Nonlinear Security Constrained Optimal Power Flow for Combined AC and HVDC Grids

power systems eehlaboratory Vasileios Saplamidis Nonlinear Security Constrained Optimal Power Flow for Combined AC and HVDC Grids Master Thesis PSL 1417 EEH Power Systems Laboratory ETH Zurich Supervisor: Roger Wiget Expert: Prof. Dr. Göran Andersson Zurich, November 19, 2014

Abstract The goal of this master thesis is the development of an algorithm for solving the non-linear Optimal Power Flow (OPF) and Security Constrained Optimal Power Flow (SCOPF) problems for combined AC and Muti-Terminal DC (MTDC) grids. Both grids are modeled using the non-linear power flow equations. Grid-level models for AC/DC converters are developed based on the Voltage Source Converter (VSC) design with non-linear power throughput constraints and current-dependent losses. A new model for DC/DC converters is proposed based on modern high power - low ratio converter designs. Such devices are used to improve the OPF solution for combined AC/MTDC grids by controlling the DC power flows. The formulation of the SCOPF problem is such that it works for preventive OPF where no corrective actions are allowed after the contingency, as well as for corrective OPF with adjustable actions limits. A contingency filtering approach to reduce the SCOPF problem size, is also developed and its impact on the convergence speed and accuracy is investigated. Several case studies are performed to test both the new SCOPF algorithm and the developed models. Both the OPF and SCOPF solutions are used for a benefit analysis of a proposed MTDC grid expansion. In another study, the impact of DC/DC converters as power flow regulators on MTDC grids is found to correlate strongly with the DC line lengths. A sensitivity analysis of the corrective actions limits on the generation cost is also performed. Generator and converter limits impact is investigated separately and the results are compared. Keywords: Contingency filtering, High Voltage DC (HVDC) Transmission, Multi-Terminal HVDC (MTDC), Optimal Power flow (OPF), Security Constrained OPF (SCOPF). iv

Acknowledgment I would like to thank the people that contributed to the fulfillment of this thesis and especially: Roger Wiget, PhD student at PSL, for our excellent collaboration and fruitful discussions during the whole period of this thesis. Prof. Dr. Göran Andersson for his support throughout my Master studies and for allowing my to work on this interesting topic. I would also like to thank all my friends in Switzerland and Greece for their support throughout the duration of my studies. Special thanks to Alina and Vassilis. Finally I would like to express my gratitude to my parents, George and Cleopatra and my sister Sissy. Without their total support and encouragement nothing of all this would be possible. Zürich, November 2014 Vasileios Saplamidis v

Contents 1 Introduction 1 1.1 Motivation and Literature Review............... 3 1.2 Goal................................ 4 1.3 Thesis Structure......................... 4 2 Models 5 2.1 Buses............................... 5 2.2 Lines................................ 6 2.3 Generators and Loads...................... 9 2.3.1 Generators........................ 10 2.3.2 Loads........................... 11 2.4 AC/DC Converters........................ 11 2.4.1 Active Power Exchange - Losses............ 14 2.4.2 Voltage Limits...................... 14 2.4.3 Power Throughput Constraints............. 15 2.5 DC/DC Converters........................ 16 3 Optimal Power Flow 19 3.1 Definitions............................. 19 3.1.1 Optimization Vector................... 19 3.1.2 Objective Function.................... 20 3.1.3 Equality Constraints................... 20 3.1.4 Inequality Constraints.................. 21 3.2 OPF Problem Formulation................... 23 3.3 SCOPF Problem Formulation.................. 24 3.3.1 Types of SCOPF Problems............... 27 4 Algorithm Improvements 29 4.1 Contingency Filtering...................... 30 5 Test Cases - Results 33 5.1 Test Cases Description...................... 33 5.1.1 9-bus AC with 5-bus MTDC.............. 33 5.1.2 IEEE 14-bus AC with 5-bus MTDC.......... 34 vi

CONTENTS vii 5.1.3 RTS-96 one area with 8-bus MTDC.......... 34 5.2 Results............................... 36 5.2.1 Improved SCOPF Algorithm.............. 36 5.2.2 Economic Benefit of MTDC Grid Expansion..... 37 5.2.3 DC-DC Converters as Power Flow Control Devices.. 40 5.2.4 Influence of the Corrective Actions Limits on the SCOPF Solution.......................... 42 5.2.5 Sensitivity Analysis of the w k Components..... 46 6 Conclusion 49 6.1 Conclusion............................ 49 6.2 Motivation for Future Research................. 50 A Test systems data 51 A.1 9-bus AC with 5-bus MTDC - System Data.......... 51 A.2 IEEE 14-bus AC with 5-bus MTDC - System Data...... 54 A.3 RTS-96 one area with 8-bus MTDC - System Data...... 57 Bibliography 61

List of Figures 2.1 Venn diagram of different bus types.............. 5 2.2 Π equivalent model of a power line............... 6 2.3 Venn diagram of different branch types............. 8 2.4 Power flows sign convention................... 10 2.5 PQ - capability curve [24].................... 11 2.6 HVDC converter station [5]................... 12 2.7 HVDC VSC station model.................... 12 2.8 VSC converter station simplified model............ 13 2.9 VSC converter losses concept.................. 14 2.10 Example of a VSC converter PQ capability curve....... 16 2.11 DC/DC converter model..................... 17 3.1 Equivalent circuit of the VSC station model AC side..... 23 3.2 SCOPF Algorithm including constraint building....... 26 4.1 Improved SCOPF algorithm - Contingency filtering method. 31 5.1 14-bus AC/MTDC system.................... 33 5.2 19-bus AC/MTDC system.................... 34 5.3 32-bus AC/MTDC system.................... 35 5.4 Severity Indexes on all contingencies in the 14-bus system.. 36 5.5 Economic benefit evaluation algorithm............. 38 5.6 Power flow on line 27-32..................... 41 5.7 Power losses on line 27-32.................... 41 5.8 Total generation cost....................... 42 5.9 Active generation dispatch.................... 43 5.10 Cost of generation........................ 44 5.11 AC bus voltages......................... 45 5.12 DC bus voltages......................... 45 5.13 Average voltage deviation.................... 45 5.14 SCOPF cost increase due to converters maximum deviation limits............................... 46 5.15 SCOPF cost increase due to generators maximum deviation limits............................... 47 viii

LIST OF FIGURES ix 5.16 Influence of both P g and P c on the % generation cost increase.............................. 48

List of Tables 2.1 Typical VSC station model parameters [21].......... 13 5.1 Comparison of the improved algorithm with the full pr-scopf 37 5.2 Generator dispatch comparison................. 37 5.3 Results of economic benefit analysis of MTDC grid expansion on the IEEE 14-bus test system................. 39 5.4 Test Cases description...................... 43 A.1 9-bus AC with 5-bus MTDC: Bus Data............ 51 A.2 9-bus AC with 5-bus MTDC: Generator Data......... 52 A.3 9-bus AC with 5-bus MTDC: SVC Data............ 52 A.4 9-bus AC with 5-bus MTDC: VSC Data............ 52 A.5 9-bus AC with 5-bus MTDC: VSC Data (cont.)........ 52 A.6 9-bus AC with 5-bus MTDC: Branch Data.......... 53 A.7 IEEE 14-bus AC with 5-bus MTDC: Bus Data........ 54 A.8 IEEE 14-bus AC with 5-bus MTDC: Generator Data..... 54 A.9 IEEE 14-bus AC with 5-bus MTDC: SVC Data........ 55 A.10 IEEE 14-bus AC with 5-bus MTDC: VSC Data........ 55 A.11 IEEE 14-bus AC with 5-bus MTDC: VSC Data (cont.).... 55 A.12 IEEE 14-bus AC with 5-bus MTDC: Branch Data...... 56 A.13 RTS-96 one area with 8-bus MTDC: Bus Data........ 57 A.14 RTS-96 one area with 8-bus MTDC: Generator Data..... 58 A.15 RTS-96 one area with 8-bus MTDC: SVC Data........ 58 A.16 RTS-96 one area with 8-bus MTDC: VSC Data........ 58 A.17 RTS-96 one area with 8-bus MTDC: VSC Data (cont.).... 59 A.18 RTS-96 one area with 8-bus MTDC: Branch Data...... 59 A.19 RTS-96 one area with 8-bus MTDC: Branch Data (cont.).. 60 x

List of Symbols Unless indicated otherwise, vectors and matrices are written with bold letters while scalar values with normal ones. Latin letters Symbol Units Description A Set including all AC buses C Set including all AC buses with AC/DC converters attached D Set including all DC buses E Set including all DC buses with AC/DC converters attached G Set including all buses with generators K A Set including all branches between AC buses K D Set including all branches between DC buses S Set including all buses with SVCs T Set including all branches with tap changing or phase shifting transformers b Ω 1 or p.u. Shunt susceptance of line c 0 $/hourmw 2 Cost coefficients for generators c 1 $/hourmw Cost coefficients for generators c 2 $/hour Cost coefficients for generators C $/hour Generation cost DF Discount factor g Ω 1 or p.u. Conductance of line i Interest rate I A or p.u. Current k v Voltage relationship factor across a VSC k Q Reactive power factor for a VSC K k Set of buses adjacent to k, including k L W or p.u. Losses N Number of... xii

LIST OF SYMBOLS xiii p $/MWh Electricity price P W or p.u. Active power flow Q VAr or p.u. Reactive power flow r Ω or p.u. Resistance of line S VA or p.u. Apparent power flow t Complex turns ratio of transformer TEB $ Total Economic Benefit U Volts or p.u. Voltage magnitude x Ω or p.u. Reactance of line y Ω 1 or p.u. Complex admittance of line z Ω or p.u. Complex impedance of line x Optimization vector 1 u Control vector 1 z State vector 1 g Equality constraints 1 h Inequality constraints 1 f Objective function w Subset of u containing active power generation and converter active power throughput 1 Greek letters Symbol Units Description α Magnitude of transformer ratio θ radians Voltage angle κ 0 MW Loss coefficients for a VSC κ 1 MW/A Loss coefficients for a VSC κ 2 MW/A 2 Loss coefficients for a VSC φ radians Angle of transformer ratio Ω k Set of buses adjacent to k, excluding k Subscripts Symbol Description b Bus(es) br Branch(es) c Converter AC side(s) cont Contingencies d Load(s) 1 For these variables, numerical subscripts refer to normal operation (0) or the k-th post contingency case (k 0).

LIST OF SYMBOLS xiv dc exp filter from g inj k kk km loss noexp nom phr ref s tf to Converter DC side(s) With MTDC grid expansion VSC station filter From bus Generator(s) Power injected on bus At 2 bus k Between bus k and ground Between 2 buses k and m, k m Losses Without MTDC grid expansion Nominal value VSC station phase reactor Reference value Static VAr Compensator(s) VSC station transformer To bus Superscripts Symbol AC DC sh y Description At the optimum (or complex conjugate depending on context) AC quantities DC quantities Shunt element Year Special symbols Symbol Description Maximum Minimum [ ] Diagonal matrix with the elements of in the diagonal Vector 2 For symbols that represent power flows, a single subscript with the bus number indicates power injection, while a double subscript indicates power flow between two buses. For symbols that represent angles, a single subscript indicates the voltage angle between this bus and the reference bus, while a double subscript is defined as the difference of the angle between two buses.

List of Acronyms AC c-scopf CF DC HVDC IEEE IGBT LCC MTDC OPF pr-scopf PCC PWM RTS SCOPF SI SVC VSC WSCC Alternating Current Corrective Security Constrained OPF Contingency Filtering Direct Current High Voltage DC Institute of Electrical and Electronics Engineers Insulated-Gate Bipolar Transistor Line Commutated Converter Multi-Terminal DC Optimal Power Flow Preventive Security Constrained OPF Point of Common Connection Pulse-Width Modulation Reliability Test System Security Constrained OPF Severity Index Static VAr Compensator Voltage Source Converter Western Systems Coordinating Council xvi

Chapter 1 Introduction The global energy demand is growing and there is a strong desire to move away from conventional energy sources towards environmentally friendly renewables. Due to uncertainties that arise from the fluctuating nature of the renewable production, one of the main limitations of renewable penetration is the capacity of the existing grids. Further, solar and wind energy sources are dependent on weather conditions and can usually not be built close to the load centers. The use of High Voltage DC (HVDC) technology for grid expansion and integration of renewable energy sources has long been suggested and is regarded as the optimal solution in many cases. Such cases include offshore wind generation further than 100km away from the nearest connection point and connection in weak Alternating Current (AC) grid nodes [1]. The advantages of DC transmission lines over the AC ones have been summarized in [2] and include lower losses due to the elimination of the skin effect and line-to-line inductance phenomena, lower charging currents and cheaper sub sea cable connections. In addition, HVDC interconnections are inherently controllable since the Voltage Source Converters (VSCs) offer a large degree of control in their power output characteristics. Voltage Source Converters are a relatively recent development in the field of power converters. Most sources agree that the VSCs will soon be the main converter type used in HVDC grids, because of their superior control characteristics over the now mainstream Line Commutated Converters (LCCs) [2 4]. The main advantages of a VSC over an LCC are its ability for simultaneous and independent AC voltage and power flow control as well as almost instantaneous power flow reversal without interruption and without changing the polarity of the interconnection. Different types of VSCs have been proposed in the literature that range from 2-level converters using custom Insulated-Gate Bipolar Transistors (IGBTs) [5] to modular multilevel 1

CHAPTER 1. INTRODUCTION 2 converters [6] as well as hybrid models [7]. Each of these different VSC technologies has advantages and disadvantages concerning harmonics injection, fault blocking capabilities and they are in different stages of development. In the scope of this thesis the actual VSC implementation is not important as long as the power and voltage controllability are provided. The Optimal Power Flow (OPF) problem is one of the main ways to analyze and study a power system. The problem is based upon an objective function which is to be minimized, by changing the values of certain control variables while respecting some relevant constraints. The minimized quantity usually has to do with the cost of generation but can also include optimal allocation of reactive power sources [8] or factors that affect the stability of the system [9]. The Security Constrained OPF (SCOPF) problem is an extension of the OPF problem which calculates the cost of steady state security. Steady state security is defined in [10] as the ability of the system to operate steadystate-wise within the specified limits of safety and supply quality following a contingency, in the time period after the fast-acting automatic control devices have restored the system balance but before the slow-acting controls have responded. The additional constraints introduced in this OPF scheme lead to a different optimum because the base-case generation dispatch is changed to satisfy all the contingency cases. Due to the lack of computational power, when these methods were first applied, only the most likely or severe outages were taken into account when formulating the additional constraints. In this thesis the SCOPF formulation of [11] is expanded and then applied with all possible contingencies including generator and AC/DC converter outages and line failures. Only single element failures are studied since most transmission systems are designed around the N-1 security concept [12]. The power flow/optimal power flow problems can be solved either in their full non-linear form or in a simplified (linearized) version. The former generally yields more accurate results and is able to calculate reactive power flows and line losses but is much more computationally intensive. The latter, also known as DC power flow, is generally much faster but due to the approximations taken in the linearization process, it is not able to calculate reactive power flows, line losses or voltage levels on the system buses. Within this thesis, all the models and methods discussed make use of only the non-linear power flow equations.

CHAPTER 1. INTRODUCTION 3 1.1 Motivation and Literature Review Several methods for solving power flow problems in combined AC and HVDC grids have been proposed in literature. Most power flow calculation methods for combined AC and HVDC grids are developed around a VSC station model. The modeling details vary among different approaches but generally include a transformer, some passive current filtering and the actual converter device which can be modeled to include the power electronics induced losses as well as voltage and power throughput limits. In [13] the converter model only includes the power throughput and voltage constraints but neglects the converter losses. A more complete model is presented in [14] which includes phase reactors and filters on the AC side as well as the transformer that connects to the AC grid. In [15] the converter losses are also taken into account by using the simple method proposed in [16] that relates the VSC converter losses to the phase reactor current. Such an approach is also made in this thesis, as the aforementioned losses model is simple enough to be used in a computationally intensive optimization process and relies on quantities that are already calculated on a grid level. The addition of multiple AC/DC converters in the system increases the problem complexity since the number of lines and buses of the system is increased. The main VSC station model can be reduced to a Π equivalent through suitable transformations in order to reduce the final system complexity [17]. The use of DC/DC converters in large DC grids has been proposed in [18]. They can be used to connect DC grids of different voltage levels, or to provide power flow control within a single Multi-Terminal HVDC (MTDC) grid. There are various DC/DC converter models proposed in literature based in control oriented approaches [19]. However, such a detailed converter model is not needed for modeling the transmission grid level influence of such devices. A simple non-linear model has been used in this thesis based on the VSC converter model of [20] modified according to the DC/DC converter characteristics proposed in [18]. An algorithm for solving the OPF problem in combined AC and HVDC systems, based on Newton s method has been developed in [13]. Feng et al. [21] have proposed a benefit evaluation method based on a combined OPF using a model similar to the one used in this thesis. Their algorithm uses the OPF solution to extract the power losses and generation cost under different scenarios. The monetary benefits are projected in the future for the lifetime of the investment to calculate the total economic benefit (or loss) of a planned MTDC system. Different MTDC topologies are also investigated. In another case study in [14], the OPF problem formulation divides buses into AC buses and DC buses which have generally different

CHAPTER 1. INTRODUCTION 4 equality constraints. AC buses are further divided into Points of Common Connection (PCC) and non-pcc. One of the converter nodes assumes the role of a slack converter. The results of this study show that the combined AC/DC system has generally better voltage profiles in the expense of a slightly increased generation cost. Finally Wiget et al. [20] have proposed a unified approach to solve the combined AC/DC OPF problem that takes converter losses into account and uses the non-linear power flow equations. The SCOPF approach on combined AC/MTDC grids is a novel concept that (to the best of our knowledge) hasn t been investigated until now. The SCOPF problem for conventional (AC) transmission grids has been extensively studied in [10]. According to [11] the main issues with the traditional SCOPF problem solution is the enormously increased size of the problem. The non-linear approach for solving the SCOPF problem in large grids while taking all possible contingencies into account demands unrealistically large computational power. In the same paper, alternative formulations and solution techniques are discussed to solve such problems, including Benders decomposition, linearization of post contingency constraints and network compression. 1.2 Goal In this thesis an algorithm will be developed for solving the OPF and SCOPF problem in mixed AC/DC grids. All the calculations assume a steady state both before and after the contingencies. The use of non-linear equations for modeling power flows enables losses to be calculated for both AC and DC grids. Devices to be modeled include tap changing transformers and phase shifters on the AC grid, AC/DC converters based on VSC technology and DC/DC converters that are controlled as equivalent DC transformers in the DC grid. Losses of all these converters will also be taken into account. The SCOPF problem will be solved by using an interior-point solver in Matlab. 1.3 Thesis Structure The rest of this thesis is organized as follows. Chapter 2 describes the basic equations that model the power flows in AC/DC grids and also the nonlinear models used for the AC/DC and DC/DC converters. In Chapter 3 the optimization problem is defined and all relevant constraints are discussed. The OPF problem is also extended to include security constraints. A contingency filtering method is proposed in Chapter 4 to reduce the running time of the SCOPF calculations. Chapter 5 describes the test cases studied in this thesis and the the obtained results. Finally, the results are discussed in Chapter 6, and suggestions are made for future research goals.

Chapter 2 Models For the power flow problem to be formulated and solved, the transmission grid must be mathematically modeled. In this Chapter, the models used for the various grid elements are presented and analyzed. The resulting equations will be used to form the Optimal Power Flow problem. 2.1 Buses In a transmission grid, buses are the connection points for all other parts, like transmission lines, loads and generators. In a mixed AC - MTDC grid there are AC (set A) and DC buses (set D). The different bus sets and their relationships are shown in the Venn diagram of Fig. 2.1. The connection of the two grids takes place through AC/DC converters placed between some AC and DC nodes (sets C and E respectively). Generators are allowed on any bus (set G) but SVCs can be located only on AC buses (set S) since they need to inject reactive power in the grid. A D S Cac G dc E AC/DC converters Figure 2.1: Venn diagram of different bus types 5

CHAPTER 2. MODELS 6 2.2 Lines An AC transmission line shorter than 300km can be described with the simplified Π model (also known as lumped-circuit model ) with enough accuracy for most power flow calculations [22]. A homogeneous power line between nodes k and m has an equivalent Π model as depicted in Fig. 2.2. I k t km I k z r jx km km km I m j k Ue k Ue k t km j k sh y km sh y km U e j m m t km :1 Figure 2.2: Π equivalent model of a power line The series impedance z km = r km + jx km models the resistive behavior of the line materials as well as any inductance phenomena on the conductors. The shunt elements ykm sh model the total shunt admittance of the line and by convention are defined as half of the total line shunt admittance each. y sh km km = j bsh 2 (2.1) This general definition of a branch also allows modeling of branch devices like transformers and phase shifters. The ideal transformer (t km : 1) of Fig. 2.2 has a complex turns ratio. t km = α km e jφ km (2.2) In this model a tap changing transformer between k and m will have φ km = 0 and α km would be a control variable, while a phase shifter will have a controllable φ km. Obviously an α km = 1 and φ km = 0 would imply a power line without any kind of transformer. In real world applications the ratio and angles of such devices can only take discreet values. In this thesis all control variables including φ km and α km are allowed to take continuous values. In most real world tap changers of phase shifters their discrete operation points are so close together that the continuous variable approximation is valid. Moreover, the non-linear nature of the problem in conjunction with the large scale implied by the SCOPF formulation would result in unrealistic running times if a discreet variable scheme was assumed (integer programming).

CHAPTER 2. MODELS 7 When modeling power flows in a transmission grid, it is often useful to work with power-voltage relationships. The apparent power flow between two buses k and m can be calculated as in (2.3). S km = ykm U ke jθ k (U k e jθ k U m e jθm ) jb sh km U k 2 S km = (g km jb km )(Uk 2 U ku m e j(θ k θ m) ) jb sh km U k 2 (2.3) Separating real and imaginary parts yields: P km = g km U 2 k U ku m (g km cos(θ km ) + b km sin(θ km )) (2.4) Q km = (b km + b sh km )U 2 k + U ku m (b km cos(θ km ) g km sin(θ km )) (2.5) where θ km θ k θ m. For a DC line, the AC related quantities are zero, and this leads to the following equation for the active power flow: P DC km = 2g kmu k (U k U m ) (2.6) The factor 2 in (2.6) comes from the assumption that the cables on the DC side have bipolar structure [20]. Most calculations in the Matlab environment are inherently faster when matrix equations are used instead of sets of individual non-linear equations. By defining the admittance matrix Y bus, such matrix equations can be constructed for the power flows and nodal power injections. Assuming that the grid contains N b buses and N br branches, the current injections at all buses (I) can be calculated as: I = Y bus U (2.7) where I is an N b 1 vector containing the current injection at each bus, U is an N b 1 vector containing each bus (complex) voltage, and Y bus is a N b N b matrix whose elements are calculated through (2.8). Y km = t km t mky km Y kk = yk sh + m Ω k α 2 km (ysh km + y km ) (2.8) Where y km = g km + jb km = 1/z km, yk sh are shunt elements connected to bus k and Ω k is the set of buses adjacent to bus k, excluding k. Y bus is known as the nodal admittance matrix and is usually sparse and (if transformers with complex ratio are used) not symmetric.

CHAPTER 2. MODELS 8 The elements of this matrix consist of the branch connections between buses and any shunt elements. The connections between the AC and DC grids through the converters do not influence the admittance matrix structure but are taken into account through the equality constraints (see Section 3.1.3). Therefore, the nodal admittance matrix of a mixed AC/DC system has the form of (2.9): Y bus = 1 2 (2.9) Where: 1 = Connections between AC buses and 2 = Connections between DC buses. The indexes pairs of the non-zero elements of the two sub-matrices of the admittance matrix define the set K A K D of bus connections (branches) shown in Fig. 2.3. The subset T K A consists of the branches that have tap changing or phase shifting transformers. DC branches with DC/DC transformers are again modeled through the OPF problem s constraints are do not affect the Y bus. K A K D T Figure 2.3: Venn diagram of different branch types By comparing (2.4) and (2.6), and knowing that a DC line has only a resistive part and no reactive components, it is apparent that the same equation can be used to describe a power flow in both AC and DC grids. The factor 2 that is introduced by the assumed bipolar structure of the DC lines is therefore included in the 2 part of the Y bus matrix whose elements are defined as in (2.10)

CHAPTER 2. MODELS 9 Y km = 2g km Y kk = 2 m Ω k g km (2.10) It can be shown that the apparent power injection on each bus can be calculated as: S inj = [U] Y bus U (2.11) Where [U] is a N b N b diagonal matrix containing the elements of U (bus voltages). The non-linear nodal power injection equations implied through (2.11) are: P k = U k U m (g km cos θ km + b km sin θ km ) (2.12) m K k Q k = U k U m (g km sin θ km b km cos θ km ) (2.13) m K k where K k is the set of buses adjacent to bus k, including k itself. Apart from the nodal admission matrix, one can use the line admittances to construct additional N br N b system admittance matrices (Y from and Y to ) such that: I from = Y from U (2.14) I to = Y to U (2.15) Where I from and I to are the currents at the from and to ends of all branches. The definition of Y from and Y to has been done in [23]. The branch power flows can now be calculated as: S from = [U] Y from U (2.16) S to = [U] Y tou (2.17) This calculation method is very useful in a programming environment like Matlab due to its ability to make fast matrix calculations. 2.3 Generators and Loads The general convention followed throughout this thesis is that power flows to a bus are positive while flows from a bus are negative. In the example

CHAPTER 2. MODELS 10 in Fig. 2.4, bus k is connected to a generator, a load and a line to bus m. With the power flow directions that are depicted, P g and Q g are positive while P km, Q km, P d and Q d are considered negative. Pg+jQg k Pkm+jQkm to bus m Pd+jQd Figure 2.4: Power flows sign convention 2.3.1 Generators Due to losses in the generator s armature winding that generate heat, the stator current cannot be higher than a certain value thus limiting the apparent power generation within a circle that forms the rightmost boundary of the curve in Fig. 2.5. The reactive power generation is further limited by the field current heating limit while the reactive power consumption for an under-exited generator is also limited by heating phenomena in the stator laminations [24]. All these limits are also dependent on the operating voltage and can be different for each generator. For the power flow calculation within this thesis this detailed approach is not needed however. Thus all the limitations described so far have been approximated by constant-value limits on the active and reactive power generation. Each generator is characterized by 4 limiting constants: The maximum and minimum active generation (P g and P g ) and the maximum and minimum reactive generation (Q g and Q g respectively). Each generator is also characterized by a cost function C i (P gi ). The cost of generation is generally dependent on the active power output and for most types of generators increases as the output increases. This is usually referred to as increasing marginal cost. In reality, cost functions are not smooth but in the majority of power system studies they are approximated either by piecewise linear curves or quadratic functions. In this thesis the quadratic cost approach is adopted as shown in (2.18). C i (P gi ) = c 2 P 2 gi + c 1 P gi + c 0 (2.18)

CHAPTER 2. MODELS 11 Q Field current heating limit Underexcited Overexcited Stator current heating limit P End region heating limit Figure 2.5: PQ - capability curve [24] 2.3.2 Loads In the scope of this thesis consumption is considered inflexible. This implies that all loads have constant complex values of S d = P d + jq d. Since all of our calculations (both pre- and post-contingency) refer to steady states of the grid, there is no load fluctuation. 2.4 AC/DC Converters While HVDC technology has been in use for almost half a century, only relatively recently advances in AC/DC converter design have made MTDC grids a realistic goal. The oldest still used technology is this of the LCC. In this type of power electronic devices the power flow is controlled through the DC voltage and the current flow is controlled to be constant. After the invention of the IGBT, Pulse Width Modulation (PWM) techniques have been applied on the control of AC/DC converters leading to the development of the VSC. This type of converter offers straightforward AC side voltage and angle control, as well as independent current flow control [25]. This means that power flow reversal can be achieved in a short time without large changes in the DC voltage. Despite LCC being still the most used technology in DC interconnections worldwide, VSC converters are very likely to take over in the near future [2]. Thus only VSC converters

CHAPTER 2. MODELS 12 are modeled and used throughout this thesis as AC and DC grid connection points. HVDC converters are usually built within a station (Fig. 2.6) that contains the converter itself complete with phase reactors, switches and transformers for the AC grid interface as well as additional filters (if needed) and the DC cables connection interface [5]. Figure 2.6: HVDC converter station [5] A VSC station model that takes into account all of the above and has been widely used in literature [21, 26] is shown in Fig. 2.7. k f c dc ztf zphr Ic Idc Uk, θk Uf, θf Uc, θc zfilter Pc Pdc Udc Ploss Figure 2.7: HVDC VSC station model The bus of the AC grid where the AC/DC converter is connected is labeled as bus k. The transformer that connects the station with the main AC grid is modeled with its equivalent impedance (typically a reactance) z tf. This transformer can generally be a controllable tap changing device but, because the voltage on bus c is assumed controllable, it is assumed that z tf represents a constant ratio transformer. On the station side of the transformer there is

CHAPTER 2. MODELS 13 an additional bus f for the AC filter (z filter ) to be connected. Such filters are used to absorb high current harmonics. In modern VSC converters however, the current waveform is so close to sinusoidal that such filters are usually considered optional. Lastly, another impedance z phr is used to model the phase reactors of the converter itself. Typical p.u. values for these parameters were found in literature and are presented in Table 2.1. Bus c is the last AC bus before the converter and represents the converter terminals. Table 2.1: Typical VSC station model parameters [21] Parameter name Value (p.u.) x phr 0.02 r phr 0.0001 b filter = Im(1/z filter ) 2 x tf 0.01 r tf 0.0001 The control variables related to the AC/DC converter are the active and reactive power injections at bus c (P c and Q c respectively) as well as the AC voltage on the same bus (U c θ c ). A Y transform of the z tf, z filter, z phr impedances can further simplify the model and eliminate the need for extra voltage and angle variables for bus F. The transformed model can be seen in Fig. 2.8 and the new impedances can be calculated by (2.19). z 1 = z tf z phr + z filter z phr + z tf z filter z phr z 2 = z tf z phr + z filter z phr + z tf z filter z filter z 3 = z tf z phr + z filter z phr + z tf z filter z tf (2.19) k c Idc dc z2 Ic Uk, θk z1 z3 Uc, θc Pc Pdc Udc Ploss Figure 2.8: VSC converter station simplified model

CHAPTER 2. MODELS 14 C DC Pc DC AC Pdc Ploss Figure 2.9: VSC converter losses concept 2.4.1 Active Power Exchange - Losses From a transmission grid perspective, an ideal VSC converter transfers the active power it receives on the AC side (P c ) to its DC side (P dc ). A nonideal converter has losses that will make the transmitted power less than the one that is entering the converter. From the sign convention introduced in Section 2.3 follows that the power balance of a lossy converter (Fig. 2.9) is calculated by (2.20). P c + P dc P loss = 0 (2.20) The losses of a VSC converter have often been modeled with a quadratic equation of the phase reactor current (I c ) [16]. This model introduces 3 losses components (κ 0, κ 1 and κ 2 ) that correspond to no load losses, losses depending on I c and losses depending on I 2 c respectively. where P loss = κ 0 + κ 1 I c + κ 2 I 2 c (2.21) I c = P 2 c + Q 2 c 3Uc (2.22) and the loss coefficients κ 0 = 11.0331 10 3, κ 1 = 3.464 10 3 and κ 2 = 5.5335 10 3 (all p.u.) are taken from [15]. 2.4.2 Voltage Limits As with every bus on a transmission grid, limits on the voltage magnitude exist to ensure system stability and over-voltage protection of the equipment (insulation limits). This is also true for the buses adjacent to an AC/DC converter with the addition of one more constraint. In converters that use PWM modulation methods to produce the AC voltage, it is important to

CHAPTER 2. MODELS 15 avoid over-modulation because it creates unwanted harmonics in the AC grid. Thus, there is a maximum AC voltage obtainable from a given DC voltage on the other side of the converter. According to [27], a three-phase VSC has a maximum AC voltage given by (2.23). 3 U c = 2 2 U dc (2.23) Where U c is the maximum line-to-line RMS value on the AC side. For a VSC in bipolar operation, the factor 2 on the denominator will cancel out with the double DC voltage and thus: 3 U c = U dc 1.225U dc (2.24) 2 Equations (2.23) and (2.24) imply that both voltages are measured in volts. Since in a power flow study the per unit system is used, the above equations must change to reflect that. The nominal voltage on both AC and DC grids is set at 1 p.u. and the maximum allowed voltage on any bus is 1.1 p.u. According to [21] if no over-modulation is required, the ratio of the maximum AC side voltage to the DC voltage can be set to 1.1 thus: 2.4.3 Power Throughput Constraints U c = k v U dc = 1.1U dc (2.25) The main limiting factor on the maximum power through an AC/DC converter is the maximum allowed current through the VSC valves (I c ). The product of this current with the DC voltage yields the maximum apparent power through the VSC. S c U c I c P 2 c + Q 2 c ( U c I c ) 2 (2.26) Values for I c can be found in literature [5] and can be used to scale the VSC in various case studies. A VSC converter is able to generate or consume reactive power in a controllable fashion since the θ c angle can be independently controlled. The phase reactor z phr is the main constraint on the reactive power flow. By applying (2.5) on the F -C branch, the reactive power flow is: Q F C = b phr U 2 c + U c U f (b phr cos(θ cf ) g phr sin(θ cf )) (2.27) Given that z phr is a reactive element we can assume 1/z phr = y phr = g phr + jb phr jb phr. In this case, the maximum reactive power flow on

CHAPTER 2. MODELS 16 the c-f branch of the full VSC model in Fig. 2.7 would be: Q c = b phr U c 2 + Uc U f b phr cos(θ c θ f ) (2.28) The minimum reactive power of the VSC is represented in this study as a percent of its nominal power (k Q ) as proposed in [21]. Both the nominal power of various AC/DC converters and the value of k Q can be obtained from [5]. Q c = k Q S nom (2.29) By applying all these power constraints to a sample VSC converter the PQ capability curve shown in Fig 2.10 is produced. A value of z phr = 0.0001 + j0.3 was used in this example. 1 0.8 0.6 Reactive Power (p.u.) 0.4 0.2 0 0.2 0.4 0.6 0.8 1 Maximum apparent power Minimum reactive power Maximum reactive power 1 0.5 0 0.5 1 Active Power (p.u.) Figure 2.10: Example of a VSC converter PQ capability curve It appears that for small angle differences the change in the maximum reactive power limit is not very significant. However, because of the strong dependency of Q c on the value of z phr, the non-linear expression of the maximum reactive power limit will be used throughout this thesis. 2.5 DC/DC Converters These types of converters are used to transform one DC voltage level to another similarly to an AC transformer. They can be used to connect two

CHAPTER 2. MODELS 17 DC systems of different operating voltages, or provide power flow control within a MT-HVDC grid. From (2.6) it follows that the (active) power transmission between two DC nodes depends on their voltage difference. While AC/DC converters are capable of controlling their DC voltage, in a true multi-terminal DC grid, there may be DC buses that aren t connected to converters. In a complex enough system this will lead to lower voltages at these buses, depending on the actual system topology and power flows. In order to influence the power flows in such a system, DC/DC converters could be used as branch devices to change the voltage difference between buses, thus effectively changing the power flow in a way a phase shifting transformer can be used to change the power flow on an AC grid. According to [18], the most promising technologies for such high-power, low-ratio devices are the Alternate Arm Converter or the Modular Multi- Level Converter. Most of these devices convert DC-AC-DC and, in the scope of this thesis, can be modeled as two AC/DC converters connected back to back as seen in Fig. 2.11. Most of the additional equipment used in a AC/DC station is not needed in this model since the AC transformation takes place internally in the converter. DC k AC k AC m DC m U dc,k P dc,k P c,k P dc,m P c,m U dc,m P loss,k P loss,m Figure 2.11: DC/DC converter model Sometimes an isolation transformer is put between the two AC/DC components that provides galvanic separation. This can be modeled with the equivalent impedance shown in Fig. 2.11. The mathematical modeling of such a device consists of the same equations as the AC/DC converter although some elements like the transformer impedance (z tf ) and filter (z filter ) are not included. The voltage transformation itself doesn t need to be explicitly formulated since the voltage relationships described in Section 2.4.2 can account for low ratio transformation.

Chapter 3 Optimal Power Flow The Optimal Power Flow problem deals with minimizing an objective function while satisfying the power flow equations and any relevant limit that is imposed by the problem. In this chapter, a generic formulation of the OPF problem will be given and then modified to suit the problems solved within this thesis. 3.1 Definitions 3.1.1 Optimization Vector The optimization vector x shown in (3.1) contains all the variables that can be used to describe the state of the system (state variables - z) and variables that will be used for control (control variables - u). Vector z contains the voltages and angles of every bus of the system, while u contains the active and reactive power generations from generators and Static Var Compensators - SVCs (P g, Q g and Q s ), active and reactive power injections from the AC/DC converters (P c and Q c ) and ratios and phase angles for tap-changing or phase-shifting transformers (α and φ respectively). Vector w u that contains only P g and P c is also defined here and it will be used in the SCOPF problem formulation. x = [ z u ] = U AC θ AC U DC P g Q g Q s P c Q c α φ, w = [ Pg P c ] (3.1) 19

CHAPTER 3. OPTIMAL POWER FLOW 20 3.1.2 Objective Function The objective function f(x) is a function of some of the variables in x. The solver tries to find the optimum by finding a combination of values in x that will minimize this function. A selection of examples of objective functions is given in the following equations. N g f(x) = c 2 Pgi 2 + c 1 P gi + c 0 (3.2) i=1 N g f(x) = P gi (3.3) i=1 N g f(x) = P gi + π i U ref U i (3.4) i=1 The objective (3.2) is the sum of the production costs as defined in (2.18) for generators and is used for the economic dispatch problem where the minimization of the total production costs is needed. The next equation is simply the sum of all active generation. By minimizing (3.3) we get the solution that will result in the minimum possible amount of generation, thus minimizing system losses regardless of the generation cost. Combinations of optimization goals are also possible as seen in (3.4) where the voltage deviations from some reference values are minimized in parallel to the active generation. This generally doesn t lead to loss minimization but depending on the values of the penalty factors π i a desired voltage profile can be achieved. The π i factors impose a penalty on these deviations by generating a cost that is included in the objective. According to the magnitude of the penalty factors π i, the maximum allowed deviations from the reference voltage can be controlled. 3.1.3 Equality Constraints Power Balance Equations Both AC and DC nodes have to fulfill the nodal balance equations (i.e. the sum of power inflow and outflow must be zero). Generators, loads, shunt elements and AC/DC converters all generate power inflow (resp. outflow) to (resp. from) the buses they are connected to. The total apparent power mismatch can be calculated by (3.5). The signs in all following equations follow the convention of chapter 2.3. N b i=1 S mismatch = (S g S d ) + (S c S dc S loss ) S inj (3.5) where: S g = P g + j(q g + Q s ) is the active and reactive generation (from generators or SVCs) and S d = P d + jq d is the active and reactive demand

CHAPTER 3. OPTIMAL POWER FLOW 21 by loads. Equation (2.20) describes the relationship between the AC and DC side power flows through the converter and the VSC losses are calculated through (2.21). Finally, S inj is calculated through (2.11). All the variables in (3.5) are in vector form. Reference Angle One AC node is used as a reference bus and its angle is set to some reference value, typically zero 3.1.4 Inequality Constraints θ slack = θ ref = 0 (3.6) The inequality constraints impose limits on the variables of the optimization vector and other quantities that are calculated through them. In this chapter, the various types of inequalities that limit the variables of the problem will be analyzed. Limits on State Vector Elements All bus voltages in x are up and down limited both for equipment protection against over-voltage and system protection against voltage instability. U AC U AC U AC (3.7) U DC U DC U DC (3.8) The limits are taken ±10% of the nominal value i.e. 0.9 and 1.1 p.u. respectively as defined in [28], but can generally be different for each bus. Bus voltage angles can also be limited but for this study they are left to take values between π and π, thus: π θ AC π (3.9) Generator output is also limited as discussed in Section 2.3.1 thus producing the following inequalities: P g P g P g (3.10) Q g Q g Q g (3.11) Q s Q s Q s (3.12) Where the maximum and minimum active generation and reactive generation are uniquely defined for each generator or SVC.

CHAPTER 3. OPTIMAL POWER FLOW 22 Tap changing and phase shifting transformers also offer a limited amount of control of their complex ratio t = αe jφ. The limits of their control variables are taken into account with (3.13) and (3.14). α α α (3.13) φ φ φ (3.14) The elements of the state vector that correspond to the power through AC/DC converters (P c and Q c ) are not bounded by fixed constraints with the exception of the lower boundary for reactive power due to (2.29). Limits on Calculated Quantities k Q S nom = Q c Q c (3.15) The first limiting factor of power flow in a transmission grid is the line capacity. Each power line km is assigned a limit S km in MVA that is the highest amount of apparent power allowed. The limit results from the limited thermal stress tolerance of the line. The power flows on each line are calculated through (2.16) and (2.17) to produce the constraints shown in (3.16) and (3.17). S f S (3.16) S t S (3.17) As discussed in chapter 2.4.3, an AC/DC converter must also obey the power throughput constraint (2.26). All the quantities in this formula are also in the state vector (3.1) since C A, and I c is a converter constant that is considered known. The maximum reactive power generation is taken into account with (2.28). This formula results from the power flow equations applied between the converter terminal - filter bus. However, as mentioned in chapter 2.4 the filter bus may be absent depending on the converter technology and the VSC station configuration. Moreover, due to the VSC model simplification presented in Fig. 2.8, the complex voltage on bus F is not a part of the system variables so it must be calculated from the adjacent AC voltages. Analysis of the equivalent circuit in Fig. 3.1 yields: Ũ f = and if the filter z filter is missing: z filterz phr Ũ k + z filter z tf Ũ c z tf z phr + z tf z filter + z filter z phr (3.18) Ũ f = Ũk (Ũk Ũc)z tf z tf + z phr (3.19)

CHAPTER 3. OPTIMAL POWER FLOW 23 If any of the other impedances are missing from the model, the equivalent filter voltage is identical to either U c or U k. Ik ztf Uf zphr Ic Uk zfilter If Uc Figure 3.1: Equivalent circuit of the VSC station model AC side The maximum AC side VSC voltage (U c ) is the minimum between the respective U k of (3.7) and the one calculated by (2.25). 3.2 OPF Problem Formulation Combining all the previous equalities and inequalities, the non-linear optimization problem for economic dispatch can be formulated as following. minimize x subject to N g c 2 Pgi 2 + c 1 P gi + c 0 i=1 Pci 2 P gi P di + P ci P dci κ 0 + κ + Q2 ci 1 3Uci U i U j (g ij cos θ ij + b ij sin θ ij ) = 0, j K i i A D (3.20a) + κ 2 P 2 ci + Q2 ci 3U 2 ci (3.20b) Q gi + Q si Q di + Q ci U i j K i U j (g ij sin θ ij b ij cos θ ij ) = 0, i A θ slack = θ ref ) ykm U ke jθ k (U k e jθ k U m e jθm jb sh (k, m) K A K D km U 2 k S km, (3.20c) (3.20d) (3.20e)

CHAPTER 3. OPTIMAL POWER FLOW 24 U i k v U j, (i, j), i C, j E (3.20f) Pci 2 + Q2 ci U cii ci, i C (3.20g) k Q S nom i Q ci b phr U i 2 + Ui U fi b phr cos(θ c θ f ), i C (3.20h) U i U i min{u i, k v U dc i }, i A D P gi P gi P gi, i G Q gi Q gi Q gi, i G Q si Q si Q si, i S α i α i α i, i T φ i φ i φ i, i T π θ i π, i A (3.20i) (3.20j) (3.20k) (3.20l) (3.20m) (3.20n) (3.20o) 3.3 SCOPF Problem Formulation The concept of SCOPF is an extension of the OPF problem to include additional constraints for possible contingencies [29]. The SCOPF solution provides the optimal operating point in the base case so that if contingencies occur they would not create security violations. Thus all definitions in Chapter 3.1 are valid, but some additional constraints need to be included. In a generic form, the OPF problem is defined as: minimize x 0 f(z 0, u 0 ) (3.21a) subject to g 0 (z 0, u 0 ) = 0 (3.21b) h 0 (z 0, u 0 ) 0 x 0 x 0 x 0 (3.21c) (3.21d) Where g 0 = 0 summarizes the equality constraints (3.20b)-(3.20d), h 0 0 summarizes the inequality constraints (3.20e)-(3.20h) and the upper and lower boundaries on x 0 summarize equations (3.20i)-(3.20o). The subscript 0 refers to the normal operation of the system i.e. all lines, generators and converters are functioning normally. A contingency is defined as a system configuration that differs from the normal one because one system element has malfunctioned and is not operating. Such elements can include generators or SVCs, AC/DC converters,