Convex Relaxations for Optimization of AC and HVDC Grids under Uncertainty

Transcription

1 P L Power Systems Laboratory Center for Electric Power and Energy Department of Electrical Engineering Andreas Horst Venzke Convex Relaxations for Optimization of AC and HVDC Grids under Uncertainty Master Thesis PSL1619 PSL Power Systems Laboratory Swiss Federal Institute of Technology (ETH) Zurich CEE Center for Electric Power and Energy Technical University of Denmark (DTU) Examiner: Supervisors: Prof. Dr. Gabriela Hug Uros Markovic Lejla Halilbasic (DTU) Prof. Dr. Pierre Pinson (DTU) Prof. Dr. Spyros Chatzivasileiadis (DTU) Zurich, March 7, 2017

2 Abstract High penetration of renewable energy sources and the increasing share of stochastic loads require the explicit representation of uncertainty in tools such as the optimal power flow (OPF). Fundamental to power system operation, the OPF problem is routinely solved for a variety of applications such as market-based dispatch or system security assessment. These OPF calculations are performed without taking uncertainty explicitly into account. As the occurring deviations in the output of renewables and load demand can be substantial, an operation without awareness of uncertainty can lead to costly corrective measures and jeopardize system security. This necessitates the incorporation of the resulting uncertainty into OPF. The main contribution of this thesis is a convex formulation of the chance constrained OPF for combined AC and HVDC grids. A convex relaxation technique is used to transform the non-linear, non-convex AC-OPF problem into a semi-definite program (SDP). In order to account for uncertainty, chance constraints are included in the OPF formulation. To ensure tractability of these constraints, an affine policy is used to model the change of the full system state as a function of the forecast errors. This allows to accurately model large deviations in the output of renewables and define a-priori corrective control policies related to active and reactive power, voltages, and HVDC converter set-points. The convex formulation of the chance constrained OPF for AC grids is proposed for two types of uncertainty sets. For a rectangular uncertainty set, it is sufficient to enforce the chance constraints at its vertices. Alternatively, assuming a multivariate Gaussian distribution of the forecast errors, an analytical reformulation of the linear chance constraints and a tractable approximation of the semi-definite chance constraints is presented. The convex formulation for a rectangular uncertainty set is extended to include HVDC grids and an HVDC converter model. i

3 In order to obtain zero relaxation gap, i. e. the minimum of the SDP relaxation corresponds to the global minimum of the non-linear, non-convex AC-OPF problem, the affine policy is modified and a penalty factor on the generator droops is introduced. Near-global optimality guarantees are obtained for the penalized OPF formulation. Additionally, an alternative approach to obtain a tractable formulation of the chance constraints is proposed. A linearization using power transfer distribution factors (PTDFs) is used to estimate the change of the line and generator loading as a function of the forecast errors, and serves as a benchmark. Several case studies on the IEEE 9 bus system, a 10 bus system with an HVDC line and the IEEE 24 bus system with a multi-terminal DC grid are presented. The proposed approach using the affine policy achieves compliance with the full set of AC power flow constraints for the entire uncertainty set for all test cases, whereas the alternative approach using PTDFs violates voltage and branch flow constraints. The corrective control policies, which are included in the convex OPF formulation with the affine policy, enable the reduction of the generation cost for most of the test cases. The distance to the global optimum is upper bounded by a minimum near-global optimality guarantee of 99.0% for the presented case studies. Key Words: Convex optimization, AC optimal power flow, semi-definite programming, uncertainty, chance constraints, HVDC grids, wind power, corrective control, power system operations ii

4 Contents List of Figures List of Tables List of Symbols vi x xii 1 Introduction Motivation Literature Review Convex Relaxations of Optimal Power Flow Chance Constrained Optimal Power Flow Optimal Power Flow for Combined AC and HVDC Grids Main Contributions Thesis Structure Optimal Power Flow Formulation for Combined AC and HVDC Grids General AC Optimal Power Flow Semi-Definite Relaxation of AC Optimal Power Flow Modeling of HVDC Converter and HVDC Grids Overview of Converter Technologies Modeling Assumptions HVDC Converter Semi-Definite Relaxation of Optimal Power Flow for Combined AC and HVDC Grids Chance Constraints for AC Optimal Power Flow Extension of Chance Constraints to HVDC Grids Convex Relaxations of Chance Constrained AC Optimal Power Flow Affine Policy Corrective Control Policies Generator Droop Control (AGC) iii

5 CONTENTS iv Generator Voltage (AVR) Control Wind Farm Reactive Power Control Tractable Formulation for Rectangular Uncertainty Set Tractable Formulation for Gaussian Uncertainty Set Convex Relaxations of Chance Constrained Optimal Power Flow for Combined AC and HVDC Grids Affine Policy Corrective Control with HVDC Converter Tractable Formulation for Rectangular Uncertainty Set Linearization using Power Transfer Distribution Factors (PTDFs) PTDFs for Chance Constrained AC Optimal Power Flow Tractable Formulation for Rectangular Uncertainty Set Tractable Formulation for Gaussian Uncertainty Set PTDFs for Chance Constrained Optimal Power Flow for Combined AC and HVDC Grids Corrective Control with HVDC converter Tractable Formulation for Rectangular Uncertainty Set 45 6 Case Studies Simulation Setup Simulation Parameters Evaluation of Constraint Violation Evaluation of Droop Penalty IEEE 9 bus system Rectangular Uncertainty Set Gaussian Uncertainty Set bus system Rectangular Uncertainty Set Rectangular Uncertainty Set with HVDC Line IEEE 24 bus system Rectangular Uncertainty Set Rectangular Uncertainty Set with MTDC Grid Extensions: Generator Droops as Optimization Variables Discussion Conclusion Summary Outlook Appendices

6 CONTENTS v A Schur Complement 90 B Combination of Scenario Approach and Robust Optimization 92 C Alternative Formulation of Second-Order Cone Constraints as Semi-Definite Constraints 94 D 10 Bus System Data 95

7 List of Figures 1.1 Yearly amount of redispatch measures in the German transmission grid from years 2010 to 2015 from a report by the German federal grid regulator (Bundesnetzagentur) [1]. The shown unit is hours per year Illustration of the concept of convex relaxation: The function f(x) resembles the original non-convex, non-linear OPF problem and f 1 (x), f 2 (x) are possible convex relaxations. The relaxation f 1 (x) is tight and achieves zero relaxation gap Venn diagram of the solution sets for the original AC-OPF and the SOC, QC and SDP relaxations reproduced from [2]. Note that these sets are not drawn to scale Flow chart depicting thesis structure. Note: AC-HVDC-OPF refers to an AC-OPF for combined AC and HVDC grids Model of HVDC VSC station based on [3] Active and reactive power capability curve of HVDC converter: The blue area denotes the possible operating region Addition of large resistance between AC and DC side of the converter to obtain zero relaxation gap Modification of affine policy: The linearization between upper and lower limit is split into two corresponding linearizations starting from the exact operating point W 0. The red line indicates the true system behavior and the dashed lines the approximation which is made with the corresponding affine policy. This modification allows to obtain the exact solution W 0, not the approximation W Generator droop control with slack variables γi u and γi l to account for non-linearity of loss change Reactive power capability of wind farm Q Wi as a function of active power P Wi vi

8 LIST OF FIGURES vii 3.4 Rectangular uncertainty set displayed for two wind farms. It is sufficient to enforce the chance constraints at the vertices of the uncertainty set. The variables B i are transformed into B i as depicted in gray. This procedure can be extended to arbitrary numbers of wind farms in the network with growing number of vertices 2 n W Normal distribution of two independent forecast errors ζ 1 and ζ 2 with standard deviations σ 1 and σ 2 and probability P. The smaller ellipsis corresponds to 95.45% of samples (two standard deviations) and the larger ellipsis to 99.73% of samples (three standard deviations) Uncertainty set resulting from a multivariate Gaussian distribution of the forecast errors and the directions of approximation for the affine policy. The circles denote the points for which the semi-definite constraint is enforced. As a result, it holds for the whole dotted rectangular shape. The indices (I) (IV) denote the four quadrants of the uncertainty set for each of which the complete set of chance constraints (3.35) (3.39) is included Corrective control with HVDC converter: A set-point is introduced for upper and lower bounds of the forecast errors and a piece-wise linear interpolation is performed between the set-points. The bus k is located at the AC side of the converter Approximation of the apparent branch flow constraint S lm with PTDFs: The variation of the reactive power flow on the transmission line is neglected Corrective control with HVDC converter by including an active power set-point change PC i for each wind infeed i. The reactive power set-point remains unchanged as in the DC linearization reactive power is not modeled Modified IEEE 9-bus system with wind farms W1 and W Eigenvalue ratios, generation cost and droop penalty as a function of the droop penalty weight µ for the IEEE 9 bus test case for rectangular uncertainty set Eigenvalue ratios evaluated on the rectangular uncertainty set for the IEEE 9 bus test case. Zero relaxation gap is obtained at the vertices and at the operating point Eigenvalue ratios, generation cost and droop penalty as a function of the droop penalty weight µ for the IEEE 9 bus test case for a Gaussian uncertainty set

9 LIST OF FIGURES viii 6.5 Eigenvalue ratios evaluated on the Gaussian uncertainty set for the IEEE 9 bus test case. Zero relaxation gap is obtained at the end-points of the ellipsis axes and the operating point. The red dot denotes the worst-case scenario bus system with two wind farms W1 and W2 located at buses 4 and 10. An HVDC line with converters C1 and C2 is added between buses 2 and 10 to relieve congestion on the parallel AC transmission line Generation cost for the 10 bus test case with two uncertain infeeds as a function of the forecast error magnitudes. A system configuration with and without HVDC line from bus 2 to bus 10 is examined Eigenvalue ratios, generation cost and droop penalty as a function of the droop penalty weight µ for the 10 bus test case. A 50% variability in wind power output is assumed bus system: Comparison of uncertainty margins of power lines. Note the difference for the congested line 5 from bus 2 to bus 10. The affine policy results in 0% uncertainty margin for the congested line bus system: Corrective voltage control with generator at node 2 as a function of the wind infeeds P W1 and P W bus system: Corrective reactive power control with wind farm at node 10 as a function of the wind infeeds P W1 and P W bus system: Line flow and voltage violations as a function of the variability for the linearization using PTDFs Eigenvalue ratios, generation cost and droop penalty as a function of the droop penalty weight µ for the 10 bus test case with an additional HVDC line. A 50% variability in wind power output is assumed bus system with HVDC line: Comparison of uncertainty margins of power lines. Note the difference for the congested line 5 from bus 2 to bus 10. The affine policy results in 0% uncertainty margin for the congested line bus system with HVDC line: Active and reactive power set-points of converters C1 and C2 for the 10 bus system with forecast error magnitude of 50%. The set-points for the linearization using PTDFs and the corrective control of the affine policy are depicted. The dashed blue lines indicate the converter reactive and apparent power limits bus system with HVDC line: Violations as a function of the variability for the linearization using PTDFs

10 LIST OF FIGURES ix 6.17 IEEE 24 bus system: In the first test case, two wind farms W1 and W2 are located at nodes 8 and 24. In the second test case, a multi-terminal HVDC grid with offshore converter C1 and onshore converters C2 and C3 is connected to the AC buses 6 and 13, respectively. The offshore converter C1 is connected via an AC line to the offshore wind farm W3. A second wind farm W4 is placed at bus Eigenvalue ratios, generation cost and droop penalty as a function of the droop penalty weight µ for the 24 bus test case. A 50% variability in wind power output is assumed Generation cost for the IEEE 24 bus system with two wind farm as a function of the variability in the output of the wind farms IEEE 24 bus system: Comparison of uncertainty margins of transmission lines IEEE 24 bus system: Violations as a function of the variability in wind farm ouput for the linearization using PTDFs Eigenvalue ratios, generation cost and droop penalty as a function of the droop penalty weight µ for the 24 bus test case with a MTDC grid. A 50% variability in wind power output is assumed. The penalty factor corresponding to W 2 is weighted with an additional factor of 2.5 to obtain zero relaxation gap IEEE 24 bus system with MTDC grid: Comparison of uncertainty margins of transmission lines Comparison of the active and reactive power set-points of the HVDC converters for 24 bus test case with MTDC grid. The upper figures correspond to the offshore converter (C1), the figures in the middle to the converter at bus 6 (C2) and the figures at the bottom to the converter at bus 13 (C3) Eigenvalue ratios, generation cost and droop penalty as a function of the droop penalty weight µ for the 24 bus test case with generator droops as optimization variable. A 50% variability in wind power output is assumed B.1 Illustration how upper and lower bounds on the forecast errors are retrieved by using the scenario approach [4]. The green circles represent scenarios

11 List of Tables 6.1 Simulation parameters IEEE 9 bus system: Linearization using PTDFs (rectangular uncertainty set) IEEE 9 bus system: Affine policy (rectangular uncertainty set) IEEE 9 bus system: Linearization using PTDFs (Gaussian uncertainty set) IEEE 9 bus system: Affine Policy (Gaussian uncertainty set) bus system: Line flow on congested line from bus 2 to bus bus system: Generator voltage set-points for rectangular uncertainty set bus system: Maximum voltage violations for linearization using PTDFs bus system: Selected penalty factors for affine policy bus system with HVDC line: Line flow on congested line from bus 2 to bus bus system with HVDC line: Maximum violations for linearization using PTDFs bus system with HVDC line: Selected penalty factors for affine policy IEEE 24 bus system: Maximum violations for linearization using PTDFs Selected penalty factor for IEEE 24 bus for rectangular uncertainty set Comparison of objective function and droop penalty for 24 bus test system including MTDC grid Comparison of line loading on line from bus 6 to bus 10 and constraint violations for 24 bus test system including MTDC grid Generator droops as optimization variable: Comparison of generation cost x

12 LIST OF TABLES xi D.1 Generator, load and transmission line parameters with perunit base power of 1000 MVA from [5]

13 List of Symbols N Set of buses in the power network L Set of lines in the power network G Set of generators in the power network c k2 Quadratic generation cost of generator k c k1 Linear generation cost of generator k c k0 Constant generation cost of generator k P Gk Active power generation at bus k Q Gk Reactive power generation at bus k V k Voltage at bus k P lm Active branch flow on line (l, m) S lm Apparent branch flow on line (l, m) Y Admittance matrix of power grid e k k-th basis vector ȳ lm Shunt line admittance of line (l, m) y lm Series line admittance of line (l, m) V Complex bus voltage vector X Real and imaginary bus voltage vector V r Real part of the voltage V i Imaginary part of the voltage P injk Active power injection at bus k Q injk Reactive power injection at bus k P Dk Active power load at bus k Q Dk Reactive power load at bus k n b Number of buses in the power network W Matrix with product of real and imaginary part of voltages W opt Optimal voltage matrix (rank-1) V opt Optimal voltage vector ρ 1 Largest eigenvalue of W ρ 2 Second largest eigenvalue of W E Eigenvector of W corresponding to largest eigenvalue ρ opt Largest eigenvalue of W opt E opt Eigenvector of W opt corresponding to largest eigenvalue Active power of converter at AC bus k P Ck xii

14 LIST OF SYMBOLS xiii Q Ck Reactive power of converter at AC bus k S Ck Apparent power of converter at AC bus k P Cs Active power of converter at DC bus s C Set of converter buses in the power network R Tf Converter transformer resistance at filter bus f X Tf Converter transformer reactance at filter bus f B f Converter filter susceptance at filter bus f R Cf Converter phase reactor resistance at filter bus f X Cf Converter phase reactor reactance at filter bus f θ Ck Converter AC terminal voltage m Voltage modulation factor of converter Ploss,k conv Active power converter loss at converter k a k Quadratic converter loss component of converter k b k Linear converter loss component of converter k c k Constant converter loss component of converter k I k Converter current magnitude of converter k SC nom k Maximum apparent power of converter k m b Factor for maximum absorbed reactive power of converter k m c Factor for maximum injected reactive power of converter k n W Number of wind farms W Set of buses with wind farms P Wi Wind infeed at bus i P f W i Forecasted wind infeed at bus i ζ i Wind forecast error at bus i ɛ Confidence interval W (ζ i ) System matrix as a function of the forecast errors W 0 Matrix with products of voltage for forecasted system state W 0 Approximated system state X Change of real and imaginary bus voltage vector B i Voltage change for forecast error i Bi u Voltage change for upper limit on forecast error i Bi l Voltage change for lower limit on forecast error i d Generator droop vector γi u Droop slack variable for upper limit on forecast error i γi l Droop slack variable for lower limit on forecast error i µ Weight for droop penalty cos(φ) Power factor τ Ratio of maximum reactive power to active power v Vertices of rectangular uncertainty set V Set for vertices v B v Matrix describing voltage change for vertex v

15 LIST OF SYMBOLS xiv ζ v Forecast error for vertex v of uncertainty set W v Matrix with products of voltage for vertex v γ v Droop slack variable for vertex v n v Number of vertices v σ Standard deviation P Probability distribution Φ 1 Inverse Gaussian function κ Limit on Gaussian forecast error n AC bus Number of AC buses in the power network n DC bus Number of DC buses in the power network PTDF lm Power Transfer Distribution Factor for line (l, m) x lm Reactance for line (l, m) B AC PTDF admittance matrix for AC grid B AC Reduced PTDF admittance matrix for AC grid Pinj i Change in bus power injections for wind infeed i P inj i Reduced change in bus power injections for wind infeed i r lm Resistance for line (l, m) B DC PTDF admittance matrix for DC grid B DC Reduced PTDF admittance matrix for DC grid z lm Resistance or reactance for line (l, m) B ACDC Reduced admittance matrix for combined AC and DC grids P C i Change in converter active power set-points for wind infeed i P T Change of active power set-point of HVDC converter ρ Ratio of second to third eigenvalue of W δ opt Near-global optimality measure V G Generator terminal voltages P G Violation of upper generation limit P G Violation of lower generation limit µ v Droop penalty factor for vertex v d v,opt k Optimal droop vector for each vertex v G droop set of generators participating in droop control n G number of generators participating in droop control N s Number of samples for scenario approach e Euler number β Confidence parameter Pk lim Upper and lower limit on active power Q lim k Upper and lower limit on reactive power Vk lim Upper and lower limit on voltage magnitude Upper and lower limit on active branch flow P lim lm

16 Chapter 1 Introduction 1.1 Motivation A significant increase in renewable energy, mainly photovoltaic and wind energy, is observed all over the world. In 2015, the installed global capacity of solar PV reached 227 GW and of wind energy 433 GW [6]. In order to meet the CO 2 reduction targets of the European Union, it can be expected that wind and photovoltaic energy will constitute a major part of the electricity mix in Europe. Both of these renewable resources are volatile, hence their actual power generation deviates from the forecast. Additionally, the number of uncertain electrical loads is growing, such as electric vehicles [7]. As the shares of uncertain generation and load increase, it is necessary to incorporate the resulting uncertainty into power system operation. Furthermore, as a consequence of the increase in renewables and growing electricity demand, the power system is operated closer to its limits [8]. Significant investment in new transmission capacity and an improved utilization of existing assets are necessary. The High Voltage Direct Current (HVDC) technology is a promising candidate for enabling increased penetration of volatile renewable energy sources and providing controllability in power system operation. In China, in order to transport bulk power, e.g. wind, from geographically remote areas to load centers, significant HVDC transmission capacity has been built. By 2013, 24 HVDC transmission projects have been commissioned or have been in operation across China [9]. These could be extended to a North-East Asian Supergrid in the future [10]. In order to balance the fluctuations in the output of renewables and transfer power over large distances, a European HVDC grid is envisioned, extending the several point-to-point HVDC connections already in operation in Europe to a multi-terminal HVDC grid [11]. 1

17 CHAPTER 1. INTRODUCTION 2 Figure 1.1: Yearly amount of redispatch measures in the German transmission grid from years 2010 to 2015 from a report by the German federal grid regulator (Bundesnetzagentur) [1]. The shown unit is hours per year. Fundamental to power system operation, the optimal power flow (OPF) problem is routinely solved for a variety of applications such as marketbased dispatch or system security assessment. These OPF calculations are performed without taking uncertainty explicitly into account. However, the increasing share of renewables in the electricity mix and the associated forecast errors necessitate the incorporation of the resulting uncertainty into OPF. As the occurring deviations in the output of renewables are substantial, an operation without awareness of uncertainty can lead to costly corrective measures and jeopardize system security. Power system operators have to deal with the resulting higher degrees of uncertainty in operation and planning. In Fig. 1.1 the yearly amount of redispatch measures in the German transmission grid is depicted for the years 2010 to Redispatch refers to actions taken by the transmission system operators (TSOs) to actively change the generation schedule of power plants in order to ensure system security. A significant increase in these measures is observed over this time span. The main reasons are the phasing out of nuclear power plants in Germany, the substantial increase in renewables and the associated increased reactive power demand [1]. These developments highlight the need to introduce an AC-OPF formulation which includes HVDC grids and is able to accurately model the effect of large deviations in the output of renewables on the power system

18 CHAPTER 1. INTRODUCTION 3 operation [12]. Furthermore, a-priori suitable control policies for active and reactive power, voltages and HVDC converter set-points should be defined, which result in a safe system operation inside defined uncertainty sets. 1.2 Literature Review This section provides a literature review of the three main topics dealt with in this thesis. First, an overview of convex relaxations of the optimal power flow problem is provided. Second, chance constraints are included in the OPF formulation to account for uncertainty in power injections. Third, the modeling of HVDC grids in the OPF formulation is discussed and relevant literature is presented Convex Relaxations of Optimal Power Flow In general, the AC-OPF is a non-convex, non-linear problem. As a result, identified solutions are not guaranteed to be globally optimal and the distance to the global optimum cannot be specified. Recent advancements in the area of convex optimization with polynomials have achieved to relax the non-linear, non-convex optimal power flow problem and transform it to a convex semi-definite or second-order cone problem [13 15]. Formulating a convex optimization problem results in tractable solution algorithms that can determine the global minimum. Within power systems, finding the global minimum has two important implications. First, from an economic point of view, it can result in substantial cost savings as even a small difference between a local and the global minimum can reflect a significant cost in absolute terms. Second, from a technical point of view the global optimum determines a lower or an upper bound of the required control effort. This property can be utilized in branch-and-bound methods for mixed integer formulations (e.g. unit commitment, or topology changes). In Fig. 1.2 the concept of convex relaxation is illustrated. The function f(x) resembles the objective function of the non-convex, non-linear OPF which can have multiple local minima besides the global minimum. By using a convex relaxation technique, this function is transformed into a convex function, e. g. f1 (x) or f 2 (x). The term relaxation is used as some of the constraints of the original problem are relaxed or omitted. As a consequence, the possible solution space is enlarged and the solution to the convex relaxation is always a lower bound on the global optimum of the original non-convex problem. The term relaxation gap denotes the difference between the minimum of the convex relaxation and the global minimum of the original non-convex problem. A relaxation is tight, if the relaxation gap

19 CHAPTER 1. INTRODUCTION 4 Cost f(x) f 1 (x) f 2 (x) Figure 1.2: Illustration of the concept of convex relaxation: The function f(x) resembles the original non-convex, non-linear OPF problem and f 1 (x), f 2 (x) are possible convex relaxations. The relaxation f 1 (x) is tight and achieves zero relaxation gap. x QC SOC AC SDP Figure 1.3: Venn diagram of the solution sets for the original AC-OPF and the SOC, QC and SDP relaxations reproduced from [2]. Note that these sets are not drawn to scale. is small. A relaxation is exact, if the relaxation gap is zero, i. e. zero relaxation gap is achieved and the minimum of the convex relaxation corresponds to the global minimum of the original non-convex, non-linear problem. Although the relaxation gap is not always zero, several relaxations of the OPF have proven to be tight for a series of problems. The three most notable convex relaxation techniques are the second-order cone (SOC) [14], quadratic convex (QC) [2] and semi-definite programming (SDP) [13,15]. In Fig. 1.3, the Venn diagram of the solution space of these relaxations from the work in [2] is shown. Note that these sets are not drawn to scale. The work in [2] shows that the QC relaxation is stronger than the SOC relaxation but neither dominates nor is dominated by the SDP relaxation. For tree networks, under some conditions, it can be proven that the SDP relaxation is always exact [16]. For mesh networks, such a proof has not been yet

20 CHAPTER 1. INTRODUCTION 5 presented. The work in [17] introduces a penalty factor on reactive power to obtain a tight relaxation for mesh networks with near-global optimality guarantees. In this thesis, the focus is on the semi-definite (SDP) relaxation technique as it is tighter than the SOC relaxation and more literature exists than for the QC relaxation Chance Constrained Optimal Power Flow Literature considers uncertainty either through stochastic formulations or in the form of chance constraints [18]. Stochastic formulations do not need to make assumptions on the type of uncertainty, but they depend on the number of considered scenarios, and they usually result in a higher computational burden. Chance constraints, on the other hand, can provide analytical guarantees about the probability of constraint violation and are usually less computationally intensive, but they often require the assumption of specific uncertainty distributions. Both approaches increase the complexity of the non-convex, non-linear AC-OPF. This thesis focuses on the chance constrained OPF. To deal with higher complexity, due to the uncertain variables, existing approaches using chance constraints either assume a DC-OPF or solve iteratively linearized instances of an AC-OPF. Note that the DC-OPF formulation is an approximation and neglects both voltage and reactive power constraints which can lead to substantial errors in the outcome of OPF [19]. Both works in [20] and [21] formulate a chance constrained DC-OPF assuming a Gaussian distribution of the forecast errors. The work in [20] relies on a cutting-plane algorithm to solve the resulting optimization problem, whereas the work in [21] states a direct analytical reformulation of the same chance constraints. This framework is further extended by the work in [22] which assumes uncertainty sets for both the mean and the variance of the underlying Gaussian distributions to obtain a more robust result. Alternatively, the works in [23 26] use the full set of AC power flow equations to describe the forecasted system state and a linearization to achieve a tractable formulation of the chance constraints. The works in [23] and [24] adopt a linearization using the DC power flow. As the operating point is not known a-priori, the linearization is performed around a flat start or no load voltage, not the actual operating point. In [24], the mean and covariance matrix of the forecast errors is updated online to obtain a more robust performance. In the work from [25], an iterative back-mapping and linearization of the full AC power flow equations is used to solve the chance constrained AC-OPF. The recent work in [26] uses an iterative procedure to calculate the full Jacobian, the exact AC power flow linearization, around

21 CHAPTER 1. INTRODUCTION 6 the operating point. In this thesis, a framework is developed for the chance constrained AC-OPF with HVDC grids which includes the full set of AC power flow equations for the chance constraints, but does not rely on an iterative procedure or a linearization around the operating point. This allows to accurately model the change of the full set of the AC state variables due to forecast errors and define suitable control policies regarding active and reactive power, voltages, and HVDC converter set-points. The work in [27] makes a first step towards an OPF formulation for AC grids which takes into account security related constraints and uncertainty. The full AC power flow equations are utilized by means of the SDP relaxation technique. The change of the system state is described as an explicit function of the forecast errors with an affine policy. To ensure tractability of the chance constraints, a combination of the scenario approach and robust optimization is used. The scenario approach is used to compute bounds for the uncertainty set. Then, the chance constraints are enforced at the vertices of the resulting uncertainty set based on [4]. The possibility for changing generator voltage set-points is included in the formulation Optimal Power Flow for Combined AC and HVDC Grids There is a growing interest in building a meshed HVDC grid to transfer electricity from offshore wind farms and remote solar plants across Europe, as it is the most viable option both technically and economically for longer distances. HVDC technology is regarded as a key technology for the further integration of renewables [28]. Several works in the literature address the challenge of integrating HVDC grids in the AC-OPF formulation. The work in [29] introduces a generalized steady-state Voltage Source Converter (VSC) multi-terminal DC (MTDC) model which can be used for sequential AC/DC power flow algorithms. In the sequential method, the power flow for each AC and DC grid is computed separately and the coupling constraints are iterated back and forth to obtain a solution. This method is available as a software package for MATPOWER, called MATACDC, which will be used in this thesis to evaluate constraint violations [30]. Several works have included security constraints in the OPF formulation for combined AC and multi-terminal HVDC grids to evaluate the potential of corrective control with HVDC [31 34]. The application of chance constraints in the context of combined AC and HVDC grids is so far mainly limited to the DC-OPF formulation [35, 36]. However, the work in [37] proposes a stochastic AC-OPF with wind farms connected by line-commutated HVDC lines. Convex relaxation techniques have been applied to the OPF problem for combined AC and HVDC grids [3, 38]. The work in [38] uses second-

22 CHAPTER 1. INTRODUCTION 7 order cone programming to obtain a convex formulation of the AC-OPF with VSC HVDC. The work in [3] introduces a convex formulation of the OPF problem for combined AC and HVDC grids, using the semi-definite relaxation technique in [13]. A converter loss model is included and penalty factors are introduced in order to obtain zero relaxation gap. In this thesis, the formulation of [3] is integrated in modified form in the OPF formulation. 1.3 Main Contributions The scope of this thesis is to introduce a convex OPF formulation for combined AC and HVDC grids which is able to accurately model the effect of large deviations in the output of renewables and load demand, can define a-priori suitable control policies for both active power and reactive power, voltage and HVDC converter set-points, and can determine the global optimum. The main contributions of this thesis are the following: A tractable formulation of the chance constrained AC-OPF using an affine policy is proposed for two types of uncertainty sets. Given an a-priori specified rectangular uncertainty set, it suffices to enforce the chance constraints at its vertices. Assuming a Gaussian distribution of the forecast errors, an analytical reformulation of the linear scalar chance constraints and a tractable approximation of the semi-definite chance constraints are presented. Corrective control policies are included in the formulation. Besides the possibility to adjust the generator voltage set-points, the reactive power capabilities of wind farms are included. The tractable formulation for a rectangular uncertainty set is extended with a model for HVDC grids and HVDC converter. A converter loss model which depends quadratically on the converter current and the reactive power capabilites of the HVDC converter are included. Furthermore, a corrective control policy for the active and reactive power set-points of HVDC converter adjusts the set-points based on the realized forecast errors. The relaxation gap of the obtained solution is investigated in detail, which to the author s knowledge has not been done for convex relaxations of the chance constrained OPF for AC grids or combined AC and HVDC grids. For this purpose, a penalty factor on power losses is introduced which allows to obtain near-global optimality guarantees. Additionally, the affine policy in [27] is modified to be exact at the operating point by splitting it into an upper and lower part. As a result, for a rectangular uncertainty set, zero relaxation gap is obtained at the vertices and at the operating point. Similarly, for a Gaussian

23 CHAPTER 1. INTRODUCTION 8 uncertainty set, zero relaxation gap is obtained at the operating point as well as at the end-points of the ellipsoid axes. The presented approaches are evaluated on several test cases. The affine policy is compared to an alternative approach based on a linearization with power transfer distribution factors (PTDFs) similar to [20 24]. It is shown that the affine policy is capable of accurately modeling large deviations in the output of renewables and of including a control response of generators, wind farms, and HVDC converter, while the optimization problem remains convex. The linearization using PTDFs fails to comply with branch flow and voltage constraints for both uncertainty sets. The proposed approach using the affine policy results in a safe operation for both uncertainty sets and complies with the full set of AC power flow constraints. Furthermore, it is shown that the droop penalty is small in practice, leading to tight near-global optimality guarantees of the obtained solution. This thesis resulted in the following publication: Andreas Venzke, Lejla Halilbasic, Uros Markovic, Gabriela Hug and Spyros Chatzivasileiadis, Convex Relaxations of Chance Constrained AC Optimal Power Flow, submitted, available: abs/ Thesis Structure The structure of this thesis is illustrated in Fig. 1.4 with a flow chart: In Chapter 2, the general AC-OPF formulation is stated and the procedure to obtain the semi-definite relaxation is explained. The general AC-OPF formulation is extended with a model for HVDC grids and the semi-definite relaxation is applied to yield a convex problem. Chapter 3 introduces the affine policy and includes corrective control policies in the convex chance constrained AC-OPF formulation. A modification of the affine policy and a droop penalty are introduced to obtain zero relaxation gap. A tractable formulation for a rectangular and a Gaussian uncertainty set is proposed. Chapter 4 extends the chance constraints to include HVDC grids and corrective control of the HVDC converter set-points. For a rectangular uncertainty set, a tractable convex formulation for the chance constrained OPF for combined AC and HVDC grids is stated. In Chapter 5, an alternative approach using the so-called power transfer distribution factors (PTDFs) is explained. Several case studies are presented in Chapter 6 and the approach using the affine policy is compared with the linearization using PTDFs. Chapter 7 provides a summary of the thesis and gives some further directions of research.

24 CHAPTER 1. INTRODUCTION 9 Chapter 2 AC Grids HVDC Grids Convex Relaxation of AC-OPF [13] Convex Relaxation of AC-HVDC-OPF [3] Uncertainty in Power Injections Chance Constraints Chapter 3 Chapter 4 Affine Policy based on [27] Explicit Formulation w. r. t. the Forecast Errors Corrective Control Voltage Generator Droops Reactive Power HVDC Gaussian Uncertainty Set (AC-OPF) Rectangular Uncertainty Set (AC-OPF) Rectangular Uncertainty Set (AC-HVDC-OPF) Tractable Formulation Chapter 5 Chapter 6 Chapter 7 Linearization using PTDFs Case Studies Conclusion Figure 1.4: Flow chart depicting thesis structure. Note: AC-HVDC-OPF refers to an AC-OPF for combined AC and HVDC grids.

25 Chapter 2 Optimal Power Flow Formulation for Combined AC and HVDC Grids This chapter states the convex formulation of the optimal power flow problem for combined AC and HVDC grids. Starting from the general formulation of the non-convex, non-linear AC optimal power flow problem, the convex relaxation from [13] using semi-definite programming is stated. The modeling of HVDC grids and HVDC converters connecting AC and DC grids is explained. Finally, the convex formulation of the optimal power flow for combined AC and HVDC grids based on [3] is presented. 2.1 General AC Optimal Power Flow The optimal power flow problem is an optimization, where an objective function, e. g. generation cost or transmission losses, is minimized subject to the physical constraints of the power network. This problem was first formulated as an economic dispatch problem in 1962 by Carpentier [39]. Let N denote the set of buses of the power network, G the set of buses with a connected generator and L the set of lines connecting the individual buses. The OPF typically minimizes the following generation cost: min k G c k2 P 2 G k + c k1 P Gk + c k0 (2.1) The terms c k2, c k1 and c k0 are the quadratic, linear and constant cost associated with power production of generator k, denoted with P Gk. At the minimum generation cost, the physical network constraints and power flow equations need to be satisfied. These can be stated in general form for each 10

26 CHAPTER 2. OPF FORMULATION FOR AC AND HVDC GRIDS 11 bus k N and line (l, m) L: P Gk P Gk P Gk (2.2) Q Gk Q Gk Q Gk (2.3) V k V k V k (2.4) P lm P lm (2.5) S lm S lm (2.6) The generator active P Gk and reactive power Q Gk are constrained by (2.2) and (2.3). The terms P Gk, P Gk, Q Gk and Q Gk denote the generator limits for minimum and maximum active and reactive power, respectively. The bus voltages V k are constrained by (2.4) with corresponding lower and upper limits V k, V k. The active branch flow P lm on line (l, m) L is limited by the maximum active branch flow P lm (2.5). The apparent branch flow S lm is limited by the maximum apparent branch flow S lm (2.6). Depending on the application, either the active or the apparent branch flow constraint is included. The following auxiliary variables are introduced for each bus k N and line (l, m) L: Y k := e k e T k Y (2.7) Y lm := (ȳ lm + y lm )e l e T l (y lm )e l e T m (2.8) Y k := 1 [ R{Yk + Yk T } I{Y k T Y ] k} 2 I{Y k Yk T } R{Y k + Yk T } (2.9) Y lm := 1 [ R{Ylm + Ylm T } I{Y lm T Y ] lm} 2 I{Y lm Ylm T } R{Y lm + Ylm T } (2.10) Ȳ k := 1 [ I{Yk + Yk T } R{Y k Yk T } ] 2 R{Yk T Y k} I{Y k + Yk T } (2.11) Ȳ lm := 1 [ I{Ylm + Ylm T } R{Y lm Ylm T } ] 2 R{Ylm T Y lm} I{Y lm + Ylm T } (2.12) [ ek e M k := T ] k 0 0 e k e T (2.13) k [ (el e M lm := m )(e l e m ) T ] 0 0 (e l e m )(e l e m ) T (2.14) Matrix Y denotes the admittance matrix of the power grid, e k the k-th basis vector, ȳ lm the shunt line admittance of line (l, m) L and y lm the series line admittance. There exist different equivalent formulations of the AC power flow equations. This section focuses on the rectangular power-voltage formulation as

27 CHAPTER 2. OPF FORMULATION FOR AC AND HVDC GRIDS 12 stated in e. g. [40] and [41]. From this formulation the semi-definite relaxation of the OPF can be naturally developed. In general, the power flow equations are modeled as a function of the complex bus voltages V. The state variables in the rectangular OPF formulation are the real (V r ) and imaginary part (V i ) of the complex bus voltages. These are summarized in the vector X: X : = [V r 1 V r 2 V r n V i 1 V i 2 V i n] (2.15) = [R{V} I{V}] T (2.16) The AC power flow equations can be stated for each bus k N and line (l, m) L: P injk = P Gk + P Dk = X T Y k X (2.17) Q injk = Q Gk + Q Dk = X T Ȳ k X (2.18) P lm = X T Y lm X (2.19) Q lm = X T Ȳ lm X (2.20) The nodal active (2.17) and reactive power flow balance (2.18) determine the injected active P injk and reactive power Q injk at each bus k N. The terms P Dk and Q Dk are the active and reactive power consumption at bus k N. The sum of generation and load has to match the net sum of injected power flows for both active and reactive power. The equations (2.19) and (2.20) describe the active and reactive branch flow (P lm, Q lm ) on a transmission line. 2.2 Semi-Definite Relaxation of AC Optimal Power Flow In this section, the convex relaxation of the OPF problem as first formulated in [13,15] is outlined. For detailed proofs of the results, the interested reader is referred to [13]. The initial step is a variable transformation. The matrix W is introduced: W = XX T (2.21) The dimension of the matrix W is 2n b 2n b, where n b is the number of buses in the power network. The resulting elements of matrix W are the

28 CHAPTER 2. OPF FORMULATION FOR AC AND HVDC GRIDS 13 product of the real and imaginary part of the complex voltages: V1 rv 1 r V1 rv 2 r V1 rv n r V1 rv 1 i V1 rv 2 i V1 rv n i V2 rv 1 r V2 rv 2 r V2 rv n r V2 rv 1 i V2 rv 2 i V2 rv n i W = Vn r V1 r Vn r Vn r Vn r V1 i Vn r Vn i V1 rv 1 i V1 rv 2 i V1 rv n i V1 iv 1 i V1 iv 2 i V1 iv n i V2 rv 1 i V2 rv 2 i V2 rv n i V2 iv 1 i V2 iv 2 i V2 iv n i Vn r V1 i Vn r Vn i VnV i 1 i VnV i n i (2.22) For the Hermitian matrix W the following equality holds for an arbitrary real matrix A R 2n b 2n b : Tr{AW } = Tr{AXX T } = X T AX (2.23) This result from linear algebra (2.23) allows to express the power flow equations (2.17) (2.20) as a function of W : P Gk = Tr{Y k W } P Dk (2.24) Q Gk = Tr{ȲkW } Q Dk (2.25) P lm = Tr{Y lm W } (2.26) Q lm = Tr{ȲlmW } (2.27) V 2 k = Tr{M kw } (2.28) The reformulated OPF problem is stated according to [13] as minimizing the objective min k G{c k2 (Tr{Y k W } + P Dk ) 2 + c k1 (Tr{Y k W } + P Dk ) + c k0 } (2.29) subject to the following reformulated constraints (2.2) (2.6) for each bus k N and line (l, m) L: P Gk P Dk Tr{Y k W } P Gk P Dk (2.30) Q Gk Q Dk Tr{ȲkW } Q Gk Q Dk (2.31) V 2 k Tr{M kw } V 2 k (2.32) P lm Tr{Y lm W } P lm (2.33) Tr{Y lm W } 2 + Tr{ȲlmW } 2 (S lm ) 2 (2.34)

29 CHAPTER 2. OPF FORMULATION FOR AC AND HVDC GRIDS 14 In order to obtain an optimization problem linear in W, the objective function (2.29) is reformulated using the Schur complement: min k G α k (2.35) [ ] ck1 Tr{Y k W } + a k ck2 Tr{Y k W } + b k 0 (2.36) ck2 Tr{Y k W } + b k 1 where a k := α k + c k0 + c k1 P Dk and b k := c k2 P Dk. In addition, the apparent branch flow constraint (2.34) is rewritten using the Schur complement: (S lm ) 2 Tr{Y lm W } Tr{ȲlmW } Tr{Y lm W } (2.37) Tr{ȲlmW } 0 1 The interested reader is referred to Appendix A for the proof of (2.37) which is not included in [13]. The constraint (2.21) can be expressed by: W 0 (2.38) rank(w ) = 1 (2.39) The previously stated formulation (2.29) (2.39) is mathematically equivalent to the initial OPF formulation provided in Section 2.1. The convex relaxation is introduced by dropping the non-convex rank constraint (2.39). The resulting optimization problem with optimization variable W can be formulated as min k G α k (2.40) s.t. (2.30) (2.33), (2.36) and (2.37) (2.41) W 0 (2.42) The work in [13] states that the relaxation gap between this optimization problem and the original non-convex, non-linear OPF is zero if and only if the resulting matrix W has rank-1, denoted with W opt. A necessary condition to obtain zero relaxation gap is the connectivity of the underlying resistive graph describing the power grid. Therefore, the addition of a small resistance of 10 4 p.u., to the transformer branches in the power grid is a necessary condition to obtain zero relaxation gap, but not a sufficient condition as shown in [42]. For the common IEEE test systems, a matrix W with rank-2 is obtained and the work [13] states a procedure to obtain the rank-1 solution. The rank-1 matrix W opt can be computed as follows: W opt = (ρ 1 + ρ 2 )EE T (2.43)