Security Constrained Optimal Power Flow in a Mixed AC-DC Grid

Transcription

1 power systems eehlaboratory Stephen Raptis Security Constrained Optimal Power Flow in a Mixed AC-DC Grid Master Thesis PSL 1305 EEH Power Systems Laboratory Swiss Federal Institute of Technology (ETH) Zurich Expert: Prof. Dr. Göran Andersson Supervisors: Emil Iggland, Roger Wiget Zurich, August 21, 2013

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3 Abstract In recent years high voltage direct current transmission systems have become popular due to the economical and technical advantages they feature for some applications. A proposal that seems very promising to be able to meet future demands of the world s power systems is that of a mixed AC-DC grid that is more heavily meshed. This mixed grid poses new challenges and aspects that have to be thoroughly addressed before it can be realized. Among the most critical issues in electric power transmission is the security and reliability of the system. A good practice for power system operation is toensurethat thesystemis N-1secure. Theefficientsolution ofthesecurityconstrained optimal power flow in AC grids is a problem with sufficient background research. On the other hand the solution of the SC-OPF in a mixed grid is a problem with inadequate insight into it. This thesis presents a formulation and solution of the SC-OPF in a linear fashion, and is an extension of the linear optimal power flow problem described in [1]. The method is applied to a test case and the results obtained are displayed and analysed. iii

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5 Acknowledgments I thank my supervisors Emil and Roger for the time and effort they put in the project, and for the valuable input I had from them the past 7 months. Zürich, August 21, 2013 Stephen Raptis. v

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7 Contents List of Figures List of Tables List of Acronyms List of Symbols x xi xiii xv 1 Introduction Motivation Structure of Report OPF and Security Constrained OPF Basic OPF Concept Security Analysis and Principles Remedial Actions Preventive Measures and Control Corrective Measures and Control SC-OPF Formulation Preventive Approach Preventive-Corrective Approach Power System Security States Trade-off between Generation Costs and Security System Modelling Linear Optimal Power Flow Calculation of Active Power Flows in the AC Grid Linear Optimal Power Flow in the Mixed Grid Calculation of Active Power Flows in the DC Grid Power Balance in the Mixed Grid Matrix Formulation of Optimization Problem Optimization Vector ξ Objective Function Equality Constraints vii

8 viii CONTENTS Inequality Constraints Security Constrained OPF in the Mixed Grid Method I: Preventive Line Outage Distribution Factors in the AC Grid Line Outage Distribution Factors in the DC Grid Generalized Generation Distribution Factors Line Constraints Formulation Method II: Preventive-Corrective Objective Function Post-Contingency Terminal Control Line Outages in the AC Grid Line Outages in the DC Grid Terminal Station Outages Generator Outages Formulation of the Complete Problem Results-Case Studies System Description - Case Study I Method I: Preventive Generation Profiles Power Flow Distribution in the Mixed Grid Method II: Preventive-Corrective Costs Comparison Interaction between AC and DC Grid Terminal Control Sensitivity Analysis DC Grid Capacity and System Performance Solving Times Case Study II Conclusion and Discussion Conclusion Future Work Appendices 69 A Numerical Example 69 B Example with Matrix Formulations 75 Bibliography 79

9 List of Figures 2.1 Security-State Diagram Line Before Outage Line After Outage Simulation of Line Outage Through Bus Injections Combined AC and DC grid [1] Generation Profile for Preventive SC-OPF Distribution of Line Flows in the AC Grid Distribution of Line Flows in the DC Grid (Preventive) Terminal Station Power Transfers (Preventive) Generation Profile for Prev-Corr SC-OPF for 100% Load Case 1: AC-DC line outages Case 2: AC-DC line and generator outages Case 3: AC-DC line and terminal outages Case 4: AC-DC line, terminal and generator outages Generation Profile for Prev-Corr SC-OPF for 160% Load Case 1: AC-DC line outages Case 2: AC-DC line and generator outages Case 3: AC-DC line and terminal outages Case 4: AC-DC line, terminal and generator outages Operational Costs for 100% Load Case 1: AC-DC line outages Case 2: AC-DC line and generator outages Case 3: AC-DC line and terminal outages Case 4: AC-DC line, terminal and generator outages Operational Costs for 160% Load Case 1: AC-DC line outages Case 2: AC-DC line and generator outages Case 3: AC-DC line and terminal outages Case 4: AC-DC line, terminal and generator outages Linking of Preventive and Corrective SC-OPF Line Flows in the AC Grid for 100% Load Case 1: AC-DC line outages Case 2: AC-DC line and generator outages Case 3: AC-DC line and terminal outages Case 4: AC-DC line, terminal and generator outages ix

10 x LIST OF FIGURES 5.12 Line Flows in the DC Grid for 100% Load Case 1: AC-DC line outages Case 2: AC-DC line and generator outages Case 3: AC-DC line and terminal outages Case 4: AC-DC line, terminal and generator outages Terminal Station Power Transfers for 100% Load Case 1: AC- DC line outages Case 2: AC-DC line and generator outages Case 3: AC-DC line and terminal outages Case 4: AC-DC line, terminal and generator outages Line Flows in the AC Grid for 160% Load Case 1: AC-DC line outages Case 2: AC-DC line and generator outages Case 3: AC-DC line and terminal outages Case 4: AC-DC line, terminal and generator outages Line Flows in the DC Grid for 160% Load Case 1: AC-DC line outages Case 2: AC-DC line and generator outages Case 3: AC-DC line and terminal outages Case 4: AC-DC line, terminal and generator outages Terminal Station Power Transfers for 160% Load Case 1: AC- DC line outages Case 2: AC-DC line and generator outages Case 3: AC-DC line and terminal outages Case 4: AC-DC line, terminal and generator outages Effects of Terminal Control on Costs: Case Effects of Terminal Control on Costs: Case Effects of Terminal Control on Costs: Case Effects of Terminal Control for Π AC = Π DC : Case Comparison of Absolute Costs for 2 levels of Π AC,Π DC : Case Effects of Terminal Control for Π AC = Π DC : Case Dependency of Operational Costs from Capacities for Base OPF Dependency of Operational Costs from Capacities for Case Projection of figures 5.23 and 5.24 for comparison of dependency on terminal and DC line capacities Calculation times for all cases IEEE RTS-96 with interconnected DC grid Effects of Terminal Control for Case A.1 Small Test Grid B.1 3 Bus Test Grid

11 List of Tables 3.1 Nomenclature Nomenclature for Preventive-Corrective Method Cost of Security for the Cases studied for 100% Load Cost of Security for the Cases studied for 160% Load Cost of Security and Solving Times for Test Case of Figure A.1 Generator Data A.2 AC Line Data A.3 DC Line Data A.4 Terminal Station Data A.5 Active Power Demand A.6 Decision Variable ξ for Base OPF and Case A.7 Line Flows for Base OPF and Case 1 SC-OPF A.8 Decision Variable ξ for Case 3 and Case A.9 Line Flows for Case 3 and Case 4 SC-OPF B.1 Nomenclature for Numerical Example xi

12 xii LIST OF TABLES

13 List of Acronyms OPF optimal power flow. DCOPF linearized optimal power flow. SC-OPF security constrained optimal power flow. LODF line outage distribution factor. GGDF generalized generation distribution factor. HVAC high voltage alternating current. HVDC high voltage direct current. VSC voltage source converter. xiii

14 xiv LIST OF ACRONYMS

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16 xvi LIST OF SYMBOLS List of Symbols A AC A DC A ACprev A DCprev B AC B DC line adjacency matrix of AC grid. line adjacency matrix of DC grid. matrix of AC line outage sensitivities for preventive method. matrix of DC line outage sensitivities for preventive method. AC network admittance matrix. DC network admittance matrix. C length of vector ξ. D D E AC E I F max km G H I K L M N N P G P max G P min G P L P T P max T,i number of DC lines. pairs of DC buses connected with a line. AC line adjacency matrix for generator outages of preventive method. matrix of generator outage sensitivities for preventive method. limit on power flowing on line km. matrix of quadratic cost entries. generator allocation matrix. number of generators. number of AC nodes. number of terminals. number of DC nodes. number of AC lines. pairs of AC buses connected with a line. generator active power output. generator maximum power output. generator minimum power output. active power demand. terminal station power transfer. limit on terminal power transfer.

17 xvii P base P km QC base power for per unit representation. power flowing on line km. generator quadratic costs diagonal matrix. R km resistance of DC line connecting bus k to bus m. S T U V W AC W DC X Y Z c f g h terminal allocation matrix for DC buses. terminal allocation matrix for AC buses. AC bus voltage magnitude. DC bus voltage. DC voltage difference penalty matrix. AC voltage angle difference penalty matrix. reactance matrix of AC network. number of inequality constraints. impedance matrix of DC network. set of single-element contingencies that are accounted for. objective function. set of equality constraints set of inequality constraints. n ij number of lines connecting bus i to j. r i,m percentage of power of generator m, taken on by generator i. u u max c vector of control variables. amount of corrective control actions that can be implemented. x km reactance of AC line connecting bus k to bus m. z vector of state dependent variables. Λ I identity matrix of size I. Λ L identity matrix of size L. Π AC Π DC Π Ter Φ iq Φ eq Ψ α β penalty for AC angle differences. penalty for DC voltage differences. penalty for changes in terminal power transfer. matrix of post-contingency generator bounds. matrix of post-contingency generator outputs. distribution vector for generator outages. linear coefficient of generator cost. quadratic coefficient of generator cost.

18 xviii LIST OF SYMBOLS γ total amount of contingencies accounted for. δ AC bus voltage angle. λ vector of linear cost coefficients. ξ decision variable.

19 Chapter 1 Introduction 1.1 Motivation Power systems nowadays operate in more strained conditions than were expected during their planning stage [2]. The development of distributed generation from renewable sources together with the introduction of electricity markets have resulted in uncertainty regarding operating conditions. Steady increase in load has to be met with appropriate upgrades of the generation and transmission systems. A scheme that looks promising in dealing with these challenges is the high voltage direct current (HVDC) transmission system. HVDC links are ideal for transferring large amounts of power over long distances and for connecting renewable power infeeds to the grid. Recent advances in the voltage source converter VSC-HVDC systems also offer potential high levels of power controllability. Therefore HVDC technology will play a significant role both in future planning and operation of power systems. As the number of these point to point HVDC connections increases it becomes more sensible to connect them directly and not through the vaster AC grid. This kind of scheme could introduce additional flexibility to power systems and facilitate power exchanges and trading between systems. These concepts give rise to the DC grid. Though there are no technology gaps in small HVDC projects, large inter-regional grids still lack adequate research regarding, among others, power flow control and system security. The secure and efficient operation of such a grid poses many difficulties. The system operator must ensure that the system is operating under the satisfactory limits. A solution to this problem is known as the Security- Constrained Optimal Power Flow (SC-OPF) and is a problem that is considered to have substantial research behind it for AC grids. On the other hand, the solution of the SC-OPF in the DC grid is a problem that lacks sufficient research. Thegoal of this thesis is to provideameans by whichthe security-constrained optimal power flow can be applied to a mixed AC-DC 1

20 2 CHAPTER 1. INTRODUCTION network. 1.2 Structure of Report In chapter 2 the formulation and basic concept of the optimal power flow is outlined. Fundamental theory regarding security analysis is then presented followed by extended formulations of the OPF problem to encompass security constraints. In Chapter 3 the modelling of the AC and DC grid is presented. The formulas for power flow calculations are derived preceded by an analytical description of the linear OPF model for the mixed grid [1]. In chapter 4 two methods for the implementation of the SC-OPF in a mixed AC-DC are developed and presented. Chapter 5 contains all results from simulations performed on a mixed grid test case. Finally chapter 6 is reserved for conclusions and evaluations. A discussion is made about future improvements and possible implementations that can be augmented to the existing models.

21 Chapter 2 OPF and Security Constrained OPF In this chapter the basic concepts of the optimal power flow and securityconstrained optimal power flow are outlined. 2.1 Basic OPF Concept Power system optimization is a field that has progressed a lot in the past century. Early in the 20 th century optimal power flow (OPF) was a problem engineers had to deal with using their experience and judgement. With gradual advances in mathematical optimization and computational tools, OPF is now a problem that is solved several times a day in control stations. Even though the problem has been addressed for over 50 years, the complexities that arise in power system operation and control always create room for further improvement and new implementations. OPF refers to the class of problems that find the optimal solution to an objective function subject to the power flow constraints and other operational constraints. It is generally a problem of best generation dispatch. The formulation of the OPF in compact notation [3] is the following: Minimize f(z, u) (2.1) subject to g(z, u) = 0 (2.2) h(z,u) 0 (2.3) The objective of function f(z,u) of equation (2.1) is usually the minimization of generation costs or of system active power losses, though different objectives can be specified depending on the application. Vectors u and z consist of the control and state dependent variables respectively. Control variables u are all quantities that can be modified to 3

22 4 CHAPTER 2. OPF AND SECURITY CONSTRAINED OPF satisfy the power balance under consideration of the system limits. The set of state dependent variables z contains all the variables that depend on the state the system is in. Equation (2.2) represents the set of equality constraints. Equality constraints express the power flow equations as well as the power balance between generation-losses-load and have to be unconditionally satisfied. The inequality constraints of equation (2.3) depict network operating limits and limits on control variables. Such limitations are for instance the thermal capacities of transmission devices or the maximum active power output of a generator. 2.2 Security Analysis and Principles Apart from operating in an economically optimal way, it is crucial for a power system to operate with respect to certain criteria that ensure security and reliability. The first concept mainly refers to the operation under the satisfactory limits to avoid damage of equipment. A reliable system operation is the situation where all loads are supplied without disruptions. For these criteria to be met the system operator has to perform security analysis in order to know how robust the system is with respect to various possible contingencies. The fundamental goal of security analysis is to ensure a steady supply of power to all the loads of a system without disruptions. In other words it is performed to guarantee that an unexpected outage of an element of the system will not lead to an uncontrollable cascading failure. The formulation of the OPF problem presented in section 2.1 does not take into account contingency events. According to ENTSO-E [4] a contingency is defined as the trip of one single of several network elements that cannot be predicted in advance. A scheduled outage is not a contingency. The optimization problem that takes into consideration contingency scenarios is called the Security Constrained-Optimal Power Flow (SC-OPF). The SC- OPF is an augmentation of the OPF through the inclusion of additional constraints that guarantee system security in the event of a contingency Remedial Actions Remedial actions are the means applied by a transmission system operator to ensure security of the power transmission grid. They refer to changes in the settings of controllable quantities of the system that achieve this goal. For the scope of the project the following types of remedial actions are going to be of concern: Preventive remedial actions Corrective remedial actions

23 2.3. SC-OPF FORMULATION Preventive Measures and Control Preventive control is a measure taken to provide for a need that might occur in the case a contingency happens, due to uncertainty of being able to handle the contingency constraints once they have occurred [4]. In other words it is an action that is made while still in normal operation. The aim is to place the power system in a state, that will not suffer violations of operating limits following the occurrence of a contingency. The preventive approach is a conservative one, since the system is positioned in a state that has higher operational costs, to guarantee it is secure against the effects of a contingency that might or might not occur. It is not concerned with possible actions that could be implemented in real-time operation after the contingency has occurred Corrective Measures and Control Corrective control describes actions needed to clear violations of physical limits after a contingency has occurred. Corrective control measures do not take into account pre-contingency conditions and settings which means that post-contingency controls must move in order to satisfy post-contingency constraints. In the corrective-secure state the system might be suffering from constraint violations, thus an immediate concern of the corrective approach is the time a corrective action requires. For the corrective actions that are implemented in the following chapters, it is assumed that the time required for such actions do not endanger the system. In other words, under no circumstances will the system enter a state of non-correctable emergency because a corrective action takes too long to have effect. 2.3 SC-OPF Formulation The concept of the SC-OPF is to augment the initial OPF problem with additional constraints that relate to contingency states or to the effects an outage of an element would have on the system. Two approaches that lead to different formulations are presented in the following sections Preventive Approach The preventive SC-OPF is the problem described by the following equations [5]: Minimize f(z 0,u 0 ) (2.4) subject to g c (z c,u 0 ) = 0 c=0,1,2,...,γ (2.5) h c (z c,u 0 ) 0 c=0,1,2,...,γ (2.6)

24 6 CHAPTER 2. OPF AND SECURITY CONSTRAINED OPF where subscript 0 represents the pre-contingency(base-case) state being optimized, and subscript c (c > 0) represents the post-contingency states for the γ contingency cases that are selected. Therefore this formulation addresses pre-contingency but also the set c of post-contingency constraints. The fact that the set of equations (2.5) and (2.6) depend on controls u 0 instead of u c, for all post-contingency states, makes this formulation a preventive SC-OPF. Control levels are restricted to their pre-contingency condition settings even in the post-contingency situations. The set of post-contingency state variables is represented by z c. Each set of contingency-related equality constraints is like the set of equality constraints of the base OPF, only it corresponds to the system with one element removed. The sets of inequality constraints of contingency cases are like the equivalent constraints of the base OPF, except that the system has one element less and the limits for line flows might be different Preventive-Corrective Approach The preventive-corrective SC-OPF can be stated in the following sets of equations [5]: Minimize f(z 0,u 0 ) (2.7) subject to g 0 (z 0,u 0 ) = 0 (2.8) max h 0 (z 0,u 0 ) h 0 (2.9) g c (z c,u c ) = 0 c=1,2,...,γ (2.10) max h c (z c,u c ) h c c=1,2,...,γ (2.11) max u 0 u c u c c=1,2,...,γ (2.12) where subscript 0 represents the pre-contingency state and subscript c (c > 0) represents the post-contingency states for the γ contingency cases that are addressed. The term corrective is derived from the fact that postcontingency controls u c are allowed to move in order to satisfy(correct) postcontingency constraints. The problem is preventive-corrective because the amount of corrective control actions that can be expended is not unlimited. It is bound by an amount u c max and also by the pre-contingency control setting u 0 as in equation (2.12). If the amount of expendable corrective actions do not suffice to satisfy all post-contingency constraints, the algorithm resorts to preventive rescheduling of pre-contingency controls u 0. The vector of maximum allowed adjustments u c max is given by the following equation: u c max = T c du c dt (2.13)

25 2.3. SC-OPF FORMULATION 7 T c isthetimeavailableforcorrectiveactionsafteracontingencyhasoccurred and duc dt is the rate of response of controls to a contingency. It is therefore evident that corrective control is heavily concerned with the amount of time an action requires to be implemented. In the case studies that are considered, the quantity u max c is going to be chosen as an input for sensitivity analysis purposes. It is the system operator s responsibility to know what amount of corrective actions is expendable or to develop mechanisms to increase this amount Power System Security States In the following chapter two methods for solving the SC-OPF are going to be formulated. The first method is fully preventive while the second is preventive-corrective. To clarify the fundamental goals of each method and link the concepts associated with them to real operating situations, figure 2.1 is presented. Several operating conditions are depicted, and have certain security-related states attributed to them. The goal when applying a preventive method is to constantly maintain the system in the Secure state. In this state it is guaranteed that the system will suffer no violations of inequality constraints or loss of load in the event of a contingency. It is the best level that can be achieved from a security perspective. Under certain circumstances the system can move from the Secure to the Alert state. An increase in power demand for instance can trigger this transition. In the Alert state the system is still not suffering from any constraint violations, but a contingency would lead to operating constraints being violated. These violations would not be able to be corrected without cutting power supply to some loads of the system. To return to the Secure state the system operator must implement preventive control actions. The preventive-corrective approach aims to keep the system in the Correctively Secure state. In this state it is guaranteed that any possible violations caused by a contingency can be quickly cleared through corrective control actions without any further consequences. The system operator performs security analysis in order to develop appropriate control strategies that can be flexibly applied under a contingent scenario. If the available amount of corrective actions is not enough to clear all post-contingency violations, the system moves to the Alert state. Preventive measures are then required to ensure the extra level of security that corrective control cannot provide. In the correctively secure state a contingency will trigger a transition to the correctively secure state. Would-be violations are rapidly cleared through implementation of corrective actions. To provide security against further contingencies though, preventive measures are needed or the system risks suffering a non-correctable emergency.

26 8 CHAPTER 2. OPF AND SECURITY CONSTRAINED OPF Cases where the system is in a restorative mode after loss of load has been suffered are out of the scope of this work and are not discussed Trade-off between Generation Costs and Security The operating costs of a SC-OPF dispatch are always larger or equal to those of a normal OPF. To achieve a level of security the operating point of a power system moves away from the most profitable point. This happens because the dispatching that is defined by a SC-OPF is more distributed among the participating generators of the system. Cheap generating units have to reduce their output to avoid congestion of lines of the system. The difference is compensated by more expensive generators and therefore lightly loaded lines of the base OPF are utilized more in the SC-OPF. Enhanced security comes at the expense of additional operating costs. If P 0, P p and P c are used to denote the standard OPF, preventive and preventive-corrective SC-OPF respectively, then the relationships for system costs are: Cost(P 0 ) Cost(P c ) Cost(P p ) The difference in cost between the standard OPF and the SC-OPF has been termed Cost of Security [6] and expresses the additional costs required to guarantee system security. It is an index of the price to pay to achieve a certain level of security and will be used later to quantify and evaluate the costs induced by the SC-OPF.

27 unintentional deliberate deliberate }unintentional 2.3. SC-OPF FORMULATION 9 Secure All load supplied. No operating limits violated. In the event of a contingency there will be no violations. Alert All load supplied. No operating limits violated. In the event of a contingency there will be violations of inequality constraints that can t be corrected without loss of load Correctively Secure All load supplied. No operating limits violated. Any violations caused by a contingency can be corrected by appropriate control action Operating limits violated. The violations cannot be corrected without loss of load. All load supplied, any violation is quickly cleared through corrective actions. Correctively Alert Non-Correctable Emergency Preventive Method Transition due to reduction in reserve margins or high probability of disturbance Transition due to a contingency event } Transition through implementation of preventive actions Preventive-Corrective Method Transition due to reduction in reserve margins or high probability of disturbance Transition due to a contingency event Transition through implementation of preventive actions Transition through implementation of preventive and corrective actions } } Figure 2.1: Security-State Diagram

28 10 CHAPTER 2. OPF AND SECURITY CONSTRAINED OPF

29 Chapter 3 System Modelling This chapter introduces the linear optimal power flow problem along with the theoretical background that supports this modelling. Its application on a mixed AC-DC grid is then presented. 3.1 Linear Optimal Power Flow The OPF problem stated in section 2.1 is a non-linear, non-convex problem. As the size of the problem increases the OPF calculation requires substantial computational effort. This fact often makes the OPF unattractive for certain applications and implementations. A good alternative in these cases is to use the so called DCOPF modelling. The DCOPF model is based on approximations that lead to a linear version of the OPF problem. The approximations and simplified power flow equations that are derived can be found in [7]. The DCOPF model features many advantages [8]. The solutions obtained from it are reliable and the complexity of the problem significantly reduced compared to the full OPF. The solvers of such problems are very powerful and fast, while they guarantee convergence to the global optimum. Active power flows that are calculated are considerably accurate and the model is especially powerful when it comes to contingency analysis. The following sections explain the fundamental concepts that lead to a linear formulation of the OPF. 3.2 Calculation of Active Power Flows in the AC Grid Power flowing on an AC line is given by the following set of non-linear equations [9]: P km = U 2 k G km U k U m G km cos(δ k δ m ) U k U m B km sin(δ k δ m )] (3.1) 11

30 12 CHAPTER 3. SYSTEM MODELLING Q km = U 2 k (B km+b sh km )+U ku m B km cos(δ k δ m ) U k U m G km sin(δ k δ m )] (3.2) U k, U m stands for the voltage magnitudeat busk,m, δ for the voltage angle, andg km, B km fortheshuntconductanceandsusceptancerespectively. With the approximations made in [7] the linear formula for power flow calculation in the AC grid is derived. These assumptions are the following: All voltage magnitudes are assumed to be equal to the base voltage, i.e U n = 1 p.u n 1,2,...K. All branch resistances are neglected (R kmac = 0). It follows that G km = 0. No reactive power quantities are considered (Q km = 0). Bus voltage angles are assumed very small to make the formula sin(α) α valid. With these assumptions the steady-state power flowing from bus k to m is given by the linear equation: P AC km δ k δ m x km (3.3) x km being the reactance of the line connecting bus k and m and δ k, δ m the bus voltage angles in radians. 3.3 Linear Optimal Power Flow in the Mixed Grid The linearized OPF that includes DC grids was conceived and formulated in [1]. The work carried out in the present thesis is based on this formulation and extends it to construct a problem that incorporates security assessments. For consistency purposes this work [1] is thoroughly documented in sections Calculation of Active Power Flows in the DC Grid The active power flowing on a DC line can be calculated through equation (3.4): P DC km = V k (V k V m ) R km (3.4) which is non-linear. To incorporate power flow equations in the DC grid to the linearized OPF problem a suitable approximation is in order. If it is presumed that all voltages at DC nodes are close to their nominal value of 1 p.u, then equation (3.4) can be approximated by the following linear equation: P DC km (V k V m ) R km (3.5)

31 3.3. LINEAR OPTIMAL POWER FLOW IN THE MIXED GRID 13 V k, V m are the voltages of DC nodes k,m and R km is the resistance of the DC line connecting them Power Balance in the Mixed Grid As is the case for the AC grid, active power losses on the DC grid are assumed to be zero. Hence the power balance equations for AC and DC nodes are: P Lk = P Gk P Lk = P Gk N P km P Tk for all AC nodes (3.6) n M P km + P Tk for all DC nodes (3.7) n P Lk,P Gk and P Tk represent the load, generation infeed and terminal power transfer at node k. The sings of terms P Tk for the AC and DC nodes can be assigned arbitrarily, as long as they have opposite signs to one another so that the power balance holds Matrix Formulation of Optimization Problem The DC optimal power flow problem can be constructed with the use of matrices in the following form: Minimize f(ξ) = 1 2 ξt G ξ +λ ξ (3.8) subject to C eq ξ = b eq (3.9) C iq ξ b iq (3.10) Problem (3.8) - (3.9) constitutes a linearly constrained optimization problem with quadratic objective function and can be solved as a quadratic programming problem Optimization Vector ξ Vector ξ in the mixed DCOPF contains variables of both grids: ξ = [P G1 P G2... P GI δ 2 δ 3... δ K P T1 P T2... P TL V DC2 V DC3... V DCM ] T [C 1]

32 14 CHAPTER 3. SYSTEM MODELLING where the subscripts introduced comply to the nomenclature of table 3.1. The set of control variables consists of all active power generations P G and terminal power transfers P T, whereas the set of state dependent variables containsallacvoltageanglesδ andall DCvoltages V DC. Tosolvethepower flow equations only the relative difference in angles and relative difference in DC voltages are needed, therefore a slack bus in both the AC and DC grid can be assigned. In the formulations presented AC bus 1 and DC bus 1 are chosen as slack buses for the AC and DC grid. This way the angle of the AC slack bus and the Voltage of the DC slack bus is removed from ξ. Table 3.1: Nomenclature K Number of AC nodes N Number of AC lines M Number of DC nodes D Number of DC lines I Number of generators L Number of terminals C I+K+L+M-2 Y 2(I+N+L+D) Length of vector ξ Amount of inequality constraints Objective Function The objective of the proposed algorithm is the minimization of generation costs. Therefore the most important component of the objective function is the quadratic and linear cost of all generators. Penalties for AC angle differences and DC voltages are introduced for stability purposes and for assessing the AC-DC interaction. I [α P Gi +β PGi 2 ]+ΠAC [δ k δ m ] 2 +Π DC [V DCk V DCm ] 2 i=1 km N N= Pairs of AC buses connected with a branch. D= Pairs of DC buses connected with a branch. α = linear coefficient of generator cost. β = quadratic coefficient of generator cost. km D (3.11)

33 3.3. LINEAR OPTIMAL POWER FLOW IN THE MIXED GRID 15 Matrix G of equation (3.8) and its components are constructed as: QC β W AC β G = QC = [L L] W DC β [C C] I J k2 J 23 J 2K k 2 J W AC 32 J k3 J 3K = k J K2 J K3 J kk k K I k2 I 23 I 2M k 2 I W DC 32 I k3 I 3M = k I M2 I M3 I km k M regarding the entries of matrix W AC : { 1 if there is a line connecting bus k and m J km = 0 otherwise [I I] [K 1 K 1] [M 1 M 1] The entries of W DC are derived in a similar way. The diagonal entry of matrix G that consists of zeros means there is no application of costs for terminal operation. Vector λ of equation (3.8) contains the generator linear coefficients: ] λ = [α 1 α 2... α I 0 [1 (C I)] (3.12) Equality Constraints In the mixed grid the equality constraints are practically the equations that express the power balance at AC and DC nodes. C eq of equation (3.9) contains the appropriate matrices to represent equality constraints and are presented below. Matrix H defines the generated in-feed at AC nodes through the allocation of generators at AC nodes: H [K I] (3.13)

34 16 CHAPTER 3. SYSTEM MODELLING H ij = { 1 if generator j is connected at node i 0 otherwise The power flowing on AC lines can be expressed in matrix form: P1 AC P AC 2. δ 1 = [B AC] δ 2. The admittance matrix B AC is constructed in the following way: B 12 B 13 B 1K B k2 B 23 B 2K k 2. B AC = B K2 B K3 B kk k K [K (K 1)] (3.14) (3.15) (3.16) B km = { 1 x km if AC nodes k,m have a line connecting them 0 otherwise (3.17) The first column of the admittance matrix is removed due to the assignment of an AC slack bus, in this case AC bus 1 is chosen as the slack bus. The power flow equations in (3.5) for the DC grid take on the following form: P1 DC P DC 2. V 1 = [B DC] V 2. (3.18) The equivalent matrix for the DC grid is constructed similarly to result in B DC[M (M 1)], where the entries are: B km = { 1 R km if DC nodes k,m have a line connecting them 0 otherwise (3.19) The next matrices to be formulated indicate the interconnection of the AC-DC grid through the allocation of the terminals at buses. T [K L] : Allocation of terminal stations at AC buses (3.20)

35 3.3. LINEAR OPTIMAL POWER FLOW IN THE MIXED GRID 17 where T pq = { 1 if terminal q is connected at AC bus p 0 otherwise S [M L] : Allocation of terminal stations at DC buses (3.21) S pq = { 1 if terminal q is connected at DC bus p 0 otherwise Matrix T allocates terminals to AC nodes and matrix S allocates them to DC nodes. For thepowerbalance tohold, elements of T ands haveopposite signed elements. The power flowing on one side of the terminal must equal the power flowing on the other side since no losses are considered. With all matrices defined the equality matrix C eq is constructed in the following way and represents equations (3.6) and (3.7): C eq = H B AC T 0 (3.22) 0 0 S B DC [(K+M) C] Constant vector b eq of equation (3.9) contains the loads connected at the buses of the mixed grid: b eq = [P L,1 P L,2...P L,(K+M) ] T [(K+M) 1] (3.23) The first row of C eq represents the power balance at all AC nodes: PGi km N B AC km [δ k δ m ] P Tk = P Li (3.24) while the second row represents the power balance at all DC nodes: Bkm DC [V k V m ]+ P Tk = P Li (3.25) km D Inequality Constraints The inequality constraints for the system modelling that is implemented are stated below: Pkm AC AC AC lines (3.26) Pkm DC DC DC lines (3.27) P T,i FT,i max terminals i (3.28) P min G,i P G,i P max G,i generators i (3.29)

36 18 CHAPTER 3. SYSTEM MODELLING To obtain these inequality constraints the following matrices are constructed: J 12 J 1K J A AC = 22 J 2K.... (3.30). J N2 J NK [N (K 1)] B AC if line i starts at node j J ij = B AC if line i ends at node j (3.31) 0 otherwise The analogous matrix for the DC grid is constructed with the same philosophy leading to A DC[D (M 1)]. Then C iq is formulated as follows: Λ I Λ I A AC A AC 0 0 C iq = (3.32) 0 0 Λ L Λ L A DC A DC [Y C] Λ I and Λ L are identity matrices. The first two rows of the matrix correspond to the generating limits of units. The 3rd and 4th rows represent the limits for AC line flows and the next two rows bound the terminal power transfers. The last two rows correspond to the limits for DC line flows. Vector b iq contains all upper and lower bounds as shown in the start of section 3.3.7, and is constructed by the following components: b max g b min g =[P max G1 P max G2... P max GI ] T [I 1] (3.33) =[P min G1 P min G2... P min GI ] T [I 1] (3.34) b l,ac =[P l1 P l2... P ln ] T [N 1] (3.35) b l,dc =[P l1 P l2... P lo ] T [D 1] (3.36) b term =[P t1 P t2... P tl ] T [L 1] (3.37) b iq = [b max g b min g b l,ac b l,ac b term b term b l,dc b l,dc ] T [Y 1]

37 Chapter 4 Security Constrained OPF in the Mixed Grid In this chapter the formulation of 2 methods for the application of SC-OPF in the mixed grid are presented, along with the methodology that is used to support the modellings implemented. The term SC-OPF will be used to refer to the linearized model of the security constrained optimal power flow. 4.1 Method I: Preventive The preventive approach to the security constrained optimal power flow is based on the notion that the effects of a possible contingency should be anticipated and dealt with while in the normal state of the system. To achieve this the operator of a power system is required to do contingency analysis and take the necessary measures to ensure system security. One way to proceed is to run a power flow simulation for every contingency scenario considered and assess the results. Any violations monitored are registered and dealt with appropriately. This approach can prove inefficient and computationally forbidding when dealing with large systems. The use of a DCOPF model offers a vantage point when it comes to contingency analysis, because the effect of a contingency can be approximately quantified in a pre-determined way. Constraints for line and generator outages can be included in a single optimization problem as linear sensitivities, providing a straightforward approach and notably reducing the computational effort required. The preventive method that is developed and presented takes into account all AC and DC line contingencies as well as generator contingencies. 19

38 20CHAPTER 4. SECURITY CONSTRAINED OPF IN THE MIXED GRID Line Outage Distribution Factors in the AC Grid WhenanAClineoutageoccursthereisaredistributionofthepowerinitially flowing on that line. The percentage of this power taken up by each of the remaining lines is calculated through the line outage distribution factors (LODF) [10]. The LODF ij,km denotes the fraction of the power initially flowing on line k m, that is flowing on line i j after line r s has been outaged. It is calculated as: LODFij,km AC = n km x km (X ik X im X jk + X jm ) n ij x ij [n rs x km (X kk + X mm 2X km )] (4.1) x ij : reactance of line i j X ij : entry in the ith row and jth column of the bus reactance matrix X n ij : number of lines connecting bus i and j The admittance matrix B AC is singular and can t be inverted. By removing the row and column that corresponds to the slack bus the reduced matrix BAC red [K 1 K 1] can be inverted. The resulting matrix is then augmented by a row and column of zeros to result in reactance matrix X. B red AC = B AC(2 : K,2 : K) (4.2) X = inv(b red AC ) (4.3) With the use of the LODFs the total power flowing on line i j after the outage of line k m has occurred can be calculated as: P ij,1 = P ij,0 +LODF AC ij,km P km,0 (4.4) where subscripts 0, 1 stand for pre and post-contingency conditions Line Outage Distribution Factors in the DC Grid Equation (3.18) can be written in the following way if multiplied with [B DC ] 1 : V = [Z] P (4.5) where V stands for the vector of DC voltage entries, P stands for the power injections in the DC grid and Z is the impedance matrix derived from B DC. Matrix B DC is singular because it refers to a set of linearly dependent equations and hence is not invertible. By declaring a reference value of voltage at a chosen DC slack bus, the row and column corresponding to that bus are removed from B DC. Inversion of the reduced matrix is then

39 4.1. METHOD I: PREVENTIVE 21 performed while a row and column of zeros are added to result in matrix Z of equation (4.5). Thus if one assigns bus 1 as the slack bus matrix Z is: 0 0 K Z =. (4.6). BDC 1 0 K To calculate changes in voltages for a given set of changes in bus power injections the next equation can be used V = [Z] P (4.7) Suppose that bus i has a 1 p.u increase in power injection that is equally compensated by a 1 p.u decrease in the reference bus. Then the V values are equal to the derivative of the voltages with respect to a change in power injection in bus k. The sensitivity factors for this case are ω = df km dp i = d dp i where: [ V k V m R km ] = 1 R km [ dv k dp i dv m dp i df km : change in flow on line km ] = 1 R km (Z ki Z mi ) (4.8) dp i : incremental change in power injection at bus i Z ki = dv k dp i : element k-i of matrix Z Z mi = dv m dp i : element m-i of matrix Z R km : resistance of line km Figure 4.1 shows line k m under normal conditions while figure 4.2 shows the same line once it has been dropped. In a situation where line k m is out the circuit breakers are open as in figure 4.2. In this case no current is flowing through the breakers and line k m is totally isolated from the rest of the system. A line outage can be modelled by adding a pair of injections at the system, one at each bus corresponding to the line that is out, as seen in figure 4.3. The breakers are closed but injections P k and P m are added to bus k and m respectively. If P k = Pkm 1 and P m = Pkm 1 then no power is flowing through the breakers despite the fact that they are closed. Therefore for the rest of the network it is as though line k m is outaged.

40 22CHAPTER 4. SECURITY CONSTRAINED OPF IN THE MIXED GRID Bus k Pkm Before Outage Bus m Figure 4.1: Line Before Outage Now equation (4.7) can be used to obtain the effects of the injections at bus k,m. V k = Z kk P k + Z km P m (4.9) V m = Z mk P k +Z mm P m (4.10) The following notations are now defined: V k,v m,p km : to be the values before the outage V k, V m, P km : to be the incremental changes Vk 1,V m 1,P1 km : to be the values after the outage So for the case where line km is out it follows that P 1 km = P k = P m (4.11)

41 4.1. METHOD I: PREVENTIVE 23 Bus k After Outage Bus m Figure 4.2: Line After Outage and Equations (4.9) and (4.10) are written P 1 km = V 1 k V1 m R km (4.12) V k = (Z kk Z km ) P k (4.13) V m = (Z mm Z mk ) P k (4.14) and with use of V 1 k = V k + V k (4.15) V 1 m = V m + δ m (4.16) P 1 km = V k V m R km + V k V m R km (4.17)

42 24CHAPTER 4. SECURITY CONSTRAINED OPF IN THE MIXED GRID ΔPk Bus k P 1 km Simulation of Outage ΔPm Bus m Figure 4.3: Simulation of Line Outage Through Bus Injections which can also be written like P 1 km = P km +(Z kk + Z mm 2Z km ) P k (4.18) Pkm 1 can be replaced by P k according to equation (4.11) [ ] 1 P k = 1 (Z P kk+z mm 2Z km ) km R km (4.19) If a sensitivity factor θ is defined as the ratio of change in voltage anywhere in the system towards the initial flow P km flowing over a line then changes in bus voltage is: θ i,km = V i P km (4.20) V i = Z ik P k + Z im P m (4.21)

43 4.1. METHOD I: PREVENTIVE 25 Using the relationships for P k and P m from equation (4.19) the sensitivity factor is: (Z ik Z im )R km θ i,km = (4.22) R km (Z kk + Z mm 2Z km ) By definition the line outage distribution factor is described by the following equation: LODF ij,km = F ij F 0 km (4.23) LODF ij,km : line outage distribution factor when monitoring line i j after the outage of line k m F ij : change in flow on line i j F 0 km : original flow on line k m before the outage expanding the expression gives: LODF ij,km = F ij F 0 km = V i V j R ij Fkm 0 = 1 R ij ( V i P km V j P km ) = 1 R ij (θ i,km θ j,km ) (4.24) and by applying equation 4.20 the final equation is reached: LODFij,km DC = R km (Z ik Z im Z jk + Z jm ) R ij R km (Z kk + Z mm 2Z km )] (4.25) By applying equation (4.25) the power flowingon line i j due to the outage of line k m is: P ij,1 = P ij,0 +LODF DC ij,km P km,0 (4.26) Generalized Generation Distribution Factors The redistribution of line flows during a generator contingency can be calculated with the use of the Generalized Generation Distribution Factors (GGDFs) [11], [12]. GGDFij m describes the fraction of generation of unit m before it goes out, that flows on line i j after the outage has occurred: GGDF m ij = 1 x ij E ij [B AC ] 1 Ψ m,i (4.27) E ij is a vector of size [1 K] that has value 1 at the column that corresponds to the starting node of the line and value 1 at the column that corresponds

44 26CHAPTER 4. SECURITY CONSTRAINED OPF IN THE MIXED GRID to the ending node of the line. Ψ m is a vector of size [K 1]: 1 if outaged generator m was connected at bus i Ψ m,i = P max i max P J if there is a generator connected at bus i j m 0 otherwise (4.28) This modelling is based on the assumption that lost power of unit m is compensated by the rest of the units in proportion to their nominal maximum capabilities Line Constraints With the implementations described in sections the inequality constraints of (3.26)-(3.29) δ k δ m x km Fmax AC,km AC lines k m V DC,k V DC,m R km Fmax DC,km DC lines k m P T,i PT,i max PG,i min P G,i PG,i max are augmented by the following inequality constraints: δ k δ m +LODFkm,ij AC x δi δj km x ij Fmax AC,km V DC,k V DC,m R km δ k δ m x km terminals i generators i I monitored AC lines k m and outaged AC lines i j (4.29) +LODFkm,ij DC VDC,i V DC,j Fmax R ij DC,km monitored DC lines k m and outaged DC lines i j (4.30) +GGDFkm s P Gs Fmax AC,km monitored AC lines k m and outaged generators s (4.31) Every AC line induces one constraint regarding its normal limit and N 1 constraints to serve as preventive measures for the outage of the line. Each generator contingency accounted for induces N constraints for flows on AC lines. Every monitored DC line induces one constraint regarding normal limits and D 1 constraints for DC line outages. N, I and D are defined in table 3.1.

45 4.1. METHOD I: PREVENTIVE Formulation To obtain a formulation of the problem as in section 3.3.3, elements C iq and b iq have to be modified. C eq, b eq and G remain the same due to the fact that the preventive method does not take into account control and state variables for contingency cases nor does it solve the equivalent power flows for themodelling that is used. Theformulation of theset of equations stated in (4.31) in matrix form is achieved with the use of two matrices. The first one is the matrix of generator outage sensitivities: E I,1 = E I = GGDF Gen1 [ E I,1 E I,2... E I,N ] T [(N I) I] br GGDFbr1 Gen GGDF GenI br1 [I I] (4.32) (4.33) Matrices E I,2,...,E I,N are constructed similarly and refer to the branch with index i = 1,2,...,N. The second matrix needed to formulate equations (4.31) is the line adjacency matrix for generator outages and is as follows: ] E AC = [E AC,1 E AC,2 E AC,N (4.34) J 12 J 1K J E AC,1 = 12 J 1K... J 12 J 1K [(N I) (K 1)] [I (K 1)] (4.35) Entries of matrix E AC,i are the entries of matrix A AC in equation (3.30) corresponding to the i th line, repeated in every row. Thus E I,1 and E AC,1 together quantify the effects of all generator outages to the power flowing on AC branch 1. For each monitored line the appropriate row of A AC is repeated over I number of times (once for every generator outage considered) to produce the desired result. Regarding the AC line contingencies and the inequality constraints of equations (4.29) the following matrix is introduced: A ACprev = [ A ACprev1 A ACprev2 A ACprevN ] T [N 2 (K 1)] (4.36) Each element of A ACprev quantifies the effects of all AC line outages to the monitored line that corresponds to the subscript of this element. Matrix

46 28CHAPTER 4. SECURITY CONSTRAINED OPF IN THE MIXED GRID A ACprev1 is going to be defined as the sum of two matrices, the rest of the elements are constructed with the same methodology: A ACprev1[N (K 1)] = P p1 +Q p1 (4.37) P 12 P 1K P p1 =..... P N2 P NK [N (K 1)] (4.38) B AC,br,1 if monitored branch 1 starts at node j, for every row i P ij = B AC,br,1 if monitored line 1 ends at node j, for every row i 0 otherwise (4.39) Q 12 Q 1K Q p1 =..... (4.40) Q N2 Q NK [N (K 1)] B AC,br,i LODF br1,bri if outaged branch i starts at node j B AC,br,i LODF br1,bri if outaged branch i ends at node j Q ij = 0 if i is equal with index of monitored branch 1 0 otherwise (4.41) Each element of A ACprev therefore contains one constraint for normal operating conditions and N 1 constraints for line contingency related situations. Inanequivalent waytheformulationof A DCprev can beobtainedandwill not be presented here. The final inequality matrix then has the following

47 4.2. METHOD II: PREVENTIVE-CORRECTIVE 29 form: Λ I Λ I E I E AC 0 0 E I E AC A ACprev 0 0 C iq,prev = 0 A ACprev Λ L Λ L A DCprev A DCprev [Y prev C] (4.42) where Y prev = 2 [I +(N I)+N 2 +L+D 2 ]. Constant vector b iq,prev is as follows: b iq,prev = [b max g b min g b l,ac,g b l,ac,g b l,ac,p b l,ac,p b term b term b l,dc,p b l,dc,p ] T [Y prev 1] Vectors b l,ac,g and b l,ac,p contain the short-term limits for AC lines and vector b l,dc,p contains the short-term limits for DC lines. With C iq,prev and b iq,prev defined the preventive SC-OPF is formulated as in section Method II: Preventive-Corrective Methodology and Implementations Contrary to the preventive method of section 4.1 this method explicitly accounts for post-contingency equality and inequality constraints. Therefore a decision variable ξ c for every contingency related state is included in the

48 30CHAPTER 4. SECURITY CONSTRAINED OPF IN THE MIXED GRID total decision variable: ξ 0 ξ corr ξ = 1 =. ξ γ P 0 G δ 0 P 0 T V 0 DC δ 1 P 1 T V 1 DC. δ γ P γ T V γ DC (4.43) The controls for the base-case are the generator power outputs P G and the terminal power transfers P T. In the contingency states only the terminal power transfers are included in the control variables. This means that post-contingency corrective control is limited to changes in terminal power transfers. For both pre and post-contingency states the state variable contains the AC bus angles and the DC bus voltages. The sections to come explain analytically the formulation of the quadratic programming problem which includes the following contingency scenarios: AC Line Outages DC Line Outages Terminal Outages Generator Outages The reason behind the choice to include these sets of contingencies is simple. They are the most significant for the given system modelling. Seeing as all outages of elements of the above sets are accounted for, the length of the decision variable and the total amount of inequality constraints can be computed. These quantities are presented in table 4.1 and they are linked to elements C,Y,D,I,N,L from table Objective Function For the preventive-corrective method the objective function presented in section is extended by adding one more quadratic term. This term is the

49 4.2. METHOD II: PREVENTIVE-CORRECTIVE 31 Table 4.1: Nomenclature for Preventive-Corrective Method C corr C+(C-I)(I+N+D+L) Length of vector ξ corr Y AC N(Y-2I) Inequality constraints invoked by AC line outages Y DC D(Y-2I) Inequality constraints invoked by DC line outages Y Ter L(Y-2I) Inequality constraints invoked by terminal outages Y Gen I Y Inequality constraints invoked by generator outages Y Control 2L(I+N+D+L) Inequality constraints for terminal control Y corr Y+Y AC +Y DC +Y Ter +Y Gen +Y Control Total number of inequality constraints terminal difference penalty cost of the form [P 0 T Pc T ]2 and is introduced for all assessed contingency scenarios. The rationale behind this implementation lies in the fact that it gives additional room for sensitivity analysis. Furthermore it is beneficial when it comes to evaluating the interaction between the AC and DC grid and it also enhances system stability. A direct assignment of costs for terminal power transfers is not performed, though an implementation of this sort could be envisioned. Thus the objective function now takes on the following form: I [α P Gs +β PGs]+Π 2 AC s=1 km N [δ k δ m ] 2 +Π DC [V DCk V DCm ] 2 +Π Ter N= Pairs of AC buses connected with a branch D= Pairs of DC buses connected with a branch Π Ter is the penalty for terminal power differences. km D γ [PT 0 PT] c 2 (4.44) To include these terms in the objective function the matrix G is constructed c=1

50 32CHAPTER 4. SECURITY CONSTRAINED OPF IN THE MIXED GRID as: QC W AC c Λ L 0 0 Λ L 0 0 Λ L 0 0 W DC G = 0 Λ L 0 0 Λ L Λ L Λ L (4.45) For clarity regarding the implementation, a case where two contingencies are considered is depicted to portray the basics of the formulation. A display of further contingency states would be rather trivial. Here Λ L is the identity matrix of size L. In this way the terms [PT 0 P1 T ]2 +[PT 0 P2 T ]2 are derived from equation (3.8). Beware that all contingency state terminal flows are related to the pre-contingency terminal flows PT 0. This means that the precontingency terms PT 0 occur squared as many times as the total number of outages accounted for. This explains why the appropriate entry of matrix G in (4.45) is c Λ L (c = 2 for the example), where c is the number of contingencies regarded. Vector λ in this case is augmented by an appropriate amount of zeros. The only linear coefficients in the objective function are the ones belonging to the generator costs: [ ] λ = α 1 α 2... α I 0 [1 (Ccorr I)] (4.46) Post-Contingency Terminal Control The HVDC scheme displays significant advantages when it comes to controllability. This fact urges for a direct implementation of post-contingency corrective control of the terminal power transfers. In the proposed SC-OPF this control takes on the form P 0 T Pc T = Pc T Pc T max (4.47) which bounds the allowed change in terminal power transfer in post contingency states. In the event of a contingency, the algorithm first attempts to satisfy any

51 4.2. METHOD II: PREVENTIVE-CORRECTIVE 33 post-contingency constraint violations through post-contingency corrective actions PT c of the terminal flows. If these actions do not suffice to clear all violations then pre-contingency control levels PG 0,P0 T must move to satisfy these constraints. Even though there is an application of costs for the PT c actions, the costs incurred by such actions are significantly less than the ones that would arise in order to achieve the same goal through preventive measures. Generation costs of the units supplying the system dominate the value of the objective function, and as such the change in the dispatching of generation will lead to a more expensive system operation as a trade-off between security and optimality. To implement control over the absolute value of the difference in terminal power, two constraints are needed for each terminal variable. The combination of : { PT 0 Pc T Pc T max PT k P0 T Pc T max (4.48) achieves this result. In matrix form this is formulated below : Ter Con ξ P c T max (4.49) Λ L 0 0 Λ L Λ L 0 0 Λ L P 0 G δ 0 P 0 T V 0 DC δ 1 P 1 T V 1 DC 0 0 PT 1 max 0 0 PT 1 max 0 (4.50) Again here I L is the identity matrix of size L, whereas P c T max is the maximum amount of change in the power transferred by terminals in contingency situations Line Outages in the AC Grid Note: for purposes of space efficiency lower bounds for generator limits, negative limits for terminal flows and AC and DC line flows are not depicted in the inequality constraints to follow. They need to be included in the inequality matrices for the correct formulation of inequality constraints as in equation (3.32).

52 34CHAPTER 4. SECURITY CONSTRAINED OPF IN THE MIXED GRID As stated earlier, constraints for post-contingency line flows are explicitly accounted for through a new set of equality and inequality constraints. The inequality matrix C iqac contains all the entries that formulate the postcontingency inequality constraints for all AC line outages: C iqac[yac N(C I)] Inequality constraints regarding the contingency states where an AC line is out can be represented as follows: C iqac[yac N(C I)] ξ AC[N(C I) 1] biq AC[YAC 1] is depicted in the following way: C iqac CiqAC CiqAC N ξ 1 AC ξ 2 AC. ξ N AC b iqac (4.51) where for a chosen indexing of AC lines, 1,...,N refers to the outage of AC line number1,..., outage of AC linenumbern. Vector ξac c is the decision variable that corresponds to the state where AC line c is out. Vector b iqac is of length N(C I) and contains the post-contingency ratings for AC and DC lines, as well as limits for terminal flows. To make it easier to comprehend, the formulation of C iqac for a single line outage is presented. The pre-contingency state is also shown in order distinguish which elements change in the post-contingency state. The augmented matrix to include additional outages is constructed in a similar manner. Λ I A 0 AC Λ L 0 C 0,1 iqac = A 0 DC 0 (4.52) 0 0 A 1 AC Λ L A 0 DC [2(Y I) (2C I)] For the contingency state the Line Adjacency matrix A 1 AC has all entries equal to zero in the row that corresponds to the outaged line. The Line

53 4.2. METHOD II: PREVENTIVE-CORRECTIVE 35 Adjacency matrix for the DC Grid stays the same, since it is obvious that an outage in the AC grid has no effect on the topology of the DC grid. The same applies for the terminal stations, where for a line outage matrix I L remains unaltered. Regarding the equality constraints the matrix that contains all entries for each contingency scenario is: C eqac[n(k+m) N(C I)] (4.53) C eqac[n(k+m) N(C I)] ξ AC[N(C I) 1] = b eqac[n(c I) 1] (4.54) is depicted in this way: C eqac ξ AC 1 0 CeqAC 2. ξ 2 AC. = b eqac (4.55) CeqAC N ξac N H 0 BAC 0 T 0 0 C 0,1 eqac = 0 0 S BDC H BAC 1 T S BDC 0 [2(K+M) (2C I)] (4.56) In the Equality matrix the entry that changes is the AC bus Admittance matrix B AC, to simulate this change in the AC grid topology. Note that the generation profile H remains the same for pre and post-contingency states to model the pre-deterministic response that is assumed for the generators Line Outages in the DC Grid The formulation of constraints for DC line contingency scenarios is as follows: C iqdc[ydc D(C I)] (4.57) C iqdc[ydc D(C I)] ξ DC[D(C I) 1] b iqdc[ydc 1] is depicted in the following way: C iqdc CiqDC ξ 1 DC ξ 2 DC. b iqdc (4.58) 0 0 C D iqdc ξ D DC

54 36CHAPTER 4. SECURITY CONSTRAINED OPF IN THE MIXED GRID and for a chosen indexing of DC lines, 1,...,D refer to the outage of DC line number 1,..., outage of AC line number D. Λ I A 0 AC Λ L 0 C 0,1 iqdc = A 0 DC 0 (4.59) 0 0 A 0 AC Λ L A 1 DC [2(Y I) (2C I)] The line adjacency matrix for the contingency state A 1 DC has all entries equal to zero in the row that corresponds to the outaged line. The Line Adjacency matrix for the AC Grid remains the same. C eqdc[d(k+m) D(C I)] (4.60) C eqdc[d(k+m) D(C I)] ξ DC[D(C I) 1] = b eqdc[d(k+m) 1] (4.61) is depicted in this way: C eqdc CeqDC ξ 1 DC ξ 2 DC. = b eqdc (4.62) 0 0 C D eqdc ξ D DC C 0,1 eqdc = H 0 BAC 0 T S BDC H BAC 0 T S B 1 DC [2(K+M) (2C I)] (4.63) Here it is the DC grid that changes topology, resulting in a different admittance matrix B 1 DC.

55 4.2. METHOD II: PREVENTIVE-CORRECTIVE Terminal Station Outages Terminal stations of an AC-DC network are very important for power flow studies and security assessments since they define the interaction between the two grids. Therefore it is essential to include terminal outages in contingency analysis. A terminal contingency is the situation where due to an unexpected disruption the power flowing through the terminal is zero, P T = 0. The implementation used to simulate a terminal outage is to change the appropriate entries of vector b iq so that the following inequality constraints are formulated: P c T 0 P c T 0 The second of these inequalities is equivalent with: PT c 0. For the 2 inequality constraints to hold the power flow PT c which refers to the outage of terminal c has to be 0. Therefore this implementation achieves the desired result of simulating a terminal station contingency. Another way that seems reasonable to simulate an outage of this sort is to change the appropriate elements of matrices S,T of equations (3.20) and (3.21). However this is not advised because an implementation like this leaves a control variable free to change and affect the optimization process. C iqter[yter L(C I)] (4.64) C iqter[yter L(C I)] ξ Ter[L(C I) 1] b iqter[l(c I) 1] C iqter CiqTer ξ 1 Ter ξ 2 Ter. b iqter (4.65) C 0,1 iqter = ξ L Ter 0 0 CiqTer L Λ I A 0 AC Λ L A 0 DC A 0 AC Λ L 0 (4.66) A 0 DC [2(Y I) (2C I)]

56 38CHAPTER 4. SECURITY CONSTRAINED OPF IN THE MIXED GRID From matrix C iqter it is clear that there is absolutely no change between elements of pre and post-contingency states. An outage of a terminal station does not affect the topology of the AC or DC grid and the methodology used necessitates no alterations for matrices T and S. The matrix that represents equality constraints also displays no change between entries for pre and post-contingency states: C eqter CeqTer ξ 1 Ter ξ 2 Ter. = b eqter (4.67) 0 0 C L eqter ξ L Ter C 0,1 eqter = H 0 BAC 0 T S BDC H BAC 0 T S BDC 0 [2(K+M) (2C I)] (4.68) Generator Outages The model that is applied for the control scheme of the generating units, allows for a useful approximation regarding the responses of these units to a generator contingency. When a generating unit is outaged, other generators are required to compensate for the lost power. The modelling that is going to be used for this situation is as follows: In the event of a generator outage, automatic control of all participating units takes effect to ensure the lost generation is compensated by these units. The portion of this additional power that each unit must supply is proportionate to the unit s nominal maximum power [11]. In equation form this translates to: where P Gi,1 = P Gi,0 +P Gm r i,m (4.69) r i,m = Pmax Gi PGj max j m (4.70) and P Gi,0, P Gi,1 is the power of unit i before and after the contingency and P Gm the power of outaged unit m. Hence the generator power outputs are not included as controls in the post-contingency states, which is already evident from equation (4.43), since their response is defined in a pre-determined way.

57 4.2. METHOD II: PREVENTIVE-CORRECTIVE 39 C iqgen[ygen I(C I)] (4.71) C iqgen[ygen I(C I)] ξ Gen[I(C I) 1] b iqgen[ygen 1] C iqgen CiqGen ξ 1 Gen ξ 2 Gen. b iqgen (4.72) C 0,1 iqgen = ξ I Gen 0 0 CiqGen I Λ 0 I A 0 AC Λ L A 0 DC 0 0 Φ 1 iq A 0 AC I L A 0 DC [2(Y I) (2C I)] (4.73) The difference between base and post-contingency case is in matrices Λ I and Φ iq, where the latter matrix is introduced to define the pre-determined outputs of generators. In the post-contingency state, these online generators are producing increased power compared to what they were producing in the pre-contingency state. These power generations though still have to be bounded to the same pre-contingency capacities that each unit has. Thus a generator contingency invokes I 1 number of constraints regarding the capacities of remaining generating units and are expressed in matrix GL I r 2, Φ 1 iq = r 3, (4.74)..... r I,1 0 1 [I I]

58 40CHAPTER 4. SECURITY CONSTRAINED OPF IN THE MIXED GRID 1 r 1, Φ 2 iq = 0 r 3, r I, [I I] (4.75) The above matrices show the build-up of matrix Φ iq for the outage of generators with indices 1 and 2, the equivalent matrices for the rest of the generator contingencies are constructed accordingly. The desired result that these entries produce is the following inequality: Φ iq P 0 G P0max G (4.76) which enforces constraints to generating units according to each one s limit. The formulation of the equality constraints for a single generator outage is depicted below: H BAC 0 T 0 0 C 0,1 eqgen = 0 0 S BDC Φ 1 eq 0 0 BAC 0 T S BDC 0 [2(K+M) (2C I)] (4.77) The difference lies between elements H and Φ eq[k I]. H denotes the generator allocation matrix stated in Here matrix Φ 1 eq[k I] is defined : Φ 1 eq,ij = { Φ 1 iq,ij if there is a generator connected at bus i 0 otherwise (4.78) Matrix Φ eq assigns the in-feeds of the remaining units to the buses they are connected Formulation of the Complete Problem The individual matrix formulations presented in the previous sections can be combined with each other to construct a single optimization problem. Depending on the types of contingencies included in the cases that are envisioned, suitable concatenations can produce the desired matrices G,C eq,c iq and vectors b eq,b iq,λ of equations (3.8) (3.9). The case where all types of contingencies are incorporated in the optimization problem is as follows : C iq,total ξ corr b iq,total

59 4.2. METHOD II: PREVENTIVE-CORRECTIVE 41 C iqbase C iqac C iqdc 0 0 ξ 0 0 C iqter 0 ξ AC 0 C iqgen ξ DC 0 Ter Con,AC 0 0 ξ Ter 0 Ter Con,DC 0 0 ξ Gen 0 Ter Con,Ter 0 0 Ter Con,Gen (4.79) The variables ξ AC,ξ DC,ξ Ter,ξ Gen are used to indicate the N AC line outage, D DC line outage, L terminal station outage and I generator outage states respectively. Every single contingency is explicitly expressed through its own decision variable leading to the following relation for the total decision variable: ξ corr = ξ 0 +ξ AC +ξ DC +ξ Ter +ξ Gen b iq0 b iqac b iqdc b iqter b iqgen C eq,total ξ corr = b eq,total C eqbase C eqac C eqdc C eqter 0 0 C eqgen ξ 0 ξ AC ξ DC ξ Ter ξ Gen = b eq0 b eqac b eqdc b eqter b eqgen (4.80) The b iq constant inequality vector changes at the entries regarding contingency states. For instance if one is considering short term ratings for AC and DC lines, then the corresponding limits of the inequality vector contain increased values. The b eq constant equality vector is a repetition of the original base-case vector for all cases since no loss of load scenarios are included. The large number of contingencies that are being addressed result in matrices that are very sparsely populated.

60 42CHAPTER 4. SECURITY CONSTRAINED OPF IN THE MIXED GRID

61 Chapter 5 Results-Case Studies In this chapter the methods that have been described in chapter 4 are applied on test cases. Simulations for sensitivity analysis then follow accompanied by comments and explanations regarding the results. The last section is reserved for conclusions and a discussion about future improvements and implementations. In the following sections the base quantity for power is S base = 100 MW since only active power quantities are considered, therefore 1 p.u refers to 100 MW. All simulations regarding the results that follow were done using the quadprog solver of the MATLAB optimization toolbox. 5.1 System Description - Case Study I The test case that is studied is a hybrid grid that was envisioned in [1], and is shown in figure 5.1. It is the IEEE 14 bus test case with an interconnected 5-bus DC grid. The technical characteristics of the AC grid can be found in the equivalent test case of [13], while for the DC grid all lines have resistance equal to R km = 2.78Ω and terminals have capacities equal to 1 p.u. The two grids are interconnected at four points, where the terminals are situated. Bus 25 of the DC grid is not connected to a terminal and is a pure DC bus. System Characteristics The power system is supplied by five generators that are connected at AC buses. Generator 1 has the smallest cost coefficients making it the cheapest option for power supply. In the range of power demand of the system generator 2 follows as the next cheapest option, while generators 3,4 and 5 have the same higher operating costs. The highest power demands are situated at buses 3 and 4, therefore the general flow of power has direction and origin from the bottom left part of figure 5.1 towards the bottom right. 43

62 44 CHAPTER 5. RESULTS-CASE STUDIES 1 12 VSC VSC VSC VSC 7 AC Line Generator Voltage source converter DC Line Transformer VSC 3 Figure 5.1: Combined AC and DC grid [1] 5.2 Method I: Preventive The first method to be assessed is the preventive SC-OPF. The model provides an optimal power flow solution that is secure against any single contingency event of an AC line, DC line or a generator. The effects of every contingency from the sets of these elements are considered, regardless of whether they result in constraint violations or are not severe at all. Simulations for the base OPF and the preventive SC-OPF for case 14 result in operating costs of $/h and $/h respectively Generation Profiles The target of the objective function is the minimization of generator costs. The optimization depends on the quadratic and linear cost coefficients of the generators, their physical limits and from the level of demand. Figure 5.2 shows the generation profiles of the base OPF and SC-OPF. In the basecase generator 1 undertakes most of the generation as the cheapest unit with the largest capacity, while generator 2 also participates with a much smaller output. In the SC-OPF though, the security constraints require for a more

63 5.2. METHOD I: PREVENTIVE Base OPF Preventive SC OPF Generator Limit Power Output [p.u] Generator Figure 5.2: Generation Profile for Preventive SC-OPF distributed power dispatch as can be seen from the figure. Generator 1 now produces 70 MW less power while units 3,4 and 5 which are shut down in the base OPF now participate in the power generation Power Flow Distribution in the Mixed Grid Figure5.3 displaysthedistributionof flowson AClines fortheopfandsc- OPF.InthesecondcasethepowerflowingontheACgridismoredistributed over all the lines. Lines that carry power of cheap generating units are less loaded so that they can operate safely under a contingent situation. An example that displays this behaviour is the set of power flows on lines 1 2 and 1 5 which are connected to the bus with the cheapest generating unit. In the SC-OPF the power flowing on these lines drops from 0.82 and 0.39 p.u for the base OPF case to and p.u. This way both lines can operate within their limits if the other line is outaged. Due to the topology of the grid if either of these lines is out, the other compensates and transfers the initial power that was flowing on the outaged line. Notice that the sum ofthepowerflowingonlines1 2 and1 5is 1p.uintheSC-OPF.Therefore in the event one of the two lines is outaged, the other line will have a power flow of 1 p.u which is the limit of all AC lines of the system. The algorithm finds the optimum solution under security constraints given the fact that the paths that transfer power from cheap generating units are congested during the outage of a single element of the system. The distribution of power flows in the DC grid is shown in figure 5.4 The topology of the DC grid and the fact that post-contingency terminal control is not performed in the preventive method, intensely handicap the

64 46 CHAPTER 5. RESULTS-CASE STUDIES 1 Base OPF Preventive SC OPF Line Limit Power Flow [p.u] AC Line from bus k to m Figure 5.3: Distribution of Line Flows in the AC Grid 0.5 Base OPF Preventive SC OPF Line Limit Power Flow [p.u] DC Line from bus k to m Figure 5.4: Distribution of Line Flows in the DC Grid (Preventive) total power flowing on the DC grid in the SC-OPF case. The total injection of power into the DC grid drops from 1 p.u to 0.5 p.u to ensure secure operation in the case of unexpected contingencies. Terminal 1 that injects power in the DC grid has to halve its transfer rate to make sure that neither of DC lines 1 25 or 31 1 suffer from overloads if one of them is outaged.

65 5.3. METHOD II: PREVENTIVE-CORRECTIVE 47 The results can be seen in figure Base OPF Preventive SC OPF Terminal Limit 0.6 Power Flow [p.u] Terminal connected at Bus Figure 5.5: Terminal Station Power Transfers (Preventive) 5.3 Method II: Preventive-Corrective In the following sections several cases are envisioned regarding the type of contingencies that are included in the preventive-corrective SC-OPF problem. These are : Case 1: SC-OPF that accounts for all AC and DC line outages. Case 2: SC-OPF that accounts for AC and DC line outages, and generator outages. Case 3: SC-OPF that accounts for AC and DC line outages, as well as outages of terminal stations. Case 4: SC-OPF that accounts for AC and DC line outages, terminal outages and generator outages. Generation Profiles The dispatching of power generation among the participating units for the cases that are considered is depicted in figure 5.6. In all cases generator 1 is required to reduce its power output. Generator 2 has a small increase in the power it is supplying, and generators 3,4 and 5 also participate in

66 48 CHAPTER 5. RESULTS-CASE STUDIES the supply of the loads. Compared to the generation profile of figure 5.2 however, the shift in generation levels between base OPF and SC-OPF cases is not so severe. In the preventive method there has to be an enforcement of preventive measures to place the system in a secure state only through changes in the generation dispatch. The corrective method provides more flexibility due to corrective actions that can be implemented in a possible contingent situation. This means that the generation dispatch is not required to change as much as in the preventive method. Notice how in case 4 of the corrective SC-OPF, where all contingency scenarios are included, generator 1 is providing the system with MW in pre-contingency steady state. In the preventive case generator 1 has an output of 150 MW, and in this case constraints for terminal outages are not accounted for. So the corrective method that includes security constraints for a greater set of contingencies offers a cheaper solution ( $/h) than the preventive method ( $/h) Base OPF Case 1 Case 2 Case 3 Case 4 Generator Limit Power Output [p.u] Generator Figure 5.6: Generation Profile for Prev-Corr SC-OPF for 100% Load Case 1: AC-DC line outages Case 2: AC-DC line and generator outages Case 3: AC-DC line and terminal outages Case 4: AC-DC line, terminal and generator outages Costs Comparison The next indicative quantity to be addressed is the operational cost of the system in each of the cases described. What is clear from figure 5.8 is that the inclusion security constraints for line outages leads to a more expensive system operation. A further augmentation to include terminal contingency

67 5.3. METHOD II: PREVENTIVE-CORRECTIVE 49 Power Output [p.u] Base OPF Case 1 Case 2 Case 3 Case 4 Generator Limit Generator Figure 5.7: Generation Profile for Prev-Corr SC-OPF for 160% Load Case 1: AC-DC line outages Case 2: AC-DC line and generator outages Case 3: AC-DC line and terminal outages Case 4: AC-DC line, terminal and generator outages scenarios further increases operational costs. In this case the total generating capacity is rather large compared to the demand and that is why the augmentation of security constraints for the generator outages does not give rise to extra costs. Thus the most severe of generator outages is not in fact causing any post-contingency violations which means that no re-dispatch is required. An increase in the total load of the system by 60% results in the operational costs of figure 5.9. The notation 100% Load refers to the original loading of the IEEE system which is 259 MW. The figure depicts the effects an increased demand has on the system costs. Unlike 5.8, here the augmentation of constraints for system security during generator outages has a notable influence on the total costs. Under heavier loading situations the generators have large power outputs, which makes the outage of a unit a more severe and stressful phenomenon. Hence the preventive measures needed in these circumstances result in an even more conservative and costly dispatch of generation. It is also interesting to point out that while Case 2 differs from Case 1 in a small amount, Case 4 differs from Case 3 much more. This highlights the dependency of system performance from terminal operation. The inclusion of terminal contingencies affects the amount of corrective actions that can be taken during a disruption as well as the actual power that is fed and extracted from the DC grid. Tables 5.1 and 5.2 list the cost of security for

68 50 CHAPTER 5. RESULTS-CASE STUDIES $/h Base OPF (Case 0) Lines (Case 1) Lines+Gen (Case 2) Lines+Terminals (Case 3) All Contingencies (Case 4) Figure 5.8: Operational Costs for 100% Load Case 1: AC-DC line outages Case 2: AC-DC line and generator outages Case 3: AC-DC line and terminal outages Case 4: AC-DC line, terminal and generator outages the various cases and two loading levels. It can be an indicative value for power system planning and development. Table 5.1: Cost of Security for the Cases studied for 100% Load Scenario Type of Contingencies included Operational Cost [$/h] Case 0 No Security Case 1 Lines Case 2 Lines,Generators Cost of Security [$/h] Case 3 Lines,Terminals Case 4 Lines,Generators,Terminals It is interesting to compare Case 2 of the corrective SC-OPF method to the preventive method since the set of contingencies accounted for is the same. Figure 5.10 shows the cost of the corrective approach with varying limits of expendable terminal corrective control. That is, the points on x-

69 5.3. METHOD II: PREVENTIVE-CORRECTIVE 51 $/h x Base OPF (Case 0) Lines (Case 1) Lines+Gen (Case 2) Lines+Terminals (Case 3) All Contingencies (Case 4) Figure 5.9: Operational Costs for 160% Load Case 1: AC-DC line outages Case 2: AC-DC line and generator outages Case 3: AC-DC line and terminal outages Case 4: AC-DC line, terminal and generator outages Table 5.2: Cost of Security for the Cases studied for 160% Load Scenario Type of Contingencies included Operational Cost [$/h] Case 0 No Security Case 1 Lines Cost of Security [$/h] Case 2 Lines,Generators Case 3 Lines,Terminals Case 4 Lines,Generators,Terminals axis depict the percentage of allowed change in terminal power transfers in post-contingency states. The maximum (100%) permissible corrective control is here equal to the the capacity of the terminal stations, which is 1 p.u. The cost of the preventive approach is plotted on the same graph and is a horizontal line since no corrective actions are allowed in the preventive

70 52 CHAPTER 5. RESULTS-CASE STUDIES method. As the amount of allowed corrective actions is reduced the cost of the corrective SC-OPF rises and as this amount approaches 0 the solution of the corrective method converges to the solution of the preventive method. This is to be expected because with no corrective actions allowed the second method is practically the same problem as described by the first method. The difference then lies in the modelling and formulations of the problems Case 2 Corrective Case 2 Preventive 7850 $/h P T /P max T Figure 5.10: Linking of Preventive and Corrective SC-OPF Interaction between AC and DC Grid Figure 5.11 shows the line flows in the AC grid. In the security constrained cases the power flows are more distributed over the lines. Lines that would usually be congested, or loaded very close to the their capacity, are less heavily loaded to avoid any violations in the event of a contingency. The difference is compensated by other lines which in the normal OPF would be lightly loaded. Be that as it may, the ability to make corrective actions through the terminals means that lines that are connected close to cheap generating units can in fact be loaded close to their limits, because in the event of a contingency the terminals can alleviate possible over-loadings. So with the limits of the AC lines at 1 p.u and given the fact that generator 1 is the cheapest unit, one would expect lines 1 2 and 1 5 to be more heavily loaded. The reason they are not is owed to the fact that the DC grid is more intensely loaded as can be seen in figure 5.12, where DC lines 1 25 and 13 1 are both at their limit of 0.5 p.u. However observe that at case 1 and 2, lines 1 2 and 1 5 are delivering 0.75 and 0.37 p.u power. If no corrective actions were implemented during an outage of either of thetwolines, theline operating wouldbe forced to deliver

71 5.3. METHOD II: PREVENTIVE-CORRECTIVE p.u power, which would be unacceptable. Through the application of corrective actions 12 MW will be shifted to the DC grid in the event of a contingency of line 1 2 or 1 5. The equivalent holds for the DC grid, where lines 1 25 and 13 1 are loaded at 0.44 p.u. An outage of one of the two lines will result in congestion of the operating line (0.5 p.u), and the additional power will be fed back into the AC grid to avoid constraint violations. Therefore the implementation of corrective actions makes it possible for generator 1, the cheapest unit, to produce significantly more power compared to a fully preventive dispatch. Another observation is that the cases which include generator outages, i.e cases 3 and 5 do not display significant differences compared to cases 2 and 4 respectively. This is justified by the fact that the total load of the system is rather small compared to the total capacity of the generators, so the augmentation of extra security for generator outages to cases 2 and 4 does not lead to a more expensive dispatch. Power Flow [p.u] 1 Base OPF Case Case 2 Case 3 Case Line Limit AC Line from bus k to m Figure 5.11: Line Flows in the AC Grid for 100% Load Case 1: AC-DC line outages Case 2: AC-DC line and generator outages Case 3: AC-DC line and terminal outages Case 4: AC-DC line, terminal and generator outages The flows in the DC grid shown in figure 5.12 present a slightly different picture than the distribution of flows in the AC grid. Here the security constrained cases have reduced line flows in almost all of the lines. With the given topology the total power flowing in the DC grid for the constrained cases is actually less, so the redistribution results in the majority of lines being less loaded. One can also observe this fact in figure 5.13 where it is evident that the terminals are feeding the DC grid with less power in the security constrained cases. However in comparison to the preventive method

72 54 CHAPTER 5. RESULTS-CASE STUDIES Base OPF Case1 Case 2 Case 3 Case 4 Line Limit Power Flow [p.u] DC Line from bus k to m Figure 5.12: Line Flows in the DC Grid for 100% Load Case 1: AC-DC line outages Case 2: AC-DC line and generator outages Case 3: AC-DC line and terminal outages Case 4: AC-DC line, terminal and generator outages and the results of figure 5.4 the amount of power flowing in the DC grid is substantially larger. Terminal corrective control can deal with clearing many of the violations that occur when a DC line is outaged. It follows that the DC grid can in fact be more heavily loaded with the corrective approach than with the preventive one. Terminal connected at AC bus 1 is operating as a rectifier in all simulations and is feeding power into the DC grid since generator 1 is the largest and cheapest unit. The rest of the terminals are operating as inverters, feeding the power flowing in the DC grid back into the AC grid to supply the loads. For a loading level of 1.6 times the original one the power flows can be seen in figures 5.14, 5.15 and 5.16 The lines in the AC grid are transferring more power compared to the case in figure The DC lines however have similar distribution and loading levels to the case of figure 5.12 because the DC grid was already stressed to its limit even for the original conditions. Therefore the AC grid takes up the slack to deliver the additional power Terminal Control Sensitivity Analysis To evaluate the effects of terminal corrective control, simulations are run for varying loading levels and different security constrained cases. For a designated loading level and case, the amount of corrective control that can

73 5.3. METHOD II: PREVENTIVE-CORRECTIVE 55 Power Flow [p.u] 1 Base OPF Case Case 2 Case 3 Case Terminal Transfer Limit Terminal connected at Bus Figure 5.13: Terminal Station Power Transfers for 100% Load Case 1: AC-DC line outages Case 2: AC-DC line and generator outages Case 3: AC-DC line and terminal outages Case 4: AC-DC line, terminal and generator outages Power Flow [p.u] Base OPF Case 1 Case 2 Case 3 Case 4 Line Limit AC Line from bus k to m Figure 5.14: Line Flows in the AC Grid for 160% Load Case 1: AC-DC line outages Case 2: AC-DC line and generator outages Case 3: AC-DC line and terminal outages Case 4: AC-DC line, terminal and generator outages be expended affects the behaviour of the system. Figure 5.17 shows the effects of corrective control on the system costs for case 1. The maximum amount of allowed change in power transfers PT max is here considered to be

74 56 CHAPTER 5. RESULTS-CASE STUDIES Base OPF Case 1 Case 2 Case 3 Case 4 Line Limit Power Flow [p.u] DC Line from bus k to m Figure 5.15: Line Flows in the DC Grid for 160% Load Case 1: AC-DC line outages Case 2: AC-DC line and generator outages Case 3: AC-DC line and terminal outages Case 4: AC-DC line, terminal and generator outages Power Flow [p.u] 1 Base OPF Case Case 2 Case 3 Case Terminal Transfer Limit Terminal connected at Bus Figure 5.16: Terminal Station Power Transfers for 160% Load Case 1: AC-DC line outages Case 2: AC-DC line and generator outages Case 3: AC-DC line and terminal outages Case 4: AC-DC line, terminal and generator outages equal to the capacity of the terminals. The simulations were performed for a set of values for P T, or equivalently u c if the notation of section is used. How these values reflect on the parameters of equation 2.13 is out

75 5.3. METHOD II: PREVENTIVE-CORRECTIVE 57 of the scope of the current work. When the amount of corrective actions that can be expended is limited, the generation costs rise. This can be seen in the first part of the graph, where tighter limits on the allowed change in terminal power transfers result in higher costs. The costs are normalized to the costs the system has when the terminals are assigned 100% flexibility. Therefore the normalized costs of the y-axis are described by the relation: Normalized Cost = Cost( P T = x) Cost(P max T = x) (5.1) Thesteepnessof thecurvedependsontheloading ofthesystem. As theload increases, the steepness of the curve decreases. This happens because the system is closer to congestion, meaning that the flexibility of the terminals hasareducedimpactsinceall linesarealready closer to theirlimits. Whatis % Load 120% Load 140% Load 160% Load Normalized Cost P T /P max T Figure 5.17: Effects of Terminal Control on Costs: Case 1 interesting is the fact that after a certain point in the graph, approximately when P T /PT max = 0.25 the increase in flexibility does not result in reduced costs. After this point the terminal corrective actions can t further decrease the cost of operation. This is due to the fact that for these simulations the penalization of the AC grid is much more severe than the penalization of the DC grid. Thus the DC grid is congested because it is the cheapest option and this creates an upper bound, after which any increase in the amount of corrective control can t further optimize the solution.

76 58 CHAPTER 5. RESULTS-CASE STUDIES Infigure5.18thesamesimulationsasinfigure5.17areperformedforcase 2 that includes line outages and terminal outages. In this case the amount of corrective controls allowed have a greater impact on system costs. This is evident from the increased normalized costs and by the fact that the region wherein the amount of corrective actions allowed affect the costs is larger than in case % Load 120% Load 140% Load 160% Load Normalized Cost P T /PT max Figure 5.18: Effects of Terminal Control on Costs: Case 2 To display the effects a congested DC grid has on terminal control and flexibility, simulations for Π AC = Π DC are carried out. Figure 5.20 refers to case 2 with an application of equal penalties for AC angle differences and DC voltage differences. A first remark to be made is that compared to figure 5.18 the relative costs are lower when the terminals are firmly constrained to their pre-contingency settings. This is to be expected seeing as the power flowingin the system is more distributed between the two grids and thus reduced loading of the DC grid doesn t affect operating costs as much. However, unlike figure 5.18 the system costs continue to decrease in proportion to increased expendable control of the terminals, almost until the maximum allowed value of P T /PT max. This behaviour is observed because there is more room for terminal corrective actions when the DC grid is not congested. This means that the system in this case can benefit from severe changes in post-contingency controls.it follows that one or more violations in post-contingency states of the system can be entirely cleared through corrective control of terminal flows instead of using preventive generation dispatching to achieve the security level desired.

77 5.3. METHOD II: PREVENTIVE-CORRECTIVE % Load 120% Load 140% Load 1.07 Normalized Cost P T /P max T Figure 5.19: Effects of Terminal Control on Costs: Case % Load 120% Load 140% Load 160% Load 1.05 Normalized Cost P T /PT max Figure 5.20: Effects of Terminal Control for Π AC = Π DC : Case 2

78 60 CHAPTER 5. RESULTS-CASE STUDIES A comparison of absolute costs invoked by these two levels of penalty costs is shown in figure 5.21 for 100% load. Figures 5.19 and 5.22 portray simulations for case 4 where all contingencies are accounted for. The results are similar to the simulations for case Π AC =30,Π DC =0.1 Π AC =Π DC = Absolute Cost($/h) P T /PT max Figure 5.21: Comparison of Absolute Costs for 2 levels of Π AC,Π DC : Case DC Grid Capacity and System Performance The amount of power the DC lines and the terminals are able to transmit undoubtedly influence the overall performance of the system. Seeing that the DC networksare still at an early stage, some insight in the way technical specifications such as capacities affect costs is really useful. The first simulation considers a large amount of combinations for the limits PT max and FDC max and is performed for the non-secure base OPF. The result is the surface plot of figure Along the x-axis the quantity that changes is the terminal capacity, while along the y-axis it is the DC line capacity. Each pair of these values induce a cost that is depicted on z-axis. Very close to the axes the cost of operation rises dramatically as was expected. For a fixed value of terminal capacity the steepness of the curves running along y-axis depict the sensitivity to DC line capacities. The same applies for curves running along x-axis for sensitivity to terminal capacities. The steepness of the former is greater, though the curves related to the DC

79 5.4. SOLVING TIMES % Load 120 % Load 140 % Load Normalized Cost P T /P max T Figure 5.22: Effects of Terminal Control for Π AC = Π DC : Case 4 line capacities even out horizontally much earlier than the curves related to terminal capacities. Thus operating costs depend more on terminal capacities, which is logical given the fact that power flowing through the terminals is a prerequisite to power flowing on DC lines. In figure 5.24 the same analysis is done as in figure 5.23, only for Case 2. The two surface plots are similar, though in Case 2 the costs continue to decrease for higher values of capacities. In figure 5.23 there is an area wherein increase in capacities does not further minimize costs, leading to a horizontal area, but with the inclusion of security measures the system can further benefit from increased capacities. This is also evident from the fact that the difference between highest and lowest cost is twice as big in the security constrained case than in the base-case. 5.4 Solving Times A significant advantage of using linear models is the fact that the solvers available obtain the solution very fast. The solution for the base OPF is obtained in 7 10 milliseconds whereas the full non-linear OPF for the same problem requires approximately 2.5 seconds to reach a solution [1]. The solving times for all cases that are studied are seen in figure 5.26

80 62 CHAPTER 5. RESULTS-CASE STUDIES 7950 Absolute Cost($/h) DC Line Capacity (p.u) Terminal Capacity (p.u) 0 Figure 5.23: Dependency of Operational Costs from Capacities for Base OPF Absolute Cost($/h) DC Line Capacity (p.u) Terminal Capacity (p.u) 0 Figure 5.24: Dependency of Operational Costs from Capacities for Case Case Study II The next test case to be studied is a larger grid compared to the one studied in sections It is the RTS96 test case [14] with an added DC grid.

81 5.5. CASE STUDY II (a) Base-Case (b) Case 2 Figure 5.25: Projection of figures 5.23 and 5.24 for comparison of dependency on terminal and DC line capacities

82 64 CHAPTER 5. RESULTS-CASE STUDIES Time[sec] Base OPF (Case 0) Preventive Lines+Gen (Case 1) Lines+Gen (Case 2) Lines+Terminals All Contingencies (Case 3) (Case 4) Figure 5.26: Calculation times for all cases The DC grid consists of eight DC buses, six of which are interconnected to the AC grid through a terminal, and sixteen DC lines. The system is seen in figure Figure 5.27: IEEE RTS-96 with interconnected DC grid

83 5.5. CASE STUDY II 65 The costs induced by the cases described in the beginning of section 5.3 are gathered in table 5.3, along with the time required for the solver to obtain the solutions. Table 5.3: Cost of Security and Solving Times for Test Case of Figure 5.27 Scenario Type of Contingencies included Operational Cost [$/h] Cost of Security [$/h] Solving Time [sec] Case 0 No Security Case 1 Lines Case 2 Lines,Generators Case 3 Lines,Terminals Case 4 All Contingencies Figure 5.28 depicts the effects of post-contingency terminal control on the operating costs. Contrary to the first case which was presented earlier, the amount of corrective control allowed has no noticeable effect on operational costs for the system under consideration. Several cheap generating units are distributed over the entire system and the buses where they are placed have many interconnecting lines with adjacent buses. This means that the system is robust. The topology of the system and its technical characteristics might contribute to the fact that operational costs are independent of terminal corrective control. It is probable that a different positioning of the terminals could lead to a situation where operational costs depend on the amount of post-contingency corrective actions. The reduced flexibility of the system is highlighted by another interesting observation, derived by comparing tables 5.1 and 5.3. If the cost of including security measures for all contingencies (case 4) is expressed in relation to the cost of the base-opf, then for the IEEE 14-bus system: Cost 14 (%) = Cost case4 Cost base Cost base Equivalently for the RTS-96 system: = = 2.4% (5.2) Cost RTS 96 (%) = Cost case4 Cost base Cost base = = 3.86% (5.3) It is proportionately more expensive to provide security for the RTS-96 system than for the IEEE 14-bus system. Solving times in this case increase a lot. But Cases 2 and 4 that require

84 66 CHAPTER 5. RESULTS-CASE STUDIES % Load 115 % Load 125 % Load 1.6 Normalized Cost P T /P max T Figure 5.28: Effects of Terminal Control for Case 4 approximately 30 seconds to reach a solution are accounting for more than 200 contingencies, so the solution is obtained fast for the size of the problem. Effective contingency screening could significantly reduce computational time, since most of the non-binding contingencies would be neglected.

85 Chapter 6 Conclusion and Discussion This chapters serves as a summary of key points and gives an outlook of possible future research challenges. 6.1 Conclusion In the current work two methods of applying the security constrained optimal power flow in a mixed AC-DC grid were presented. The preventive SC-OPF is a tool that could prove to be very useful in the future of power system operation. A robust formulation and solution to this problem is a state of the art challenge since an effective method could significantly enhance the stability and reliability of power systems. The additional operating costs that are induced by a preventive dispatch are insignificant compared to the prospect of a guaranteed supply. Extreme failures like major blackouts can be avoided, along with the severe economic consequences these situations impel. The preventive-corrective method generally poses as a cheaper alternative to the fully preventive method. Corrective actions implemented after the occurrence of a contingency are usually attributed zero costs because they only have to be implemented rarely. However these actions have to be performed based on a set of viable control strategies developed by the system operator. If these strategies prove to be poor, preventive measures needed to ensure security could be more expensive than the equivalent measures of a preventive method. The importance of post-contingency terminal control with respect to operating costs was analytically described in the previous chapter. Flexible changes in the power exchanged between the AC and DC grid are suitable to clear a substantial amount of post-contingency constraint violations. These corrective actions provide a cheaper solution than a preventive MW dispatch of generation. For such control schemes to exist, the appropriate strategies have to be 67

86 68 CHAPTER 6. CONCLUSION AND DISCUSSION planned and the physics behind such real time operations of VSC terminals have to be studied extensively. It can be argued that with the development of liberalized power markets the level of security of power systems has been weakened [2]. Especially with the introduction of intra day markets, there is substantial pressure not to interfere with the market. Under these circumstances the use of corrective control is favoured over that of preventive rescheduling. 6.2 Future Work A future extension of the proposed methods would be a formulation of the SC-OPF that encompasses non-linear effects. A non linear OPF problem formulation for the mixed grid is shown in [15]. Further development to include security constraints for this more precise problem seems promising.

87 Appendix A Numerical Example In this section a numerical example of the preventive-corrective SC-OPF in the mixed grid is presented. The test case is a small custom system that is shown in figure A.1. All system quantities are presented in the tables that follow: Table A.1: Generator Data Gen No At AC Bus Pmax [p.u] Pmin [p.u] Linear Cost Coefficient Quadratic Cost Coefficient Table A.2: AC Line Data Line No From AC Bus To AC Bus Reactance [p.u] Rating [p.u] Table A.6 contains the decision variable ξ for the base OPF and the precontingency settings ξ 0 of Case 1 that includes security constraints for AC and DC line outages. The distribution of power flows is registered in table A.7. The results for Cases 3 and 4 can be found in tables A.8 and A.9 69

88 70 APPENDIX A. NUMERICAL EXAMPLE G2 PL3 3 G3 PL2 VSC T3 3 T2 VSC T1 VSC 1 PL1 G1 Figure A.1: Small Test Grid

89 71 Table A.3: DC Line Data Line No From DC Bus To DC Bus Resistance [Ω] Rating [p.u] Table A.4: Terminal Station Data Terminal No AC Bus DC Bus Rating [p.u] Table A.5: Active Power Demand P L1 [p.u] P L2 [p.u] P L3 [p.u] P L4 [p.u]

90 72 APPENDIX A. NUMERICAL EXAMPLE Table A.6: Decision Variable ξ for Base OPF and Case 1 Base OPF Case 1 SC-OPF Cost of Operation $/h Cost of Operation $/h G G Generation Profile AC Bus Angles Terminal Transfers DC Bus Voltages G Generation Profile G G G δ 1 0 δ 1 0 δ δ AC Bus Angles δ δ δ δ P T P T P T Terminal Transfers P T P T P T V 1 0 V 1 0 V DC Bus Voltages V V V Table A.7: Line Flows for Base OPF and Case 1 SC-OPF Base OPF Case 1 SC-OPF Power Flow Power Flow AC Line DC Line AC Line Power Flow Power Flow DC Line

91 73 Table A.8: Decision Variable ξ for Case 3 and Case 4 Case 3 SC-OPF Case 4 SC-OPF Cost of Operation 6981 $/h Cost of Operation $/h G G Generation Profile AC Bus Angles Terminal Transfers DC Bus Voltages G Generation Profile G G G δ 1 0 δ 1 0 δ δ AC Bus Angles δ δ δ δ P T P T P T Terminal Transfers P T P T P T V 1 0 V 1 0 V DC Bus Voltages V V V Table A.9: Line Flows for Case 3 and Case 4 SC-OPF Case 3 SC-OPF Case 4 SC-OPF Power Flow Power Flow AC Line DC Line AC Line Power Flow Power Flow DC Line

92 74 APPENDIX A. NUMERICAL EXAMPLE

93 Appendix B Example with Matrix Formulations To get a clear idea of the way the various matrices are formulated, a simple 3 AC bus 3 DC bus system of figure B.1 is studied. All three AC buses are interconnected with a DC bus through a terminal station. The matrices that are needed to formulate the fully preventive SC-OPF problem are going to be explicitly defined. The dimensions of the matrices to follow are according to the nomenclature of table 3.1: The equality constraints, which only account Table B.1: Nomenclature for Numerical Example K=3 Number of AC nodes N=3 Number of AC lines M=3 Number of DC nodes D=3 Number of DC lines I=2 Number of generators L=3 Number of terminals C=9 Length of vector ξ Y prev =58 Amount of inequality constraints for the pre-contingency state as documented in section 4.1.5, are constructed as follows: H = , T = , S =

94 76 APPENDIX B. EXAMPLE WITH MATRIX FORMULATIONS G2 3 PL2 VSC T3 3 T2 VSC T1 VSC 1 PL1 G1 Figure B.1: 3 Bus Test Grid B AC = , B DC =