HYDRO power plant (HPP) turbine governing has been studied extensively in the

FACTA UNIVERSITATIS (NIŠ) SER.: ELEC. ENERG. vol. 23, no. 2, August 2010, 191-198 Suboptimal Design of Turbine Governors for Low Head Hydroturbines Dušan B. Arnautović and Dane D. Džepčeski Abstract: The paper considers a low head hydro units control problem by using the modern control theory for turbine governor design. The design of optimal and suboptimal turbine governor is performed by using the optimal control theory and projective controls method. Implementation of modern control theory in contemporary turbine governors is fairly easy to obtain due their realisation in digital technology. Comparison for the closed-loop system dynamic performances, shows that better results are obtained with optimal and suboptimal governors as compared to conventional turbine governors. Keywords: Hydro power plant control, turbine governor, suboptimal control. 1 Introduction HYDRO power plant (HPP) turbine governing has been studied extensively in the past. These studies have shown that the verification of the stability and quality of speed regulation always represents an important phase in the design of HPP. In recent years, considerable attention has been directed toward the problem of designing turbine governors, because of its influence on the dynamic performance of HPP. Classical control theory has been widely used in the conventional approaches of turbine governor design. On the other hand, recent reasearches, emphasize the significance of modern control theory. Therefore, the application of modern control techniques to HPP control, particularly turbine governing, is an area of great interest. Development of new turbine governor design, incorporating digital components, are motivated in part by the expectations that new types of governor will Manuscript received on February 28, 2010. The authors are with Nikola Tesla Institute, Automation and Control Department, 11000 Belgrade, Koste Glavinića 8a, Serbia (e-mail: darnautovic@eieent.org). 191

192 D. Arnautović and D. Džepčeski: improve the quality of speed regulation [1 4]. Numerous new features in control are enabled in turbine governors that are realised in digital technology, due to their flexibillty and their capabillity to implement more complex algorithms [2, 5 8]. Flexibillity of digital technology, in both, hardware and software, enables at short notice development, adjustment and verification of applied control approach. Complexity of control algorithm, now is a concern of software design, hardware limits in number of inputs and outputs, amount of memory or software execution time are rearly exceeded. In the reasearches presented in this paper, the method of projective controls is used to solve the problem of synthesis of Kaplan or bulb turbine governors with an arbitrary number of inputs and two outputs (the regulation of Kaplan or bulb turbine is performed by two means guide vane and runner blade). It is shown that the developed methodology for synthesis of static and low-order dynamic output regulators can be directly applied for synthesis of low head hydroturbine governors. This procedure enables the choice of structure and adjustment of parameters of turbine governor, all in order to improve the dynamic performance of HPP. Realisation of choosen governor structure, structure and governor parameters modification and theirs setting to particular use, by means of actual digital technology, is considerably simple. Complexity of structure and structure variations are only limited by the knowledge and experience of turbine governor designer. 2 Mathematical Model of the Hydro Generator With a Low Head Turbine The first step in determining suboptimal control is to develop an adequate HPP model [9 12]. For turbine governor design purposes, to assure stability of regulation, linearized models are normally used to describle dynamics about an operating point. The basic components involved in the representation of a single low head turbine generator connected through a single transmission line to an infinite bus are shown in the block diagram in Fig. 1. The equations describing the plant behaviour are summarized below. All variables are in per-unit deviations. When modelling the hydro unit with a low head turbine, because of the relatively short pipelines, the water and pipes are taken to be incompressible. With this commonly used assumption [1 3] the stiff waterhammer equation can be presented as: dq ξ = T W (1) dt where T W is water starting time and q is turbine flow.

Suboptimal Design of Turbine Governors for Low Head Hydroturbines 193 Fig. 1. Block diagram of low head turbine-generator system. The electromechanical equations of the system can be expressed as: dδ dt (T a + T p ) dω dt = ω 0 ω = m T m L Dω (2) where ω 0 is rated machine speed, T a and T p are time constants that reperesent inertia of machine and water rotary masses, m T and m L are turbine and load torque respectively and D is damping coefficient. For Kaplan turbines T p is small comparing to T a, and usualy is neglected. However, for bulb turbines, where T a is small, it is necessary to take into consideration the influence of water masses, which rotate together with turbine. This is achieved by adding T p on T a, where T p is function of runner blade angle. Relation T p /T a for bulb turbines is usualy in range from 0.35 to 1.2. The turbine flow q and torque m T in Kaplan and bulb turbines are, in general, nonlinear functions of several variables: head h, guide vane opening a, machine speed ω and runner blade position φ : q =q(h,a,ω,φ ) m T =q(h,a,ω,φ ) (3) When small deviations of flow and torque, from an operating point takeplace, it is considered that these curves may be represented by the following Taylor-series

194 D. Arnautović and D. Džepčeski: expansion φ : q = q q q h+ a+ h a ω ω + q φ φ = T 1 h+t 2 a+t 3 ω + T 4 φ m T = m h m m h+ a+ a ω ω + m φ φ = T 5 h+t 6 a+t 7 ω + T 8 φ (4) The fact that the regulation of water flow in Kaplan and bulb turbines is performed by two means, both the guide vane opening and the runner blade position must be taken into consideration. The corresponding servomotor equations are da dt = u c a T gv dφ = u c φ dt T rb (5) where u c is regulator output, T gv and T rb are guide vane and runner blade time constants respectively. The complete model is obtained by joining the above equations. The model in state space form can be written as ẋ =Ax+Bu y =Cx (6) where x T = [δ, ω q, a, φ ] and the nonzero elements of matrices A and B are: a 12 = ω 0 T 5 a 23 = T 1 (T a + T p ) a 32 = T 3 T 1 T w a 35 = T 4 T 1 T w b 41 = 1 T gv p tie a 21 = T a + T p a 24 = T 5T 2 T 1 T 6 T 1 (T a + T p ) a 33 = 1 T 1 T w a 44 = 1 T gv b 51 = 1 T rb a 22 = T 3T 5 T 1 T 2 + T 1 D T 1 (T a + T p ) a 25 = T 4T 5 T 1 T 8 T 1 (T a + T p ) a 34 = T 2 T 1 T w a 55 = 1 T rb c 12 = 1. The machine output vector y can be chosen arbitrarilly and is composed of one or more locally available measurements (machine speed, derivative of speed, integral of speed, turbine torque, electric power output, etc.), which can be used as input variables to the turbine governor.

Suboptimal Design of Turbine Governors for Low Head Hydroturbines 195 3 Problem Formulation The problem is to design a turbine governor in the hydro power plant so as to achieve a satisfactory damping of speed and faster responses. This problem can be turned into the following output regulator problem. For the given state space description of a hydro power plant (6) the problem is to design a static turbine governor, with feedback gain K, of the form u = Ky (7) such that the closed-loop system has satisfactory dynamic characteristics measured by the quadratic criterion J = 1 2 0 (x T Qx+u T Ru)dt, Q 0, R > 0 (8) Therefore, for the general case of synthesis it is necessary to determine: (1) the regulator dimension, (2) regulator parameter, and (3) feedback gains so that the closed-loop system has satisfactory dynamic characteristics. The design procedure by projective controls is given in [7]. 4 Case Study For HPP D- erdap 2 The proposed methodology was applied to the design of turbine governors on HPP D- erdap 2. System data are given in Table 1. Table 1. Hydro power plant D- erdap 2 data T a + T p (s) 3.06 T 3 (p.u.) 0.48 T w (s) 4.97 T 4 (p.u.) 0.471 T gv (s) 0.2 T 5 (p.u.) 1.301 T rb (s) 1. T 6 (p.u.) 0.672 D(p.u.) 1. T 7 (p.u.) 0.814 T 1 (p.u.) 0.235 T 8 (p.u.) 0.33 T 2 (p.u.) 0.85 The weighting matrices used to define the reference dynamics were Q = diag{11111}, R = [140] in all examples. A variety of methods are available for choosing these matrices. In practice, elements of Q and R are chosen so that the resulting control meets the control performance specifications and requirements. In this study, the problem is solved by considering the following factors: (i) minimization of the machine speed deviation is the most important criterion, and (ii) by

196 D. Arnautović and D. Džepčeski: increasing the weightings on control the system becomes slower, which is necessary because of both, guide vane and runner blade servomotor opening and closing speed limits. Solving the Riccati equation, we find the optimal closed-loop system matrix F, possessing the reference spectrum given in Table 2. The reference spectrum is the same for all analyzed cases. Table 2. System eigenvalues C o n t r o l l e r s Λ (F) static, r = 1 static, r = 2 Λ(Ac) Λ(Ac) 1.01 ± j11.20 0.76 ± j11.19 1.01 ± j11.20* -5.18-5.58 1-4.91-1.09-1.09-0.96-0.76-0.76-0.90 * retained eigenvalues Various types of turbine governors were analyzed, but only two are represented here: (1) static type of turbine governor with one available measurement (machine speed), and (2) static type of turbine governor with two available local measurements (machine speed and derivative of speed). In case (1) only r = 1 eigenvalue of the reference dynamics can be retained. There are only three candidate solutions because of one pair of conjugate complex eigenvalues. All three posible candidate solutions for the system studied were investigated. Retaining λ 4 = 5.178 produces the best solution. Eigenvalues of the closed-loop system matrix Ac for this solution are given in Table 2. The feedback gain for this case is K = [ 0.205]. In order to imrove the obtained solution, case two will be studied, when the static turbine governor with two available measurements is introduced, i.e. y T = [ωώ] and r = 2 eigenvalues of the optimal solution can be retained. There are four possible combinations for the suboptimal solution λ 12, (λ 3,λ 4 ), (λ 3,λ 5 ) and (λ 4,λ5). Examing all four, we find that the best solution is obtained when the pair λ 12 is retained. Eigenvalues of the closed-loop system matrix Ac for this solution are given in Table 2. In this case the system is also stable. The results show that a strong stabilizing effect can be obtained by introducing two measurements in the turbine governor. The solution is better than in case (1). The feedback gain for this case is K = [ 0.646 0.064]. The hydrounits in HPP D- erdap 2 are equipped with electronic achelerotachometric type of turbine governors with temporary speed droop. It is of interest to compare the proposed suboptimal static turbine governors with the existing conventional turbine governor. The responses of the characteristic varibale ω of the closed-loop system, due to an impulse disturbance are shown in Fig. 2. An impulse

Suboptimal Design of Turbine Governors for Low Head Hydroturbines 197 change of load torque, which is introduced into the electromechanical equation (2), is used as disturbance. As shown in Fig. 2 all machine speed responses return to the null steady state. From the obtained results, it can be seen that turbine governors designed by using the proposed method are suitable for enhancing the stabillity and yield a better performance compared to that obtained by using the conventional governor. Fig. 2. Machine speed responses: (1) conventional governor, (2) static governor, r = 1, (3) static governor, r = 2. Peak values and setting times for all studied cases are given in Table 3. Applied control strategies enabled the suboptimal solutions to reduce speed overshoot from 4.5% to 10% and to provide faster responses to load shedding from 41.87% to 53,12% as compared to the conventional electronic turbine governor. Table 3. Characteristic time response values. Type of governor Peak value Setting time (p.u.) (s) conventional 0.090 16. static, r = 1 0.086 9.3 static, r = 2 0.081 7.5

198 D. Arnautović and D. Džepčeski: 5 Conclusions The described methodology permits the design of optimal and suboptimal multivariable static and low-order dynamic turbine governors in hydropower plants with Kaplan or bulb turbines. Methodology is based on projective controls and provides a systematic design of the control loop with an increased number of available measurements and increased dimension of the controller. Digital technology offers, to the turbine governor designer, various reasources to implement and to prove modern control theory on considerably simple way. Comparison for the closed-loop system performances shows that better results are obtained with suboptimal turbine governors as compared to conventional turbine governors. References [1] T. Sontakke and T. Basu, Improvement in the transient response of a hydrosystem by compensation and stabilizing signal, Electric Power System Research, no. 4, pp. 213 223, 1981. [2] P. Kopacek and E. Zauner, Governing turbines by microcomputers, Water Power & Dam Construction, no. 9, pp. 26 30, 1982. [3] S. Zongshou, Z. Chengang, and F. Wenyuan, Waterhammer pressure compensation in turbine regulating systems, Water Power & Dam Construction, no. 9, pp. 25 28, 1986. [4] C. Sanathanan, Double loop control of hydro turbine unit, Water Power & Dam Construction, no. 12, pp. 25 27, 1986. [5] D. Findlay, H. Davie, T. Foord, A. Marshall, and D. Winning, Microprocessor-based adaptive water turbine governor, IEEE Proc. Pt.C, vol. 127, pp. 360 369, 1980. [6] P. Ham and N. Green, Developments and experience in digital turbine control, in IEEE/PES 1988 Winter Meeting, 88 WM, New York, USA, 1988, pp. 236 2. [7] D. Arnautović and D. Škatarić, Suboptimal design of hydroturbine governors, IEEE Trans. on Energy Conversion, vol. 6, no. 3, pp. 438 444, Sept. 1991. [8] P. Schniter and L. Wozniak, Efficiency based optimal control of kaplan hydrogenerators, IEEE Trans. on Energy Conversion, vol. 10, no. 2, pp. 348 353, June 1995. [9] M. Brezovec, I. Kuzle, and T. Tomisa, Nonlinear digital simulation model of hydroelectric power unit with kaplan turbine, IEEE Trans. On Energy Conversion, vol. 21, no. 1, pp. 235 241, Mar. 2006. [10] E. D. Jaeger, N. Janssens, B. Malfliet, and F. Van De Meulebroeke, Hydroturbine model for system dynamic studies, IEEE Trans. on Power Systems, vol. 9, no. 4, pp. 1709 1715, Nov. 1994. [11] IEEE Working Group, Hydraulic turbine and turbine control models for system dynamic studies, IEEE Trans. on Power Systems, vol. 7, no. 1, pp. 167 179, Feb. 1992. [12] N. Kishor, R. P. Saini, and S. P. Singh, A review on hydropower plant models and control, Renewable and Sustainable Energy Reviews, vol. 11, pp. 776 796, 2007.