Optimized Risk Limits for Stochastic Optimal Power Flow

Transcription

1 power systems eehlaboratory Andrew Morrison Optimized Risk Limits for Stochastic Optimal Power Flow Master Thesis PSL1521 EEH Power Systems Laboratory Swiss Federal Institute of Technology (ETH) Zurich Examiner: Prof. Dr. Göran Andersson Supervisor: Sidhant Misra, Line Roald Zurich, March 29, 2016

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3 Abstract The increasing amount of uncontrollable renewable infeeds in the power system demand new security methods beyond the N-1 security conditions to ensure network stability. Chance constrained optimal power flow (CCOPF) is an optimization program designed to dispatch generators and reserves while considering the random fluctuations of uncontrollable power infeeds. The additional constraints of CCOPF bring several advantages over the traditional optimal power flow (OPF) formulation, such as the ability to limit line flow and generator overload violations to a specified value ɛ. However, choosing an appropriate value of ɛ is an iterative process which requires many simulation runs and extensive knowledge of the system being tested. This thesis proposes methods for implementing ɛ as a decision variable in the optimization in order to minimize both operational cost and system risk. In addition, a new type of chance constraint is proposed which reduces the number of decision variables and constraints while producing a similar result to the full formulation. Methods for introducing risk into the cost function are introduced, including a method which can reduce the cost of redispatching generation by over 36%. These methods are also computationally efficient, solving a 24-bus network on an ordinary desktop computer in less than five seconds. iii

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5 Acknowledgements This thesis is dedicated to my mother and father, who always motivated me to learn and made countless sacrifices to help me get to where I am today. v

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7 Contents List of Acronyms List of Symbols ix xi 1 Introduction Notation Optimal Power Flow Chance Constrained Optimal Power Flow Weighted Chance Constraints The Reserve Procurement Problem Epsilon in the Cost Function Weighting in the Objective Function Uniform Weighting Optimal Redispatch Weighting Multiobjective Optimization Formulations Standard Chance Constraint Formulation Combined Constraint Linear WCCs Formulation Combined Constraint Quadratic WCCs Formulation Combined Constraint Matrix α Formulation sum(r) Formulation CCOPF with Security Chance Constraints Implementation MATLAB Implementation Julia Implementation Cutting Planes for Weighted Chance Constraints Case Study Test System System Analysis with the ɛ Method vii

8 viii CONTENTS 6.3 Uniform vs. Optimal Redispatch Weighting Solo vs. Combined Constraints Tradeoff Between Line and Generator Security Linear vs. Quadratic WCCs Performance of CCOPF with Security Constraints Conclusion Appropriate Constraints for Lines and Generators Concluding Remarks A Appendix A - Formulations 71 A.1 Optimal Redispatch Weight Calculator Bibliography 73

9 List of Acronyms RES TSO OPF ACOPF SCOPF CCOPF CC WCC CDF Renewable Energy Sources Transmission System Operator Optimal Power Flow Alternating Current Optimal Power Flow Security Constrained Optimal Power Flow Chance Constrained Optimal Power Flow Chance Constraint Weighted Chance Constraint Cumulative Distribution Function ix

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11 List of Symbols ɛ p i d i v i ω i σ w B bus B f p ij c i α i,j M G V W E Φ(x) Security Parameter in the CCOPF Problem Scheduled Power Output of Generator i Scheduled Power Demand at Node i Forecasted Power Output of Fluctuating Infeed i Zero-Mean Fluctuating Component of Fluctuating Infeed i Standard Deviation of ω i Bus Susceptance Matrix Line Susceptance Matrix Line Flow Limit on Line ij Cost Per Unit of Energy from Generator i Response of Generator i to a Unit Deviation of Fluctuating Infeed j Matrix which Relates Nodal Power Injections to Line Flows The Set of all Generators in the System The Set of all Nodes in the System The Set of all Fluctuating Infeeds The Set of all Edges (Lines) in the System CDF of Standard Normal Distribution xi

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13 Chapter 1 Introduction The integration of renewable energy technologies such as wind turbines and solar panels has become an increasingly important field in grid studies today. While the prospect of environmentally clean and practically free energy is appealing, there is also an enormous drawback to these so-called Renewable Energy Sources (RES); namely, that their infeed cannot be controlled or predicted exactly. This leads to uncertainties in line flows and reserve capacities, which has lead to the grid being operated closer to its limits than ever before. Currently, Transmission System Operators (TSOs) use a set of software tools to calculate information about the behavior of the grid in order to aid in the decision making process. Optimal Power Flow (OPF) is one of these tools which uses optimization techniques to find the economically optimum generator dispatch to meet demand which does not violate generator or line limits. Several variations of the original Alternating Current Optimal Power Flow (ACOPF) have been formulated, including Security Constrained Optimal Power Flow (SCOPF), which incorporates the N-1 security conditions as constraints, and DCOPF, which reformulates the problem as a linear set of equations by using a set of assumptions about voltage magnitudes and angles throughout the system. ACOPF, SCOPF, and DCOPF are all deterministic problems, meaning that they cannot take into account the randomness of RES infeed. With the increasing penetration of RES, however, it has become more important to account for this uncertainty in the OPF problem. Already in the 1970 s [1] a method was developed to investigate nondeterministic infeeds and their effect on line flows in a power system. However, many of the equations developed were nonlinear and therefore very difficult to compute with the hardware available at the time. Today s hardware is much faster and the solution of nonlinear equations within optimization problems has now become feasible. In [2], the Chance Constrained Optimal Power Flow (CCOPF) method was developed, which is able to limit the 1

14 2 CHAPTER 1. INTRODUCTION probability of line and generator violations to a specified probability ɛ when non-deterministic infeeds are present. A further development to CCOPF was the introduction of Weighted Chance Constraints (WCCs) in [3], which use weighting functions to penalize greater limit violations more than smaller ones. This development, however, meant that the parameter ɛ no longer represented a probability, but a limitation of a parameter in varying units depending on how the constraints were weighted. One drawback to all of the CCOPF methods is that the specified parameter ɛ must be known before the optimization can be run. This paper aims to develop methods for solving the CCOPF problem with ɛ as an additional decision variable and show appropriate methods for penalizing ɛ in the cost function of the optimization. This is equivalent to finding the optimum trade-off between low operational cost and low risk. This thesis is organized as follows: Chapter 1 will outline the notation and the basics of optimal power flow. Chapter 2 will introduce the CCOPF formulation and its variations. In Chapter 3, strategies for implementing risk in the cost function will be discussed. In chapter 4, full formulations of the CCOPF variations used in this thesis will be introduced. Chapter 5 will detail the implementation and comment on the difficulties of implementation. The test system and results will be given and analyzed in Chapter 6. Chapter 7 concludes the thesis.

15 1.1. NOTATION Notation Presented here is an introduction to the notation used in this thesis. Vectors are denoted as lowercase letters (specific elements are denoted using subscripts) and matrices are denoted by bold case letters. ɛ p i d i v i ω i σ w B bus B f p ij c i α i,j M G V W E Φ(x) List of Symbols Security Parameter in the CCOPF Problem Scheduled Power Output of Generator i Scheduled Power Demand at Node i Forecasted Power Output of Fluctuating Infeed i Zero-Mean Fluctuating Component of Fluctuating Infeed i Standard Deviation of ω i Bus Susceptance Matrix Line Susceptance Matrix Line Flow Limit on Line ij Cost Per Unit of Energy from Generator i Response of Generator i to a Unit Deviation of Fluctuating Infeed j Matrix which Relates Nodal Power Injections to Line Flows The Set of all Generators in the System The Set of all Nodes in the System The Set of all Fluctuating Infeeds The Set of all Edges (Lines) in the System CDF of Standard Normal Distribution Often in this thesis, the matrix α i,j will be reduced to α i, which means that each generator responds simply to the sum of deviation of all fluctuating infeeds. Additionally, the matrix M is calculated by the product of the inverse of the bus susceptance matrix B bus and the line susceptance matrix B f [3]. This matrix, when multiplied by the nodal power flows, produces a vector of line flows. 1.2 Optimal Power Flow Optimal power flow is a tool used extensively in power system operational planning. As with any optimization problem, OPF includes an objective function to be minimized, a set of decision variables, and inequality and equality constraints. The objective function in the case of OPF is traditionally a sum of linear or quadratic generation costs which are calculated from the decision variables, which are typically the output power values of the generators in the system. The constraints can simulate various power system phenomena, such as thermal limits of lines and generators, as well as control policies. The N-1 conditions can also be formulated as constraints

16 4 CHAPTER 1. INTRODUCTION in an OPF, which makes it a very useful tool in the domain of power system security. The original formulation of the OPF problem is the ACOPF, which includes real power, voltage angle, voltage magnitude, and reactive power among its decision variables. This problem is highly nonlinear and nonconvex; researchers are still searching for a quick, globally optimal solution to the ACOPF problem for large networks. In this thesis, the so-called DC OPF was implemented, meaning that node voltages were assumed to be uniform, losses in the system were neglected, and the voltage angles between nodes were assumed to be small. This is done in order to speed up computation times since the DCOPF problem can be formulated using completely linear constraints and objective while the standard OPF cannot. The DCOPF problem can then be stated as the following, ignoring any fluctuation of infeeds: minimize p subject to c i p i (1.1) i G p i d i + v i = 0 (1.2) i V p i p max i, i G (1.3) p i p min i, i G (1.4) M ij, (p d + v) p ij, {ij} E (1.5) M ij, (p d + v) p ij, {ij} E (1.6) The objective function 1.1 aims to minimize the sum of generation costs across all generators, which is calculated for each generator as the product of a constant c i and the output power of the generator p i. Equation 1.2 enforces power balance across the system. The infeeds of traditional generators p and wind generators v must cover the load d. Equations 1.3 and 1.4 enforce the power generation limits for each generator i. Equations 1.5 and 1.6 enforce the upper and lower line flow limits for each line ij. The global optimum to this problem can be found efficiently, even for very large networks, as it includes only linear constraints, a linear objective, and is convex. There are several disadvantages to solving the DCOPF problem, however. It does not take into account any contingencies, reactive power effects, or non-deterministic infeeds. A development to the DCOPF is the SCOPF, which takes into account the N-1 conditions using additional constraints. Using sensitivity factors, the outage of a generator or line is compensated for by the remaining generators or lines. After this compensation, the limits of the remaining lines and generators must still be met. Such a set of constraints is added to take into account the outage of all generators and lines, which means that n (n 1) total constraints are added. The

17 1.2. OPTIMAL POWER FLOW 5 problem is still linear and efficiently solvable, though the N-1 conditions are not comprehensive enough to cover all likely contingencies. In addition, any non-deterministic infeeds are still not accounted for. If the randomness of wind infeeds is to be considered, the linearity and convexity of the DCOPF will be altered, a result which will be seen in the next chapter.

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19 Chapter 2 Chance Constrained Optimal Power Flow In order to model the effects of fluctuating wind infeeds, the variable ω needs to be introduced. Wind deviation ω is a zero-mean random variable which alters the total wind infeed ṽ such that ṽ = v + ω, (2.1) where v is the forecasted infeed. In this thesis, the wind deviation ω is modeled as a Gaussian random variable with mean zero and standard deviation σ w. However, ω can be modeled as other random variable types, as shown in [4]. A wind deviation /omega and standard deviation σ w can be defined for each wind infeed in the system, however in this thesis only a single total wind deviation from forecast ω and corresponding σ w will be used. If it is known that there is a random element to infeeds or loads in the system, a set of generators providing reserves can be modeled with the parameter α, which is a vector that dictates what percentage of wind deviation ω a generator i will compensate for, also known as participation factor. If the wind infeed deviates by an amount ω then the response of generator i (the change in generation) is α ij ω. The introduction of a random variable into the DCOPF means that we cannot ensure security by enforcing constraints for the forecasted operating point only. The optimization problem from the previous chapter changes into 7

20 8 CHAPTER 2. CHANCE CONSTRAINED OPTIMAL POWER FLOW min p,r subject to c i (p i + r i ) (2.2) i G p i d i + v i = 0 (2.3) i V α i = 1 (2.4) i G p i + r i p max i, i G (2.5) p i r i p min i, i G (2.6) Risk[ r i α i ω > 0] < ɛ i, i G (2.7) Risk[M (ij, ) (p αω d + v + ω) > p ij ] < ɛ ij, {ij} E (2.8) Risk[M (ij, ) (p αω d + v + ω) < p ij ] < ɛ ij, {ij} E (2.9) The objective 2.2 of the optimization is the minimize the cost of generation and procuring reserves. In this thesis, the costs for generation and reserve procurement are assumed to be the same. The first constraint 2.3 reflects that power balance must be maintained for pre-deviation operation, while the second constraint 2.4 ensures that balance is maintained between the total wind deviation ω and the generator response αω. The next two constraints 2.5, 2.6 ensure that power generation p and reserves r meet generator limits. The final three constraints are chance constraints, which mean that they constrain the risk of a line or generator violation to a certain value ɛ. The first chance constraint 2.7 limits the risk of the reserves at a generator r i being exceeded by the wind deviation compensation at that generator α i ω. Only one chance constraint is required for each generator because the reserve variables r i dictate both up and down regulation. The final two chance constraints 2.8, 2.9 limit the risk of power flow on line ij exceeding the maximum line flow p ij in either direction to a specified value ɛ ij. As will be seen in the next section, the chance constraints for each line and generator are defined by a mean and standard deviation which reflects how the line flow or generation changes depending on the random variable ω. If wind deviation ω is Gaussian with mean 0 and standard deviation σ w, then the line overloads will also be Gaussian with µ u ij = M (ij, ) (p d + v) p ij, (2.10) (σ u ij) 2 = M (ij, ) (I α)σ 2 w(m (ij, ) (I α)) T. (2.11) A similar expression can be derived for the line underloads. µ u ij = p ij M (ij, ) (p d + v), (2.12)

21 9 (σ u ij) 2 = M (ij, ) (I α)σ 2 w(m (ij, ) (I α)) T. (2.13) For generators, there are two methods for defining the mean and standard deviation of the violations. The original method uses the same technique as for line flows, which results in a unique mean and standard deviation for each generator. The generator violations are also Gaussian if ω is Gaussian. µ i = r i, (2.14) (σ i ) 2 = α i σ 2 wα T i (2.15) The second method, developed for this paper, requires a slight alteration to the generator chance constraint presented in Equation 2.7. It is known as the sum(r) method. Instead of constraining the risk for each generator, it is possible to constrain the total system risk for all generators with a single chance constraint as follows: Risk[sum(r) < ω] < ɛ g. (2.16) This constraint limits the risk of the total system reserves sum(r) being exceeded by the wind deviation to a specified value ɛ g. Using this type of constraint has multiple advantages over the original. First, the number of decision variables is reduced. Instead of having an ɛ i for each generator, the risk on all generators in the system is constrained by a single ɛg. If risk ɛ is a decision variable, then the total number of decision variables is reduced. In addition, the α variables are eliminated since the deviation given to each generator is not considered. This value is still needed for the line chance constraints, but it can be calculated simply as the proportion of reserves at generator i compared to the total system reserves. α i = r i sum(r) (2.17) A further advantage is the reduction of constraints, since only one constraint is used for all generators instead of m constraints. As will be seen later, these constraints can be highly non-linear, so any reduction in constraints is a large advantage. The one drawback to this method is that each generator cannot be secured separately, that is, priority for security of one generator over another cannot be given, if desired. The total system violation given by the sum(r) constraint will also be Gaussian as long as the wind deviation ω is Gaussian. The mean and standard deviation of this total system violation is given as µ = r 1 r 2 r m, (2.18) σ = σ w. (2.19) The means and variances given in this section will be input to the chance constraints derived in the following section.

22 10CHAPTER 2. CHANCE CONSTRAINED OPTIMAL POWER FLOW Risk 1 Risk 1 Risk Overload y Overload y Overload y Standard f(x)=χ(y>0) Linear f(x)=yχ(y>0) Quadratic f(x)=y 2 χ(y>0) Figure 2.1: Possible weighting functions for chance constraints. [3] 2.1 Weighted Chance Constraints Regardless of the type of constraint used, these chance constraints must be reformulated in order to be used in an optimization program. This section will give a short overview of this reformulation. A generalized chance constraint, first shown in [3], is of the form f(y(ω))p (ω)dω ɛ (2.20) The function P (ω) represents the multivariate distribution function of the fluctuations. The function y(ω) denotes the magnitude of overload and varies depending on the component which the chance constraint is applied to. For example, the function y(ω) for the upper limit of power flow on line ij is y(ω) = p ij (ω) p max ij (2.21) where p ij (ω) is the adjusted line flow after a deviation. For the lower limit of power flow on line ij the function y(ω) becomes y(ω) = p min ij p ij. (2.22) Similar expressions can be found for generator chance constraints as well. If y > 0, the limit on the specified component is violated, while if y < 0 the component is in the normal operating range. The weighting function f(.) can be implemented in many ways, and three examples can be seen in Figure 2.1. Standard chance constraints are implemented using a step function χ(y > 0) as the weighting function f(.). Thus Equation 2.20 reduces to which further simplifies to 0 P (y)dy ɛ (2.23) µ + Φ 1 (1 ɛ)σ 0 (2.24)

23 2.1. WEIGHTED CHANCE CONSTRAINTS 11 where Φ 1 is the inverse normal CDF. The means and variances from the previous section can be used in this equation to produce a chance constraint for each component, or in the sum(r) case, for the set of all generators. In the case where ɛ is not a decision variable, this constraint is linear. The weighting function f(.) in the case of standard chance constraints demonstrates that all overloads will be penalized equally, regardless of size. This is reflected in the unit of risk ɛ, which for standard chance constraints is unitless. Thus the chance constraint limits the probability of an overload on the specified component. This type of constraint would see no difference between a 4MW violation and a 1MW violation. Linear weighted chance constraints are implemented using a linear function yχ(y > 0) as the weighting function f(.). Equation 2.20 in this case reduces to 0 yp (y)dy ɛ. (2.25) This integral can be reduced further with the assumption that y is a Gaussian random variable. It then becomes the expectation of a truncated Gaussian random variable, which is given as [3] ( µ 1 Φ ( µ σ ) ) + σ 2π e 1 2 ( µ) 2 σ ɛ. (2.26) where Φ is the CDF of the standard Gaussian. Regardless of whether or not ɛ is a decision variable, this constraint is highly non-linear. However, it has the advantage of penalizing larger violations more, as can be seen from the weighting function in Figure 2.1. The unit of risk ɛ in this case is [MW], which means that risk is given in the linear chance constraint case as an expectation of overload on a component. A 4MW violation is penalized the same as four 1MW violations. Quadratic weighted chance constraints are implemented using a quadratic function y 2 χ(y > 0) as the weighting function f(.). This allows Equation 2.20 to reduce to 0 y 2 P (y)dy ɛ. (2.27) Using the same reasoning as for linear weighted chance constraints, this integral reduces to the second moment of a truncated Gaussian random variable [3]. ( (µ 2 + σ 2 ) 1 Φ ( µ σ ) ) + µσ 2π e 1 2 ( µ) 2 σ ɛ (2.28)

24 12CHAPTER 2. CHANCE CONSTRAINED OPTIMAL POWER FLOW Similarly to linear weighted chance constraints, this constraint is highly non-linear. Quadratic weighted chance constraints also penalize larger violations more than smaller violations, but to a greater degree than linear weighted chance constraints. The unit of risk ɛ in this case is [MW 2 ], which reflects this. A 4MW violation is then penalized the same as sixteen 1MW violations. 2.2 The Reserve Procurement Problem The nonlinearities of the linear and quadratic WCCs make it difficult see the relationship between the participation factor α given to a generator and the amount of reserves r i allocated at the generator. Assuming a single participation factor α i per generator which responds to the overall wind deviation ω (with standard deviation σ w ), for the standard chance constraint it is quite clear that for fixed ɛ the relationship between α i and r i is linear: r i Φ 1 (1 ɛ i )σ w α i (2.29) If a larger participation factor α in the reserve provision is demanded of a generator, then its amount of reserves r i must increase. Also, if the risk limit ɛ i of a generator decreases, then the value of the inverse CDF will increase and the amount of reserves must increase. In reality, this will likely be treated as an equality constraint in the optimization, since the optimizer wants to minimize the cost of reserves and will do so up to the point that the constraint above is tight. For linear and quadratic WCCs, this relationship and its consequences are not so clear. In order to investigate this relationship, the α i and r i values were graphed against each other for several values of fixed epsilon. For standard constraints, the results can be seen in Figure 2.2. As expected, the α i -r i relationship is a linear one, and generators which participate in balancing must be paid for reserves r in return. The slope of the relationship is determined by ɛ, with a steeper slope meaning lower ɛ, which means a higher security level and therefore that more reserves need to be procured for a given participation factor α i. For linear and quadratic constraints, the results are plotted in Figures 2.3 and 2.4, respectively. As can be seen from the plots, the α i -r i relationship is certainly not linear. An unexpected result of these tests is seen in the x-intercepts of the graphs, which should be at (0, 0) in order to reflect that a generator having a non-zero α i should be providing reserves. However, this is not what is seen in the figures. A non-zero intercept indicates that generators that have a non-zero α i are not paid for reserves, which is unfair and an unwanted result of WCCs. To investigate the reserve procurement issue further, a value of zero for r i can be input to the linear and quadratic constraints. The linear WCC case is shown below.

25 2.2. THE RESERVE PROCUREMENT PROBLEM ε = 0.05 ε = 0.10 ε = 0.15 ε = 0.20 r [MW] α [] Figure 2.2: α i vs. r i in CCOPF with standard constraints ε = 0.05 ε = 0.10 ε = 0.15 ε = r [MW] α [] Figure 2.3: α i vs. r i in CCOPF with linear WCCs.

26 14CHAPTER 2. CHANCE CONSTRAINED OPTIMAL POWER FLOW ε = 0.05 ε = 0.10 ε = 0.15 ε = 0.20 r [MW] α [] Figure 2.4: α i vs. r i in CCOPF with quadratic WCCs. ( r i 1 Φ ( r i α i σwα 2 i T ) ) + α i σ 2 wα T i 2π e 1 2 ( r i α i σ 2 wα T i ) 2 ɛ First, a value of zero is zero is input for r i and the equation is simplified. α i σ w 2π ɛ α i ɛ 2π σ w (2.30) Equation 2.30 shows that a generator can indeed take a portion of the wind plant deviations without actually providing any physical reserves. A similar derivation shows the free balancing power given in the quadratic constraint case. 2ɛ α i (2.31) σ w These equations show that both quadratic and linear WCCs allow this behavior to happen, however in different amounts. To investigate how much free balancing power is given over various ɛ magnitudes, the two values are plotted against each other in Figure 2.5 for both quadratic and linear

27 2.2. THE RESERVE PROCUREMENT PROBLEM 15 constraints. The σ w value was taken from a later simulation. While the linear WCCs will give more free balancing power per ɛ for large ɛ, the figure illustrates that for small epsilon quadratic WCCs give more. In this thesis, ɛ values of less than 0.1 are typically seen, which is the range where quadratic WCCs can provide more than double the free balancing power of linear WCCs Quadratic Linear α [] ε [] Figure 2.5: ɛ vs. α i in CCOPF for linear and quadratic WCCs. The discovery of this reserve procurement issue is very important since taking a portion of the wind deviation reaction without scheduling reserves will cause many small generator limit violations. These will occur when a generator is producing either at minimum or maximum power, which is possible because no reserves were procured. The values of ɛ seen in typical examples of this thesis are quite small and therefore the free balancing power given would also be quite small, but it is nevertheless important to be able to explain a discrepancy and take steps to fix it to make the method more robust. There are several approaches which can be taken to prevent the distribution of free balancing power. One solution is to add an accompanying standard chance constraint with a given ɛ value. In Switzerland, the accepted value for generator overload probability is 0.001, which will be used in the rest of this thesis. An example of this constraint and its effect on the α i -r i relationship is shown in Figure 2.6. A zoom of Figure 2.6 near the origin can be seen in Figure 2.7. As seen in the first figure, for generators having a larger value of α i, which are more crucial to system stability, the

28 16CHAPTER 2. CHANCE CONSTRAINED OPTIMAL POWER FLOW relationship is constrained by the linear WCC and the probability of overload is lower than In addition, the value of epsilon associated with the generator will be the expected value of overload in MW. As seen in the zoomed-in figure, for generators having a low value of α i, the relationship is constrained by the standard constraint and the probability of overload is fixed at A second solution to this reserve procurement issue is to use the sum(r) formulation, which eliminates the α decision variables entirely Linear WCC, ε = 0.01 Linear WCC, ε = Standard, ε = r [MW] α [] Figure 2.6: α i vs. r i for combined standard and linear WCCs.

29 2.2. THE RESERVE PROCUREMENT PROBLEM Linear WCC, ε = 0.01 Linear WCC, ε = Standard, ε = r [MW] α [] x 10 3 Figure 2.7: α i vs. r i for combined standard and linear WCCs (zoomed in at the origin).

30 18CHAPTER 2. CHANCE CONSTRAINED OPTIMAL POWER FLOW

31 Chapter 3 Epsilon in the Cost Function 3.1 Weighting in the Objective Function It is a relatively straightforward task to determine the cost of generation in a system, however determining the cost (or value) of security in the system is a larger challenge. It can be seen that the generation cost cannot decrease if additional limits are added to lines and generators since fewer generation schemes are available. The goal in balancing generation cost and security is finding the operating point where lines and generators that are at risk are protected, while not over-restricting lower risk lines and generators. With ɛ as a decision variable, the risk at each generator and line can be clearly seen as a result of the optimization. The aim of this section is to exhibit techniques for pricing ɛ in a way that maintains system security and keeps generation costs low Uniform Weighting In the scaled optimization problem described in the previous chapter, the average price of generation (per MW) is one. Thus it may be of interest to price the scaled value of ɛ in the same way, with a uniform price of one for each variable. This means that the emphasis on security for each line and generator will be the same, which reflects an emphasis on system security rather than the security of individual lines or generators. This is an intuitive approach, yet the problem still remains as to how to scale this uniform price in relation to the price of generation in order to achieve the optimal balance between generation cost and security. This issue as well as other pricing schemes will be investigated in this chapter Optimal Redispatch Weighting The use of uniform weights makes the assumption that all generators and lines are equally valuable and should be protected with the same price for 19

32 20 CHAPTER 3. EPSILON IN THE COST FUNCTION risk. However, it is possible that certain lines and generators are of higher importance to the operation of the system. Thus it would be desirable to put a higher price on risk for these particular generators and lines. To determine the relative values of generators or lines against each other, a technique known as optimal redispatch can be used. For relative weights of lines, consider a dispatcher s response to observing that a line is overloaded. They would most likely change generation slightly in a way that would change the flow on that particular line so that its limits are no longer violated. The cost of this action can be simulated with the assumption that the price to change a line flow by one MW is independent of the current state of the system (this is true if generation prices are linear). One way of attempting this calculation for a single line is to look at the cheapest way to redispatch all generation in order to change the line flow by one MW either up or down. The objective would be to minimize the sum of the changes in up regulation P g + and down regulation Pg multiplied by the cost of these changes. minimize c + P + g + c P g (3.1) In this optimization, there are three sets of decision variables. The first two are the changes in up regulation P g + and down regulation Pg. The third set of decision variables are the changes in line flow P L. The changes in up regulation P g + and down regulation Pg are allowed to be in the range [0, Pg max ], however it is expected that they will not reach the upper limit since the change in line flow desired is only one MW. The values for changes in line flow P Li are not restricted in the variable bounds. The decision variables are summarized in Table 3.1. In this formulation, m represents the number of generators and n the number of lines. Table 3.1: Decision Variables for the Optimal Redispatch Weight Calculator Name Limits Description P 1 + P m + 0 P i + p max i Positive change in generation of generator i P1 P m 0 Pi p max i Negative change in generation of generator i P ij P ln P L Change in line flow for line ij There are three sets of linear constraints used in the optimization. The first set of constraints ensures that the changes in up and down regulation are reflected in changes in line flows. P L = M( P + g P g ) (3.2) The second set of linear constraints ensures that the total amount of generation in the system does not change (i.e. power balance is maintained). P + g = P g (3.3)

33 3.1. WEIGHTING IN THE OBJECTIVE FUNCTION 21 The final linear constraint specifies that the line flow on line ij must change by at least one MW. P ij 1 (3.4) If it is desired to calculate for a negative change in flow, the previous constraint must only be changed slightly. P ij 1 (3.5) If this optimization is done for each line, then relative cost weights can be found and applied to the ɛ variables for lines. However, this is not a very realistic way to redispatch line flows. In reality, an operator would not redispatch all generators to change a line flow. More likely is that they would use only one generator y for up regulation and one generator z for down regulation. To simulate this scenario, only two sets of linear constraints must be added to make sure that other generators do not participate in the redispatch. p + gj p k = 0, j y (3.6) = 0, k z (3.7) With these two sets of linear equations, a more realistic redispatch can be simulated. A downside to this method is that for each line, each pair of generators must be simulated. In the first method, only one simulation is necessary per line. One must be careful in this case when simulating pairs of generators for specific lines, as not all pairs in a system will be able to successfully redispatch the specified line. In addition, there will be many successful redispatches that are extremely expensive. One reasonable way to account for this is for each line to find all of the possible redispatch costs and then take the average of the cheapest ten or twenty. The optimal redispatch weight calculating method is summarized in the Appendix. It is not as intuitive to find a weighting for generators that will reduce the redispatch cost. For generators, the effect of overload is different than on lines. An overload on a line can be tolerated for some time and simply redispatched within a certain amount of time, however a generator overload will likely result in the shutdown of the generator, which can be quite expensive. However, it is unlikely that an operator would allow a generator to overload, even if it was necessary to take a portion of the wind fluctuation at limit. More likely is that additional reserves would be found at higher cost. Thus it is likely that the optimal redispatch weights for generators are the same as uniform weights. If it is still desired to increase security at the most important generators, one way to find weights is to see how the total cost of generation changes when a specific generator is removed. This can be done by performing successive OPFs, each time with a different generator

34 22 CHAPTER 3. EPSILON IN THE COST FUNCTION removed, and then comparing these costs to the cost of the OPF when all generators are available. 3.2 Multiobjective Optimization When introducing ɛ as a decision variable, inevitably the problem of integrating it into the cost function and scaling it to a level appropriate with the cost of generation must be solved. In essence, there are two cost functions, c g corresponding to the generation cost and c ɛ corresponding to the cost of insecurity. There are optimization methods, however, which can help investigate this relationship and even find the optimal solution without initially knowing the optimal scaling between c g and c ɛ. These methods are known at multiobjective optimization methods, which are explored in detail in [5]. Only two simple multiobjective methods were explored in this thesis, the ɛ method (not to be confused with the CCOPF decision variable), and the weight method. The ɛ Method The ɛ method chooses one of the objective functions to be the main objective. In the case of this thesis, the cost of generation c g was used for this purpose. The other objective function(s) is then added as a constraint in the optimization and limited to a value τ (in the formal method, this symbol would be ɛ, but for clarity, it is replaced). This limiting value τ can then be altered in order to explore the relationship between the objective functions. When τ is small, this is as if you are simulating a high weighting on the cost function represented by the constraint. When τ is large, then the problem is weighted towards the main objective. Formulating the ɛ penalty as a constraint comes with the benefit of the ability to use functions other than a simple sum of ɛ as the cost function c ɛ, even non-linear functions. Another choice of penalty function is max(ɛ), which would limit the most unsecured generator or line. In order to investigate the effects of both sum(ɛ) and max(ɛ), both can be used as constraints with separate τ sum and τ max values. By varying these values it can be seen how the dispatch is changed when only a specified amount of ɛ is allowed.

35 3.2. MULTIOBJECTIVE OPTIMIZATION 23 The CCOPF problem with these new constraints can be seen below. min p,r subject to c i (p i + r i ) (3.8) i G ɛg + ɛ l τ sum (3.9) maximum(ɛ g, ɛ l ) τ max (3.10) p i d i + v i = 0 (3.11) i V α i = 1 (3.12) i G p i + r i p max i, i G (3.13) p i r i p min i, i G (3.14) Risk[ r i α i ω > 0] < ɛ i, i G (3.15) Risk[M (ij, ) (p αω d + v + ω) > p ij ] < ɛ ij, {ij} E (3.16) Risk[M (ij, ) (p αω d + v + ω) < p ij ] < ɛ ij, {ij} E (3.17) The two new constraints 3.9 and 3.10 are the constraints for the sum of risk variables and the maximum risk variable, respectively. The rest of the formulation is unchanged with respect to the original given in Chapter 2. The ɛ method is very useful for seeing which lines and generators are most essential in the system, however it can also have a practical purpose. The total amount of risk in the system can be specified, and the maximum amount of risk on a single component can also be specified. This is an alternative to finding a realistic cost for ɛ which balances perfectly the generation costs in the system. The Weight Method In contrast to the ɛ method, the weight method of multiobjective optimization can help find the optimal balance in cost between generation and security. The multiple objectives are combined into a single objective using weights w i, one for each of the objective functions. In the CCOPF case, the composite objective function is made up of the generation cost function and the security cost function. minimize p w 1 c i (p i + r i ) + w 2 ( ɛ g + ɛ ij ) (3.18) i G ij E i G Typically, w 1 is set to one and only the other weights are varied. By sweeping a range of weight values, a curve can be generated that shows the

36 24 CHAPTER 3. EPSILON IN THE COST FUNCTION tradeoff between the objectives. As the weight for the security cost function is increased, generation cost should also increase due to a higher emphasis on security than on low generation cost. The resulting graph will then show generation cost versus some metric of security. This metric can be redispatch cost, probability of overload, total amount of overload in MW, or any of a number of other options. In this thesis, linear WCCs will be investigated and the metric will be total amount of overload in MW over 100,000 random wind scenarios. It is desirable to find a knee in the curve which shows where the tradeoff between objective functions is optimal.

37 Chapter 4 Formulations In this chapter, fully explained formulations for several CCOPF variations will be given with the risk limit ɛ as a decision variable. For some formulations, such as standard chance constraints, the formulation varies greatly from the equations seen in Chapter 2 and extended explanations will be given. For others, such as linear WCCs, the final formulation uses the same equations from that chapter and only a brief explanation will be given. 4.1 Standard Chance Constraint Formulation At first glance, standard chance constraints appear to be the simplest of the three weightings. However, the introduction of the risk limit ɛ to the set of decision variables creates a non-linear term in the chance constraint. Also, the constraint is not guaranteed to be convex since the weighting function used, a step function, is not convex. In order to make the problem tractable with ɛ as a decision variable, several changes to the problem must be made. Repeated below is the standard generator chance constraint. r i + Φ 1 (1 ɛ i )α i σ w 0 (4.1) There are several issues with this equation that need to be addressed. First is its non-convexity. This can be improved by changing this continuous function into a lookup table into the function with binary variables. For example, if five binary variables b k are introduced and each has a corresponding c k which gives the value of Φ 1 (1 ɛ) at five chosen values of ɛ, Equation 4.1 becomes: r i + b 1 c 1 α i σ w + b 2 c 2 α i σ w + b 3 c 3 α i σ w + b 4 c 4 α i σ w + b 5 c 5 α i σ w 0. (4.2) This equation is still not convex; instead, it transforms the CCOPF into a mixed-integer problem. The product of b k and α in Equation 4.2 means that this equation is non-linear, and therefore further work is needed. Typically, 25

38 26 CHAPTER 4. FORMULATIONS McCormick relaxations [6] can be found for products of two variables, and luckily, for a product of a binary and continuous variable this relaxation is exact. It requires the introduction of a new variable which represents the product and three new constraints which ensure that the new variable does represent the product of the two. This must be done for each product, which means in this example, the introduction of five new product variables prod k and 15 new constraints (3 for each prod k ). The new linear constraint set can be seen below. r i +prod 1 c 1 σ w +prod 2 c 2 σ w +prod 3 c 3 σ w +prod 4 c 4 σ w +prod 5 c 5 σ w 0 (4.3) prod k α i, 1 k 5 (4.4) prod k α i α max (1 b k ), 1 k 5 (4.5) prod k α max b k, 1 k 5 (4.6) In the above formulation, α max, which would normally be one, is the same as maximum(pg max ) due to the variable scaling. With all of these new constraints, the generator CCs are finally solvable by a computer. Several problems still remain, however. First, one must choose which five ɛ values will be represented by the binary variables b k and corresponding c k variables. With the discretization of the inverse normal CDF function, a severe limitation has been placed on the problem in the fact that only a discrete number of epsilons can be chosen by the generators. One would certainly want to include an ɛ value of zero for generators that do not take any α, however, this is not possible with the constraints formulated above. The constant c k associated with the binary variable for zero epsilon would need to be infinity, since that is the value of Φ 1 (1). One solution to this problem is to simply take a value very close to zero for the lowest ɛ choice, however, this was found to cause numerical issues the closer the value approached to true zero. In order to truly implement a zero ɛ choice, the constraint set above needs to be altered. This new constraint needed to ensure that a generator could not take any α or r if the value of the zero ɛ binary variable b 1 was chosen. The improved formulation can be seen below. r i + prod 2 c 2 σ w + prod 3 c 3 σ w + prod 4 c 4 σ w + prod 5 c 5 σ w 0 (4.7) r i + α i + 2b 1 maximum(pg max ) 2maximum(PG max ) (4.8) prod k α i, 2 k 5 (4.9) prod k α i α max (1 b k ), 2 k 5 (4.10) prod k α max b k, 2 k 5 (4.11) The new constraint shown in Equation 4.8 brings an advantage in addition to implementing a true zero ɛ: the product variable prod 1 for each

39 4.1. STANDARD CHANCE CONSTRAINT FORMULATION 27 generator is unnecessary and can be eliminated from the formulation. The reasoning behind this constraint is simple: when the zero epsilon binary variable b 1 is chosen, then both α i and r i for the generator must also be zero. When b 1 is zero, then both α i and r i are allowed up to the maximum limit, which for each variable is maximum(pg max ) due to scaling. The question remains as to how to choose the best ɛ values for the other binary b k variables. The ɛ choices must be chosen specifically for each system and specific problem, with these recommendations: one choice must be zero, the largest choice should be the largest acceptable ɛ value for the system (as determined by the operator or analysis) and the three middle values should be closer to zero than to the largest value. This is because usually α is distributed to only a small number of generators in large amounts, and these generators will need the highest ɛ choice. Most generators will choose zero and have no α i, and the few generators left will only need a very small amount of ɛ. The formulation above successfully implements standard constraints for generators. The question of implementation still remains for lines. The biggest challenge in implementing standard constraints for lines is that α i is not easy to pull out of the square root it is under in the equation, and therefore it is very difficult to use the same product rule used for generators. To illustrate this, the standard constraint for lines (upper flow limit) is shown below. M (ij, ) (p d + v) p ij Φ 1 (1 ɛ ij ) M (ij, ) (I α)σw(m 2 (ij, ) (I α)) T (4.12) It is clear that this equation will be much harder to manipulate than Equation 4.2 for generators, though it is possible using a variable replacement for the standard deviation term. Plus, the motivation for standard constraints for lines does not exist; the only reason standard constraints were implemented for generators was because of the concept of free balancing power with WCCs. There are no such issues with using WCCs for lines. The full formulation for standard chance constraints is given below. Either linear or quadratic WCCs can be used; in this example linear WCCs for lines are implemented. In this formulation, m represents the number of generators in the system, n represents the number of lines, z represents the number of wind infeeds, q represents the number of nodes, and k represents the number of binary variables per generator. The constants c i represent the chosen mappings from binary variables to values of the inverse normal CDF.

40 28 CHAPTER 4. FORMULATIONS Table 4.1: Total Number of Decision Variables: (k + 2)m + n Name Limits Description p i p min i p i p max i Output power of generator i α i 0 α i 1 % of ω compensated by generator i r i ri min r i ri max Reserves procured at generator i ɛ ij 0 ɛ ij Security level for line ij prod 12 prod mk 0 prod max(p max g ) Product of binary and r variables b 11 b mk 0, 1 Binary generator security levels Linear Constraints (Total = m(3 + 3(k 1)) + 2) 1) Guarantees that the sum of all alpha variables is equal to one. (One Constraint) α 1 + α α m = 1 2) Guarantees that the amount of power generated is equal to the amount of power consumed. (One Constraint) p 1 + p p m + v 1 + v v z = d 1 + d d q 3) Guarantees that only one binary variable at generator i is active (m Constraints) b i1 + b i2 + + b im = 1 4) Restricts the sum of generation and reserves at generator i to p max i. (m Constraints) p i + r i p max i 5) Restricts the difference of generation and reserves at generator i to p min i. (m Constraints) p i + r i p min i 6) Chance constraint for generator i related to zero-ɛ binary variable. (m Constraints) r i + α i + 2b i1 maximum(p max G ) 2maximum(P max G ) 7) Chance constraint for generator i related to other (k 1) binary variables. (m Constraints) r i + prod i2 c 2 σ w + prod i3 c 3 σ w + + prod ik c k σ w 0 8) McCormick relaxation equations for product variable prod ij. (3m(k 1) Constraints)

41 4.1. STANDARD CHANCE CONSTRAINT FORMULATION 29 prod ij α i prod ij α i α max (1 b ij ) prod ij α max b ij Non-linear Constraints (Total = 2n) 1) Linear WCC for line ij for positive line flow. (n Constraints) µ u ij = M (ij, )(p d + v) p ij (σ u ij )2 = M (ij, ) (I α)σ 2 w(m (ij, ) (I α)) T µ u ij ( 1 Φ ( µu ij σ u ij ) ) 1 + σu ij 2 e 2π ( µ u ij ) 2 σij u ɛ ij 2) Linear WCC for line ij for negative line flow. (n Constraints) µ l ij = p ij M (ij, ) (p d + v) (σ l ij )2 = M (ij, ) (I α)σ 2 w(m (ij, ) (I α)) T µ l ij ( 1 Φ ( µl ij σ l ij ) ) 1 + σl ij 2 e 2π ( µ l ij ) 2 σ l ij ɛ ij