figure 6.4. Figure 6.3. Time-domain illustration of small frequency fluctuations. Ts is the sampling time. The frequency change process from point A to B is a frequency fluctuation, as used in this thesis. From intuitive observations of Figure 6.3, the frequency oscillation can be roughly divided into two components : (1) very low frequency fundamental (with long period, larger than 10~20 s); (2) high frequency harmonic (with short period, less than 1~2 s depending on the sampling time). The fundamental is generally a random signal, while in recent years, the grid frequency oscillations with some specific long periods appear in different power systems, e.g. Nordic power grid (with the period around 60 s) [129, 151], Colombian power grid (with the period around 20 s) [18]. Besides, long period oscillation of grid frequency in Great Britain and Turkey are presented in [57] and [59] respectively. For the short-period harmonic, this work defines a frequency fluctuation as a monotonic frequency changing process between a local maximum (or minimum) and a neighbouring local minimum (or maximum), as shown in Figure 6.3. From Figure 6.3, one could have an intuition that, for the frequency changing process, direction variations happen every one or two sampling periods. The intuition is further verified by Figure 6.4. The green columns are only for the frequency fluctuations with small values which are within ±2.5 mhz. Figure 6.4 demonstrates two important features: (1) the values of frequency fluctuations are mostly very small, within ±2.5 mhz which equals to pu; (2) the time lengths of small fluctuations are also extremely small. In short, the results indicate that the power system frequency experiences both long period fundamental oscillations and harmonic fluctuations with small amplitude and high frequency. The frequency input would lead to the unfavourable amount of small GV movements. More data and discussions can be found in Paper XI and Paper XII. 85

86 (a) (b) Figure 6.4. For the frequency data with 1 s sampling time in the month (March 2012): (a) Histogram of values of frequency fluctuations; the total amount of frequency fluctuations is The total amount of small frequency fluctuations is (b) Histogram of time lengths of the small frequency fluctuations Analysis on influencing factors In this part based on Paper XI, the GV movement is analysed by theoretical analysis based on ideal sinusoidal frequency input and simulations with real frequency records. The influences on wear and tear of different factors, e.g. governor parameters, PF mode and nonlinear governor factors, are explored Method and model Numerical simulations of PFC are conducted under both OF and PF, by applying Model 1. The engineering case here is HPP 1. Detailed settings are given in Appendix B. In terms of theoretical analysis, basic analytical formulas based on idealized frequency deviation signals are deduced. For idealized frequency deviation signals as described in f Af sin 2 t Tf Af sin( t), (6.1) the following formula is proposed to estimate the accumulated movement distance (D y ): Ttotal D y 4 GPI Gm A. (6.2) f T f Here, A f is the amplitude of the sinusoidal input frequency signal, T total and T f represent the total time and a period respectively. G PI is the gain of the PI controller and G m is the product of the gains of mechanical components (backlash and lag), as shown below: 86

87 G PI 1 b p G Kp 1 T f Ki 2 1bK p p 1 T f bpk i backlash G lag 2 2, [pu/ pu], (6.3) BLgv 1, [pu/ pu], (6.4) A 1 in T 2 y T f (6.5) Here, A in is the input amplitude, while BL gv represents the value of backlash. T y stands for the lag in the main servomotor. Here, a simpler representation of the backlash is applied, comparing to Equation (3.42) that is based on the describing function method. Additionally, the response time (section 4.2), i.e. the time it takes for the opening to reach 63.2 % ( 1-e -1 ) of its final value after a step disturbance is T 1 bk p p 1ln(1 b K ) bk, [s]. (6.6) r p p p i The response time indicates the rapidity of PFC Results: influencing factors on wear and tear Here, the influence of different factors is discussed, based on Equation (6.2) and simulations under sinusoidal input signal, Δf with amplitude Hz ( pu), and with a real record of frequency deviation in the Nordic power system. As shown in Table 6.1, the governor parameters have essential influence on the GVO movements. The theoretical formulae for ideal input reflects the trend for real movements well, as can be seen from the comparison between the formula calculation and simulation results of movement distance. In Table 6.1, under different parameter settings, the change tendencies of movement distance under ideal and real frequency are in good agreement, for both OF and PF modes. Therefore the formulae are effective to achieve a good tendency estimation. Note that in the formula, the gain and movement distance are directly determined by the period. However, the period of real frequency is changing all the time and hard to get an approximate value. This brings a difficulty of applying the formula to estimate the real condition, but it will not influence the tendency prediction. 87

88 Table 6.1. Influence of governor parameters under OF and PF. For the movement distance, the total calculation and simulation time is two hours. Movement distance under OF and the response time are calculated by formulas; other values are from simulation. PF 75% and 100% mean that the GVO set point is at 75% and 100% of rated value, respectively. 4. Groups 1 to 4 are Vattenfall standard sets of parameters. Movement distance (Dy) under Sinusoidal frequency (with Movement distance b P K P K I Response backlash) under real frequency No. [pu] [pu] [s -1 ] time (s) OF PF-100% T f = T f = T f = T f = PF - PF - OF 60s 120s 60s 120s 75% 100% In terms of the application of PF, as shown in Table 6.1, the influence is normally not too large; however the difference could be relatively substantial under large gain conditions. Besides, PF may lead to either increase or decrease of movement, comparing with OF. More discussions on nonlinear factors and two main influence factors under PF, operation set point and surge (water level fluctuation in surge tank), can be found in Paper XI. 6.2 Controller filters for wear reduction considering frequency quality of power systems Aiming at the aforementioned problem in section 6.1 and the initial results, in this section, applying a suitable filter in the turbine controller is proposed as a solution for wear reduction. However, the controller filters impact the active power output and then affects the power system frequency. Therefore, the purpose of this section is the trade-off between two objectives: (1) reducing the wear and tear of the turbines; (2) maintaining the regulation performance, reflected by frequency quality of power systems. The widely-used dead zone is compared with a floating dead zone and a linear filter, by time domain simulation and frequency domain analysis. The filters are introduced in section

89 6.2.1 Method and model The governor system (Figure 3.8) in Model 2-L (section 3.2.1) is applied to simulate the GV movements, based on a measured one-day grid frequency data. Then, the frequency under the influence of the different filters can be simulated and compared by using Model 2-L, based on a certain load disturbance. The mean value and standard deviation (SD) are chosen as the indicators to evaluate the frequency quality [152, 153]. However, load disturbances in power systems are unknown. Therefore a grid inverse [129] model is built, as shown in Figure 6.5, to compute a load disturbance from the existing measured frequency. The transfer function of the grid inverse model is shown in Ms D Gr () s ts p 1. (6.7) To avoid high amplification of high frequency noise in the grid frequency signal, a pole with time constant t p (set to 0.1) is added [135]. Figure 6.5. Simulink model of a grid inverse for computing the load disturbance, as highlighted by red In terms of the theoretical analysis, the describing functions (section ) and Nyquist criterion (section ) are adopted to examine the frequency response and stability of the system with different filters. Model 3-L is applied, as described in section On-site measurements Here, on-site measurements and its comparison with simulations are presented. The measurement was conducted in HPP 8. The measurement length is 6800 s, and Figure 6.6 shows a 3000 s period of comparison between the simulation and the measurement. The detailed information is shown in Appendix B. As shown in Figure 6.6, the simulation matches the measurement well in time domain. 89

90 Figure 6.6. Comparison of the simulated GVO and the GVO measured in HPP 8. The measurement noise is demonstrated in the small plot inside the right figure. Table 6.2. Statistics of the one-day GVO movements under different conditions of the servo (Ty) and backlash Movement distance (full strokes) Number of movements Backlash Ty = 0 Ty = 0.1 Ty = 0.2 Ty = 0 Ty = 0.1 Ty = % % % % % % Time domain simulation Further results of time domain simulation are presented here. In the following simulations, the governor parameter setting adopts Ep3 (see Table 14.2 in Appendix B), which leads to relatively large gain and GV movements. Other settings are also given in Appendix B. Firstly, the cases without any filters are presented in Table 6.3, showing that without any filters, the governor system (especially the actuator) inherently filters the majority of the frequency fluctuations. This can also be observed from the measurements above: The time length of a GV movement is relatively long, the average value is 35.1 s (6800 s divided by 194 movements). The period value is 60~70 s, corresponding to the period of fundamental frequency fluctuation in Nordic power grid. This further shows that the GV movement is mainly determined by the fundamental component of the input frequency, and the influence of the harmonic component is not significant. The performances of different filters are presented in Table 6.3. It is shown that the traditional filter, dead zone, indeed reduces the movement distance, but not the movement amount. In contrast, the floating dead zone has a good 90

91 performance for the distance and especially for the movement amount. The linear filter could also decrease both two indicators, however the effect is not obvious. Table 6.3. Statistics of the one-day GVO movements and the frequency quality with different filter types. The GVO movement is simulated by the governor model in Model 2-L, under two frequency input data with different sampling times. The frequency quality is computed by Model 2-L. T s is the sampling time of the data. In the column 50-Mean, the values refer to the mean frequency deviations; for example, the value Hz means the mean value of the frequency without any filter is Hz. SD stands for the standard deviation. Filter type Value setting (pu) Guide vane opening Frequency quality Ts = 1 s Ts = 0.02 s distance amount distance amount 50 - Mean SD No filter(abs) \ strokes strokes Hz Hz No filter (pu) \ 100.0% 100.0% 100.0% 100.0% 100.0% 100.0% ±0.01% 89.9% 99.5% 89.9% 99.1% 121.0% 108.9% Frequency ±0.02% 80.4% 96.8% 80.4% 96.6% 142.7% 118.1% filter - Dead ±0.05% 54.6% 83.9% 54.5% 84.6% 221.1% 146.2% zone (E dz) ±0.1% 22.4% 43.8% 22.3% 43.1% 354.1% 193.8% Frequency % 96.5% 87.7% 99.4% 97.0% 105.4% 103.7% filter % 88.5% 71.7% 93.6% 81.1% 103.5% 112.9% Floating % 56.1% 34.5% 62.5% 40.9% 114.9% 149.1% dead zone 2 0.1% 17.2% 4.7% 19.7% 6.5% 104.2% 229.7% Frequency % 96.4% 99.0% 97.0% 100.0% 102.0% filter % 93.5% 97.4% 92.2% 100.1% 106.3% Linear (T f1) % 89.8% 95.3% 88.9% 100.1% 125.1% ±0.1% 98.7% 99.7% 98.7% 99.7% 102.6% 101.0% GVO filter - ±1.0% 87.6% 92.8% 87.6% 93.3% 124.8% 110.7% Dead zone ±2.0% 77.2% 85.7% 77.1% 85.7% 149.8% 121.5% (E dz) ±5.0% 46.7% 55.0% 46.7% 55.1% 232.4% 153.9% GVO filter - 0.1% 94.4% 83.6% 94.4% 83.5% 100.0% 101.0% Floating 0.5% 77.6% 56.8% 77.6% 56.5% 99.6% 107.1% dead zone 1.0% 62.5% 40.9% 62.5% 40.8% 101.2% 117.3% (E fdz) 2.0% 43.0% 23.5% 43.0% 23.4% 97.6% 143.7% % 96.4% 99.0% 97.0% 100.0% 102.0% GVO filter % 93.5% 97.4% 92.2% 100.1% 106.3% Linear (T f2) % 89.8% 95.3% 88.9% 100.1% 125.1% The frequency quality, the crucial trade-off factor, is analysed under different filters here. As demonstrated in Table 6.3, the dead zone leads to a poor frequency quality. By contrast, the linear filter only slightly increases the standard deviation. The floating dead zone results in a medium performance of the frequency quality. 91

92 The effects of these three types of filters are obvious from the histogram in Figure 6.7. The floating dead zone and the linear filter increase the standard deviation, but the distribution is still single-peaked even with a large parameter setting. However, the dead zone leads to a very unfavourable doublepeaked distribution. Figure 6.7. Histogram of the one-day frequency simulated by the Nordic power system model with three types of filters Frequency domain analysis: stability of the system Here, the Nyquist stability criterion is applied to test the stability of the system with different filters. Firstly, the nonlinear filters are tested, as shown in Figure 6.8. The system is stable with the dead zone under these parameter settings. In contrast, the floating dead zone leads to a limit cycle oscillation in the system with the Ep3 parameter setting; while when the system adopts the Ep0 setting, the oscillation is avoided. However, the describing function framework only gives indication of the system stability, therefore the result is verified by a time-domain simulation of a load step change ( pu), as shown in Figure 6.8 (b). Nevertheless, in section 6.2.3, even with the limit cycle oscillation under the Ep3 setting, the frequency quality under the floating dead zone is still acceptable. Then, the influence of the linear filter is discussed, as described in Figure 6.9 (a). For the system with the Ep3 parameter setting, the critical value of the filter constant, T f1 or T f2, is approximately 4.0 s. It is validated clearly by timedomain simulation, as shown in Figure 6.9 (b). 92

93 (a) (b) Figure 6.8. (a) Frequency-domain result: the Nyquist curve of the open-loop system under different governor parameters and the negative reciprocal of the describing functions. (b) Time-domain result: the system frequency after a load step change ( pu), simulated by Model 2-L. (a) (b) Figure 6.9. (a) Frequency-domain result: gain margin and phase margin of the system with the linear filter. (b) Time-domain result: the system frequency after a load step change ( pu), simulated by Model 2-L Concluding comparison between different filters The main conclusion of this section is shown in Table 6.4. It suggests that the floating dead zone, especially the GVO filter after the controller, outperforms the widely-used dead zone on the trade-off between the wear reduction and frequency quality. 93

94 Table 6.4. Comparison between different filters Filter Advantages Disadvantages Dead zone 1. Deceasing the GV movement distance effectively 2. No obvious influence on system stability Two fatal points: 1. Worst frequency quality: the double-peaked distribution 2. Not very effective in reducing the GV movement amount, which is the main goal Floating dead zone Linear filter 1. Well reducing the GV movement amount 2. Deceasing the GV movement distance effectively 3. Good frequency quality, especially with the GVO filter 1. Best frequency quality 2. Can decrease both the movement distance and amount to some extent Might cause limit cycle oscillation (however it can be avoided, i.e. by tuning the governor parameters; even with the limit cycle oscillation, the frequency quality is still acceptable) 1. Cannot obviously decrease both the movement distance and amount 2. Might cause system instability (however it can be avoided, i.e. by tuning the governor parameters) 6.3 Framework for evaluating the regulation of hydropower units In this section, a framework is proposed as shown in Figure 6.10, combining technical operation strategies with economic indicators, to obtain relative values of regulation burden and performance of PFC. Figure Framework for quantifying and evaluating the regulation of hydropower units. Efficiency loss as well as wear and fatigue are adopted to represent the burden; regulation mileage and frequency quality are applied to evaluate the regulation performance. For the quantification, Model 2-K is applied and calibrated with measurements from HPP 6 and HPP 7. Kaplan turbines are studied here, since they are more complicated in terms of control. Hence, the methodology and results can 94

95 easily be simplified and extended to other turbine types. Burden relief strategies and their consequences are discussed, under two idealized remuneration schemes for PFC, inspired by the ones used in Sweden and in parts of the USA. They differ in the underlying pricing philosophy mainly in that the Swedish one does not take actual delivery into account, but rather compensates for the reserved capacity The framework In the framework, burden is represented by efficiency loss and wear/fatigue, and regulation performance is evaluated using regulation mileage and frequency quality. The technique to quantify burden and regulation performance and the corresponding indicators that serve as the main outputs of the numerical simulations are introduced in subsection Optimizing the regulation conditions of hydro units is the key to easing their incurred burden. Various regulation conditions are comprehensively compared by varying the turbine governor parameters (Ep1 Ep3 in Table 14.2 in Appendix B), operating set-points (seven points in Figure 6.11) and regulation strategies (subsection 6.3.2). (a) (b) Figure Illustration of the combinator table for the turbine in the HPP 6 (a) and HPP 7 (b). In each figure, seven on-cam operating points are highlighted by blue scatters, they are within the maximum efficiency range. Under different conditions, the following two idealized pricing schemes of regulation payments are concisely analysed: strength payment and mileage payment that are inspired respectively by the ones used by the TSO SvK in Sweden and by PJM Interconnection LLC (PJM), a regional transmission organization in the USA [154]. The schemes are detailed in subsection Various simulations are conducted to test the above-mentioned indicators based on HPP 6 and HPP 7, as illustrated in Figure

96 Figure Illustration of the simulation structure. The models 2-K-1 through 2-K- 3 are introduced in subsection The blocks with dashed outline represent the selections in simulation setting based on different conditions: i varies from 1 to 3, for three sets of governor parameters Ep1 Ep3; j varies from 1 to 4, for three regulation strategies (S1 S3) and an ideal on-cam case (S0); k varies from 1 to 2 for two HPPs; n varies from 1 to 7 for seven operating set-points. The set in the parenthesis with simulation presents different cases conducted in the model. In total, there are 168 ( ) and 24 ( ) simulation cases conducted in Model 1 and Model 2 respectively. The terms in the parenthesis with the output variables show the actual needed set of results for analysing the corresponding indicator in this paper Methods Here, the detailed methods are introduced. It is worth noting that the burden (efficiency loss, wear and fatigue) is discussed from a physical perspective. Further economic modelling to obtain the gain or loss of profit from regulation is not included, while it is necessary in the future to fully characterize the effects of PFC on system economics Numerical models As shown in Figure 6.12, three models are applied. Model 2-K-1 and Model 2-K-2 are presented in section 3.4, and Model 2-K-3 is introduced here. In subsection 6.2.1, a method of simulating and evaluating the frequency quality for units with Francis turbines is introduced. Here, the method is improved and extended for Kaplan turbines. Model 2-K-3 is for computing the unknown sequence of one-day load disturbance from the original measured frequency, as shown in Figure The grid inverse model is shown in Equation (6.7) in subsection 6.2.1, and the detailed parameters are given in Table 14.5 in Appendix B. The value of time constant t p is set to 0.1 s in this study. 96

97 Figure Block diagram of Model 2-K-3 with the grid inverse model for computing the load disturbance; the lumped HPP block is described in the part with dashed outline in Figure (a) (b) Figure Comparison of measurement and simulation during a period of normal PFC. (a) GVO (y) and RBA (α), the deviation value is shown. (b) Active power. In Model 2-K-3, all the regulating HPPs in the Nordic power grid are lumped into one scaled HPP with the scaling factor (K 3 ). The model of the lumped HPP is introduced in subsection HPP 6 and HPP 7 are applied as engineering cases. One unit of each HPP is taken as the study case. The detailed parameter values of the two HPPs are shown in Table 14.6 in Appendix B. For HPP 6, the dynamic processes under normal PFC is simulated by Model 2-K-1 and compared to the measurements under the governor parameters Ep1 (Table 14.2), as shown in Figure The simulation of the GVO, the RBA and the power output has a good agreement 97

98 with the measurements, showing that the model can yield trustworthy simulation results Regulation strategies The following operation strategies S1 S3 and an ideal case S0 are analysed. (1) S1: normal PFC in which GV and RB regulate without any artificial filter (widely implemented); (2) S2: PFC with a floating dead zone filter for reducing the movement of runner blades; (3) S3: PFC with the runner blades being totally fixed (no RB movements); (4) S0: Normal PFC in an ideal on-cam condition, and it is unrealistic and only implemented to identify the off-cam loss in normal PFC Method of quantification Here, the method of quantification of the burden and quality of regulation is introduced. (1) Efficiency loss The losses are classified into the following compositions, as illustrated in Figure Figure Compositions of efficiency losses analysed in this work The loss in steady state operation, -Δη st, is st 1 st, [pu]. (6.8) A negative value of the efficiency change Δη indicates an efficiency loss. η st is the on-cam steady state efficiency that is a constant value taken from the interpolation, and it varies for different operating points. In this section, the main object is the extra efficiency loss due to regulation, which is given as Sj st Sj, [pu]. (6.9) Here, η Sj is the average value of the instantaneous efficiency during the operation period (one day) under a specific strategy (S j ). More specifically, the efficiency loss in transient due to deviation from the set-point (on-cam) can be obtained as, [pu]. (6.10) S0 st S0 98

99 The loss due to off-cam in normal PFC is achieved by the difference between Δη S0 and Δη S1. For example, the extra loss due to off-cam condition under strategy S2 and S3 is Δη S0 - Δη S2 and Δη S0 - Δη S3, respectively. (2) Wear and fatigue For quantifying the wear and fatigue of turbines, we use the two indicators introduced in subsection for both GV and RB: the movement distance and the movement amount. The movement distance can be described in N YGV, dist yis yis 1 is1. (6.11) N YRB, dist ais ais 1 is1 Here, N is the total amount of samples and is means the sample number. y and a represent GVO and RBA respectively. (3) Regulation mileage The regulation mileage is introduced to quantify the amount of work hydropower units expend to follow a regulation signal, as described in N MR Pm rated pm, is pm, is1, [MW]. (6.12) is1 Here, p m is the active power in per unit; P m-rated is rated power of the Kaplan turbine, and its unit is MW. (4) Frequency quality The frequency quality is evaluated to comprehensively reflect the regulation performance of the hydropower unit. As presented in subsection and in the lower part of Figure 6.12, the core idea is comparing the new frequency sequence of the power system under different regulation conditions, to examine whether the frequency quality is deteriorated. The frequency quality is mainly evaluated through the root mean square error (RMSE with respect to the rated frequency 50 Hz) of the frequency sequence. In section 6.2, the frequency quality reflects the influence of a lumped unit that represents all the generating units in the grid. By contrast, the method here examines the influence of the regulation from a single Kaplan unit on the frequency of the whole grid Regulation payment In this section, only relative values of payment are considered. Clearing prices are not considered in the quantification. The strength payment Pay strength, inspired by SvK, is computed as Pstep SR, [MW/Hz] 0.1. (6.13) SR Paystrength, [pu] SR base 99

100 Here S R is the regulation strength; P step is the increase in output power caused by a frequency step change that is selected as from 50 Hz to 49.9 Hz by SvK. S R-base is a base value for normalizing the payment, and here it is set to the regulation strength (41.08 MW/Hz) of the unit in HPP 6 under Ep1 and S1. The payment will depend on governor parameterization, turbine characteristics, burden relief strategy, and operating point. The mileage payment Pay mile, inspired by PJM, is based on the mileage in power output M R Pay, [pu]. (6.14) mile M R base M R-base is a base value for normalizing the payment, and here it is set to the regulation mileage (449.5 MW in Table 6.5 below) of the unit in HPP 6 under Ep1 and S Setting of simulations For all the simulations, the length is 24 hours and time-step is 0.02 s. The input signal of Model 2-K-1 and Model 2-K-3 is a sequence of measured one-day (24-hour) Nordic grid frequency, and its sampling time is 1.0 s. The implementation of the strategy S2 and S3 is by setting the value of the filter that is the floating dead zone for RBA. For the S2 and S3, the value of the floating dead zone is set to 0.03 pu and 1.0 pu respective; while for the S1, this filter value is set to 0. For the ideal case S0, all the mechanical components in runner blade part (after the combinator in Figure 3.17) are removed to achieve a purely on-cam operation. The initial steady-state value of water head is set to the rated value (1.0 pu). (1) Settings for simulating the GVO, RBA and the efficiency loss The adopted model is Model 2-K-1, without the engagement of the power grid components. Hence the scaling factor (K) is set to 1.0. As presented in Figure 6.12, there are 168 ( ) simulation cases conducted. (2) Simulating and evaluating the frequency quality Firstly, Model 2-K-3 is applied for computing the load disturbance. Currently, the Nordic TSOs require a fixed amount of regulating power of PFC in the whole grid: 753MW for 0.1 Hz frequency deviation, hence the total regulation strength S RT is 7530 MW/Hz, and it is normalized as S RT-pu that is 10.0 pu [129]. Consequently, in Model 2-K-3, the product of the governor static gain 1/b p and K 3 is 10.0 pu [129]. Here, the governor parameter set is selected as Ep1 for computing the load, and the droop in Model 2-K-3 is noted as b p3 (b p0 = 0.04 pu). Therefore, the value of K 3 is set to 0.4 pu (10 b p3 pu). Then, with the simulated load as input, Model 2-K-2 is applied to simulate new frequency sequences. In Model 2-K-2, the regulation power in the grid is provided by the Kaplan unit (Model 2-K-1) that is the examining object as well as the rest of units (Francis unit in the lumped HPP). In Model 2-K-2, the values of two scaling factors (K 1 and K 2 ) are described in 100

101 SR1 SR2 SRT (6.15) SR1pu SR2pu G1K1 G2K2 SRT pu Here the S R1 and S R2 are the regulation strength of the Kaplan unit and the lumped HPP respectively; their corresponding values in per unit are represented as S R1-pu and S R2-pu. G 1 and G 2 are the gains from frequency deviation to power deviation for the Kaplan unit and the lumped HPP, as presented in SR 1 G1 10fstep Pm rated, [pu]. (6.16) 1 G2 bp 2 Here, Δf step is the step change value of frequency 0.1Hz (i.e pu); P m-rated is the rated power of the Kaplan unit, b p2 means the droop in the lumped HPP. In this section, the regulation strength S R2 is kept as constant. The S R1 of the Kaplan turbine changes with various conditions and influences the frequency quality. For each HPP, the new frequency sequences are compared to the frequency sequence from Ep1 and S1. The operating point has little influence on the frequency quality, hence the operating point stays the same (selected as point 5) in all simulations, as shown in Figure Burden quantification In this section, the efficiency loss as well as wear and fatigue due to regulation are discussed. The overall results under various operation conditions are shown in Table 6.5, and detailed results of efficiency loss are presented in Table 6.6. It is shown that the governor parameter directly influences the efficiency loss: the lower the droop (the higher the static gain), the more efficiency loss. However, the loss is not linearly related to the droop due to the complexity and nonlinearity in the system. The compositions of efficiency loss in regulation are studied by comparing the results in different strategies. The difference between efficiency losses under S1 and S0 is very small, indicating that the loss due to normal PFC is mainly caused by the trajectory deviation from the set-point. The extra loss due to off-cam operation is negligible for normal PFC, but not for wear reduction strategies. The operation strategy S3 leads to a considerable efficiency loss that is larger than 1.0 % under the high gain setting Ep3 and mainly caused by off-cam operation, showing the economic drawback of the strategy. While the strategy S2 only causes a slight increase (approximately 0.02 %) of efficiency loss compared to the S1. 101

102 Table 6.5. Overall results of different operation conditions from one-day simulation. The bar in each cell indicates the relative magnitude of the values with the same color. The results of efficiency loss are condensed from Table 6.6. The results of GV movement are based on HPP 6, because there is little difference on the indicators between two HPPs. The operating point does not influence much on movement of GV and RB, hence only the results from point 5 are shown. For frequency quality, the values of the change of root mean square error are shown; negative and positive values are shown with green and grey bars respectively, and positive values indicate better frequency quality. The regulation mileages are shown with purple bar, and the detailed results are in Figure Burden Regulation performance Payment Parameter Ep1 (b p = 0.04) Ep2 (b p = 0.02) Ep3 (b p = 0.01) Strategy S1 S2 S3 S1 S2 S3 S1 S2 S3 Avg-HPP % % % % % % % % % Efficiency Min-HPP % % % % % % % % % change [pu] Avg-HPP % % % % % % % % % Min-HPP % % % % % % % % % GV Distance movement [/] Amount Dist.-HPP RB Dist.-HPP movement Amount-HPP [pu] Amount-HPP Frequency HPP % -0.28% 0.52% 0.45% -0.07% 1.56% 1.48% 0.38% quality [pu] HPP % -0.24% 0.43% 0.38% -0.04% 1.31% 1.25% 0.38% Mileage HPP [MW] HPP Strength HPP % 79.7% 33.5% 204.6% 191.1% 54.6% 421.9% 422.3% 43.6% [pu] HPP % 67.5% 25.6% 162.9% 150.2% 32.1% 317.3% 308.4% < 0 Mileage HPP % 51.0% 46.3% 243.6% 127.5% 107.2% 536.7% 294.6% 221.7% [pu] HPP % 46.2% 41.6% 212.4% 112.6% 98.4% 450.9% 257.4% 213.3% It is found that the operating set-point has a considerable influence not only on the steady state efficiency but also on the efficiency loss in transients, as shown in Table 6.6. The differences between the maximum and minimum of efficiency changes under seven points are considerable (e.g %, 1.087% and 1.610% under Ep1 through Ep3 respectively for HPP 7). This causes complexity in the quantifying and evaluating process. In some cases, the efficiency even increases compared to the steady state value, as highlighted by blue in the table cell. The main explanation is that the on-cam efficiency of a set-point (e.g. point 1) is not the global maximum. 102

103 Table 6.6. Detailed simulation results of efficiency change (Δη Sj ). The seven operating points of each HPP are shown in Figure Negative values mean efficiency loss. The bar in each cell is a graphical representation of the value, and the negative and positive values are highlighted by red and blue respectively. The average values of seven points are selected as the representative results listed in Table 6.5. The value of Max-Min is the difference between the maximum and minimum of the efficiency changes of seven points. Para. Operation point Ep1 Ep2 Ep3 Δη S0 Δη S1 HPP 6 HPP 7 Δη S2 Δη S3 Δη S0 Δη S1 Δη S2 Δη S3 (S3) (S0) (S1) (S2) (S3) (S0) (S1) (S2) % % % % % % % % % 0.019% 0.014% % 0.301% 0.301% 0.269% 0.205% % 0.112% 0.092% 0.023% % % % % % 0.045% 0.039% 0.003% 0.235% 0.235% 0.212% 0.159% % % % % 0.371% 0.371% 0.312% 0.123% % % % % % % % % % 0.075% 0.030% % 0.057% 0.056% 0.041% 0.006% Average % % % % % % % % Max - Min 0.309% 0.309% 0.299% 0.241% 0.774% 0.774% 0.734% 0.666% % % % % % % % % % 0.031% 0.022% % 0.502% 0.502% 0.483% 0.040% % 0.185% 0.171% % % % % % % 0.042% 0.022% % 0.215% 0.215% 0.183% % % % % % 0.305% 0.304% 0.259% % % % % % % % % % % % % % % % % % Average % % % % % % % % Max - Min 0.555% 0.555% 0.565% 0.403% 1.087% 1.087% 1.092% 1.043% % % % % % % % % % % % % 0.511% 0.511% 0.482% % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % % Average % % % % % % % % Max - Min 0.643% 0.643% 0.638% 0.582% 1.610% 1.610% 1.608% 1.053% In terms of wear and fatigue, the governor parameters has a direct influence, as shown in Table 6.5. A lower droop leads to longer distance and larger amount of movements. For the GV movements, results do not differ from S1 to S3, because different strategies only affect the runner blade side. The burden relief strategy S2 leads to a significant decrease of both the distance and amount of RB movements, and strategy S3 totally diminishes the RB movement. Besides, the influence from different operating points is small, and this decreases the complexity in analysis. 103

104 6.3.4 Regulation performance Here, the regulation mileage and frequency quality are discussed as crucial trade-off perspectives of the burden. As shown in Table 6.5 and Figure 6.16, there is a high correlation between regulation mileage and the static gain (1/b p ). As to the operating strategy, the wear reduction strategy S2 and S3, especially S3, lead to decreases in mileage. The results are not influenced much by the operating set-points, and this is conducive to simplify the evaluation. (a) (b) Figure Regulation mileage of two HPPs for 24 hours. For each case, results from all seven operating points are close, hence the scatters overlap each other; the average values of seven points in various conditions are presented in Table 6.5. In terms of frequency quality, the results are presented in Table 6.5. A larger value indicates a better frequency quality. Overall, lower droop that corresponds to higher regulation strength of the unit results in better frequency quality, based on our setting that the regulation strength of the rest of the units in the grid is unchanged. Besides, the influence of the off-cam operation is shown clearly: when the strategy S2 is applied, the frequency quality worsens compared to the performance of S1 under the same Ep setting. The deterioration of the frequency quality is more obvious under S Regulation payment Evaluating consequences of regulation payment schemes is of great importance for both power producers and TSOs. Here, the two idealized schemes, inspired by the ones used by SvK in Sweden and PJM in the USA, are analysed through their relative payment values. Relative values of payments under different conditions are presented in Table 6.5. Both strength payment and mileage payment lead to compensation overall aligned with performance, as it correlates well with RB movement and frequency quality. Compared to strategy S1, strategy S3 leads to a considerable compensation decrease while causing an obvious drop in efficiency and poor frequency quality. Therefore, S3 may not be a promising choice for producers. 104

105 Alternatively, strategy S2 can be suitable for producers, because it does not result in significant compensation decline (especially for HPP 7) and can largely decrease the RB movements without deteriorating the efficiency and frequency quality too much. An important finding is a large difference in two payments under burden relief strategies S2 and S3. This illustrates the influence that payment schemes might have on operating strategies chosen by producers. (a) (b) Figure Strength payments based on the strength obtained from the -0.1 Hz frequency step change and a Hz frequency step change. (a) HPP 6; (b) HPP 7.For each frequency change case, the relative values are normalized with the respect to the strength under Ep1 and S1 for each HPPs. For HPP 7, the strength under S3 with Ep3 is negative, because the turbine efficiency sharply decreases in the off-cam operation and the power output after the frequency change is even smaller than the original value. However, the regulation performance of the case is positive (with +0.38% in frequency quality and MW in mileage) since the average frequency deviation is considerably smaller than the 0.1 Hz used to determine strength. This demonstrates a limitation of the strength payment: it ignores the performance of actual delivered regulation. As shown in Figure 6.17, the negative strength is avoided by varying the frequency change to Hz from the original value 0.1 Hz: the payment increases to 63.5 % from -9.9% under S3 with Ep3 for HPP 7. This reflects the complexity of achieving an appropriate implementation of strength payment. 105