Flexible Transmission in the Smart Grid: Optimal Transmission Switching

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1 Flexible Transmission in the Smart Grid: Optimal Transmission Switching Kory W. Hedman, Shmuel S. Oren, and Richard P. O Neill Abstract There is currently a national push to create a smarter, more flexible electrical grid. Traditionally, network branches (transmission lines and transformers) in the electrical grid have been modeled as fixed assets in the short run, except during times of forced outages or maintenance. This traditional view does not permit reconfiguration of the network by system operators to improve system performance and economic efficiency. However, it is well known that the redundancy built into the transmission network in order to handle a multitude of contingencies (meet required reliability standards, i.e., prevent blackouts) over a long planning horizon can, in the short run, increase operating costs. Furthermore, past research has demonstrated that short-term network topology reconfiguration can be used to relieve line overloading and voltage violations, improve system reliability, and reduce system losses. This chapter discusses the ways that the modeling of flexible transmission assets can benefit the multi-trillion dollar electric energy industry. Optimal transmission switching is a straightforward way to leverage grid controllability; it treats the state of the transmission assets, i.e., in service or out of service, as a decision variable in the optimal power flow problem instead of treating the assets as static assets, which is the current practice today. Instead of merely dispatching generators (suppliers) to meet the fixed demand throughout the network, the new problem co-optimizes the network topology along with generation. K.W. Hedman (*) School of Electrical, Computer, and Energy Engineering at Arizona State University, Tempe, AZ, USA kory.hedman@asu.edu S.S. Oren Industrial Engineering and Operations Research Department, University of California at Berkeley, Berkeley, CA, USA oren@ieor.berkeley.edu R.P. O Neill Federal Energy Regulatory Commission, Washington, DC, USA richard.oneill@ferc.gov A. Sorokin et al. (eds.), Handbook of Networks in Power Systems I, Energy Systems, DOI / _21, # Springer-Verlag Berlin Heidelberg

2 524 K.W. Hedman et al. By harnessing the choice to temporarily take transmission assets out of service, this creates a superset of feasible solutions for this network flow problem; as a result, there is the potential for substantial benefits for society even while maintaining stringent reliability standards. On the contrary, the benefits to individual market participants are uncertain; some will benefit and other will not. Consequently, this research also analyzes the impacts that optimal transmission switching may have on market participants. Keywords Mixed integer programming Optimal power flow Power generation dispatch Power system economics Power system reliability Power transmission control 1 Introduction The physics that govern the flow of electric energy across the electric transmission network create a complex and unique network flow problem. The flow of electricity across the network follows Kirchhoff s laws. These unique physical laws imply that changing a transmission asset s impedance changes how the power flows throughout the network. Moreover, electric energy is instantaneously consumed and it is currently too expensive to store. These factors, along with the many stability constraints, reliability constraints, generator dispatch constraints, etc., make this a very difficult network flow problem. However, the mathematical modeling of the network is not as complex as it could be and various control mechanisms have yet to be acknowledged as well as harnessed within the optimization formulation. Traditionally, the system operator treats transmission assets (lines or transformers) as static assets within Optimal Power Flow (OPF) problems, which are the network flow problem for the electrical grid. The OPF dispatches generators to minimize cost subject to satisfying the fixed demand throughout the network, ensuring that reliability standards are met, and satisfying all of the network flow constraints for the transmission network problem. This traditional view does not describe transmission assets as assets that operators have the ability to control. However, it is acknowledged, both formally and informally, that system operators can and do change the grid topology to improve voltage profiles, increase transfer capacity, and even improve system reliability. These ad-hoc procedures are determined by the system operators, rather than in an automated or systematic way. Furthermore, such flexibility is not incorporated into dispatch optimization problems today. This is a shortcoming regarding today s electric grid operations; due to the physics that govern the flow of electric energy and due to the complexities within this network flow problem, it is extremely unlikely that there is a single optimal network topology for all periods and possible market realizations over a long time horizon. The electric grid is built to be a redundant network in order to ensure mandatory reliability standards and these standards require protection against worst-case

3 Flexible Transmission in the Smart Grid: Optimal Transmission Switching 525 scenarios. However, it is well known that these network redundancies can cause dispatch inefficiency and, furthermore, a network branch that is required to be built in order to meet reliability standards during specific operational periods may not be required to be in service during other periods. Consequently, due to the interdependency between network branches (transmission lines and transformers), it is possible to temporarily take a branch out of service during certain operating conditions and improve the efficiency of the network while maintaining reliability standards. Past research has identified how the control of transmission assets can be used to benefit the network. These papers have generally focused on the switching of transmission lines when a line is overloaded, when there are voltage violations, as well as other factors related to using the control of transmission assets to alleviate an active network constraint. These approaches, however, do not attempt to use the control of transmission assets optimally by co-optimizing the network topology with generation to improve the dispatch efficiency during steady-state operations. Optimal transmission switching formally introduces the control of transmission assets into the classical formulation of dispatch optimization problems common in system and market operation procedures employed by vertically integrated utilities and Independent System Operators (ISOs). There is currently a national push to model the grid in a more sophisticated, smarter way as well as to introduce advanced technologies and control mechanisms into grid operations. In particular, there are national directives that call on researchers to examine topics in this general area of research. The US Energy Policy Act of 2005 includes a directive for federal agencies to encourage...deployment of advanced transmission technologies, including optimized transmission line configuration. 1 This research is also in line with FERC Order 890: to improve the economic operations of the electric transmission grid. It also addresses the items listed in Title 13 Smart Grid of the Energy Independence and Security Act of 2007: (1) increased use of... controls technology to improve reliability, stability, and efficiency of the grid and (2) dynamic optimization of grid operations and resources. This research examines the smart grid application of harnessing the control of transmission assets by incorporating their discrete state into the network optimization problem and it analyzes the benefits and market implications of this concept. The rest of this chapter is broken down to include six main sections. The following section provides a thorough overview of the literature that is relevant to this research as well as a discussion on current industry practices that demonstrate the benefit of transmission control. Section 3 discusses the impact that transmission switching has on the feasible set of dispatch solutions, its affect on reliability, and how it differs from transmission expansion planning. Section 4 then presents a mathematical overview of OPF problems and optimal transmission switching. Section 5 presents results on the potential economic savings as a result of optimal transmission switching. Section 5 also focuses on the market implications 1 See Sec.1223.a.5 of the US Energy Policy Act of 2005.

4 526 K.W. Hedman et al. when these optimization models are modified to include the control of transmission assets. While optimal transmission switching can improve economic efficiency of grid operations, the implementation of this new technology may have unpredictable distributional effects on market participants and undermine some prevalent market design principles that rely on the premise of a fixed network topology. Section 6 provides an overview of future research topics and Sect. 7 concludes this chapter. 2 Literature Review 2.1 Transmission Switching as a Corrective Mechanism Past research has explored transmission switching as a control method for a variety of problems. The primary focus of past research has been on proposing transmission switching as a corrective mechanism when there is line overloading, voltage violations, etc. While this past research acknowledges certain benefits of harnessing the control of transmission, they do not use the flexibility of the transmission grid to co-optimize the generation along with the network topology during steady-state operations. Such co-optimization, as will be shown by this research, can provide substantial economic savings even while maintaining N-1 reliability standards. Furthermore, the use of transmission switching as a corrective mechanism to respond to a contingency has been acknowledged in some of the past research to have an impact on the cost of generation rescheduling due to the contingency. However, it has not been acknowledged that such flexibility should be accounted for when solving for the steady-state optimal dispatch. Glavitsch [1] gives an overview of the use of transmission switching as a corrective mechanism in response to a contingency. He discusses the formulation of such a problem and provides an overview on search techniques to solve the problem. Mazi et al. [2] propose a method to alleviate line overloading due to a contingency by the use of transmission switching as a corrective mechanism and use a heuristic technique to solve the problem. Gorenstin et al. [3] study a similar problem concerning transmission switching as a corrective mechanism; they use a linear approximate Optimal Power Flow (OPF) formulation and solve the problem based on branch and bound. Bacher et al. [4] further examine transmission switching in the AC setting to relieve line overloads; however, they assume that the generation dispatch is already determined and fixed thereby not capturing the benefit of co-optimizing the network topology with generation. Bakitzis et al. [5] examine transmission switching as a corrective mechanism both with a continuous variable formulation for the switching decision as well as with discrete control variables. Schnyder et al. [6, 7] proposed a fast corrective switching algorithm to be used in response to a contingency. The benefit of this algorithm over past research is that they simultaneously consider the control over the network topology and the ability

5 Flexible Transmission in the Smart Grid: Optimal Transmission Switching 527 to redispatch generation whereas other methods would assume that the generation is fixed when trying to determine the appropriate switching action. Due to the complexity of this problem for its time, this method does not search for the actual optimal topology but rather considers limited switching actions. Rolim et al. [8] provide a review of past transmission switching methods, the solution techniques used, the objective at hand, etc. Shao et al. [9] continued previous research on the use of transmission switching as a corrective mechanism to relieve line overloads and voltage violations. They propose a new solution technique to find the best switching actions. Their technique employs a sparse inverse technique and involves a fast decoupled power flow in order to reduce the number of required iterations. In Shao et al. [10], a binary integer programming technique is used for the same motivation: to use switching actions as a corrective mechanism to relieve line overloads and voltage violations. Granelli et al. [11] propose transmission switching as a tool to manage congestion in the electrical grid. They discuss ways to solve this problem by genetic algorithms as well as deterministic approaches. However, their approach does not consider the impact the topology has on the choice of steady-state dispatch solutions. 2.2 Transmission Switching To Minimize Losses In Bacher et al. [12], they propose switching to minimize system losses. This paper demonstrates that, contrary to general belief, it is possible to reduce electrical losses in the network by temporarily opening a transmission line. Fliscounakis et al. [13] proposed a mixed integer linear program to determine the optimal transmission topology with the objective to minimize losses. Unlike past research, this model does search for the optimal topology but it does not co-optimize the generation with the network topology in order to maximize the market surplus. It is even possible that the solution that maximizes the market surplus has an increase in losses but by accounting for the influence between generation and transmission, the overall costs may still be lower. In contrast to these approaches, the optimal transmission switching concept maximizes the market surplus by co-optimizing the transmission topology along with generation. 2.3 Ad-Hoc Transmission Switching Protocols One of the most common industry practices of transmission line switching involves the common protocol to switch specific lines offline during lightly loaded hours. The capacitive component of a transmission line is the predominant component during low load levels whereas the reactive component is predominant at higher load levels. Consequently, during low load levels there can be situations where a

6 528 K.W. Hedman et al. transmission line causes voltage violations in the network, i.e., the voltage levels are too high. Therefore, one simple protocol that operators are aware of is to select key transmission lines that are not currently needed for reliability considerations and they take these lines out of service. This reduces the capacitance and can help alleviate voltage violations. Such a protocol is acknowledged as a procedure within the PJM network and by Excelon. Likewise, the Northeast Power Coordinating Council includes switch out internal transmission lines in the list of possible actions to avoid abnormal voltage conditions, [14, 15]. Another ad-hoc transmission switching protocol that is at times used by grid operators is to identify key transmission lines that can be taken out of service in order to improve the transfer capability on other high voltage transmission lines. This is a protocol implemented in the PJM network. 2.4 Implementation of Transmission Switching in Special Protection Schemes Special Protection Schemes (SPSs), also known as special protection systems or remedial action schemes, are becoming a mainstream protocol in electric grid operations. Grid operators identify specific grid conditions where it can be advantageous to implement an automatic, predetermined corrective action in response to specific abnormal grid operations. SPSs can be used to solve a variety of issues from maintaining voltage stability to a corrective action that is taken once a specific contingency occurs; the main motivation is to maintain proper reliable operations of the grid. These actions may involve changes in generation, reduction in load if necessary, as well as grid topology modifications. The PJM system uses SPSs to implement transmission switching protocols; this includes both pre-contingency transmission switching as well as post-contingency transmission switching. There can be situations where the operator will take a line out of service temporarily during steady-state operations but may switch the line back into service once a specific contingency occurs. Likewise, there are situations where opening a transmission line once there is a contingency can help the system recover from the contingency without causing a blackout. Further information on SPSs that implement transmission switching can be found in [16]. 2.5 Transmission Line Maintenance Scheduling The focus of past transmission line maintenance scheduling was on the effect on reliability. However, just as transmission lines affect reliability they also affect the operational costs of the electrical grid. Operators are now acknowledging the importance of transmission line maintenance scheduling not only regarding its affect on reliability but on operational costs. For instance, the Independent System

7 Flexible Transmission in the Smart Grid: Optimal Transmission Switching 529 Operator of New England (ISONE) recently released a report stating that they saved $72 million in 2008 by considering the impact of transmission line maintenance scheduling on the overall operational costs, [17]. This study, however, was done by estimating prices instead of determining an optimal maintenance schedule, which further emphasizes the need for research on network topology optimization. 2.6 Optimal Transmission Switching The initial concept of a dispatchable network was first proposed by O Neill et al. [18]. Fisher et al. [19], further developed and examined the concept of incorporating the control of transmission assets into dispatch optimization formulations. This chapter is based on past and current research by the authors on this concept; further information can be found in [18 25]. 3 Discussion of Optimal Transmission Switching, Feasible Sets, Reliability, and Transmission Planning 3.1 Transmission Switching s Impact on the Feasible Set Even though this research is based on examining optimal transmission switching with the DCOPF and not the ACOPF, optimal transmission switching still has the ability to provide substantial benefits by providing more control to the operator. One simple way to demonstrate this fact is by understanding how optimal transmission switching affects the feasible set of dispatch solutions for any OPF. For simplicity, the following example is based on the DCOPF. However, the set of feasible solutions for any optimal power flow problem, be it the DCOPF, the ACOPF, or a security constrained OPF, depends on the characteristics of the network branches. This example demonstrates what happens in the DCOPF by changing the characteristics of a line (opening a line is equivalent to changing the impedance to infinity) and based on Kirchhoff s laws it is known that a similar result can be demonstrated for any OPF. If a transmission asset s impedance is changed, this changes the feasible set of dispatch solutions. Since optimal transmission switching allows for the selection of any network topology, this gives the operator the choice to choose any dispatch that is feasible for any of these individual topologies instead of being restricted to choosing a dispatch that is feasible for only the static topology. As a result, optimal transmission switching creates a superset of feasible dispatch solutions and, therefore, it improves the operational flexibility of the grid in order to potentially improve reliability, stability, and/or economic operations. Obviously, optimal transmission switching will not provide additional flexibility to the operator in

8 530 K.W. Hedman et al. situations where the electric grid is not congested, i.e., there are no active network constraints (except the node balance constraints), since all dispatch solutions that satisfy the generator constraints are feasible. Figure 1 provides a simple three-bus example; each transmission line has the same impedance but their thermal capacity limits differ. The feasible sets in Fig. 2 are defined by the thermal transmission constraints. For the original topology, there are three equations that represent the network constraints, (1)- to (3). Opening any line will change these constraints and, thus, change the feasible set. With all lines closed, i.e., in service, the feasible set is defined by the set of vertices {0, 1, 2, 3} in Fig. 2. If line A-B is opened, i.e., out of service, the feasible set is {0, 4, 5, 6} G A 1 3 G B 80 (1) G A þ 2 3 G B 80 (2) G A 50 $/MWh A 80MW Z B G B 100 $/MWh 50MW Z Z 80MW G C 200 $/MWh C 200MW Fig. 1 3-Node example Fig. 2 Feasible sets for Gen A and Gen B

9 Flexible Transmission in the Smart Grid: Optimal Transmission Switching G A þ 1 3 G B 50 (3) The advantage of optimal switching is that it gives the operator the choice to choose any dispatch solution defined by the set {0, 1, 2, 7, 5, 8, 3}, which is the nonconvex union of the nomograms corresponding to the two operating states for the line between buses A and B. Though this example does not enforce reliability standards and though it is not based on the ACOPF model, the example demonstrates the flexibility that optimal transmission switching give to the operator. No matter what type or form of constraint that is a part of an OPF problem, harnessing the flexibility of whether to keep a transmission asset in service or not will create a superset of feasible dispatch solutions. It is possible that even with this control the operator would choose the original, static topology. However, due to the nature of the electrical grid and its complexity over a wide range of operating conditions over a long time horizon, it is highly unlikely that there is one single topology that is always preferred no matter the current operating state of the grid. An electrical transmission network can have over 10,000 transmission assets, thereby creating 2 10,000 possible network configurations. Obviously, many of those configurations would result in an infeasible dispatch solution; however, many configurations would still be possible and it is near impossible for there to be one perfect topology for every operating condition. This is further confirmed by the well-known practice to open high-voltage lines during the night to improve the voltage profile. With optimal transmission switching, there is a guarantee that the solution will not be worse off than before since the original network configuration can always be chosen. For this example, if the objective were to minimize the total dispatch cost, the original solution would be located at {2} where the corresponding dispatch is 20 MW from GEN A, 110 MW from GEN B, and 70 MW from GEN C for a total cost of $26,000 per hour. By opening line A-B, there is a new feasible solution, {5}, where the corresponding dispatch is 50 MW from GEN A, 80 MW from Gen B, and 70 MW from GEN C. This places the total cost at $24,500, which translates into a $1,500 savings. 3.2 Transmission Switching and Reliability Transmission switching would not be implemented if it were to violate established reliability standards. The previous example in Sect. 3.1 shows graphically how optimal transmission switching adds flexibility to the dispatch choice for congested networks. Even though that stylized example does not enforce N-1, the conclusions would not change if reliability constraints were enforced in the OPF. Optimal transmission switching adds another layer of control to the OPF and, thus, it creates a superset of possible dispatch solutions.

10 532 K.W. Hedman et al. Even still, it is often thought that reliability must diminish if you take a line out of service, that reliability is something that is judged purely on the topology of the network. This is, in fact, not so. The research papers and practical examples on the use of transmission switching in the literature review section demonstrate that, during certain operational states, the system reliability can improve by the removal of a line. For instance, transmission switching is used today as a post-contingency corrective action in some SPSs, [16]. Reliability cannot be judged purely on the network topology alone. Reliability depends on the network topology but it also depends on the generation s commitment schedule, ramping capabilities, and their available capacity. With optimal transmission switching, it is possible to switch to a topology that has fewer available paths to transfer energy but that the different generation schedule, which can only be obtained if the topology is altered, provides more available capacity and these generators are, overall, faster than the generation dispatch solution that would have been chosen if the topology was not altered. In Sect. 2.3 of [24], there is an example demonstrating this possibility. It is then possible that the combination of changing the topology with generation improves system reliability, which again emphasizes that the grid topology itself cannot be used as the only indicator to examine system reliability. Furthermore, the true issue is not whether the system reliability in general diminishes; rather, the concern is whether the required reliability standards are met. The objective of the grid operator is to serve the load at least cost subject to maintaining set reliability standards as well as satisfying the operational constraints. Thus, no preference is given to solutions that improve system reliability beyond required levels; rather, with multiple solutions that satisfy these requirements, the operator chooses the least cost solution. Optimal transmission switching is consistent with this conventional policy to serve load at least cost while maintaining established reliability requirements as it can improve the market surplus while meeting the required reliability standards. If the network is initially N-3 2 reliable but with the optimal transmission switching solution it is only N-2 reliable, then no required reliability standard has been violated. The correct decision is, therefore, to implement the optimal transmission switching concept to be able to obtain superior economic dispatch solutions that also meet set reliability standards, e.g., N-1. Furthermore, with a truly smarter, more advanced electrical grid, transmission lines that are temporarily taken out of service during no-contingency operating states can be switched back into service if there is a contingency. This would bring the grid back to its redundant state during a contingency state; this concept is further discussed in Sect As was discussed in Sect. 2.4, similar corrective actions are used today by ISOs as there are SPSs that implement precontingency and post-contingency switching actions, [16]. 2 N-k reliability means that the system can survive the simultaneous failure of any k elements without violating any constraints on the surviving network and without the need for load shedding.

11 Flexible Transmission in the Smart Grid: Optimal Transmission Switching Optimal Transmission Switching and Transmission Planning The previous sections demonstrate how optimal transmission switching creates a superset of feasible dispatch solutions and how this added flexibility in dispatch choice can be used to improve dispatch efficiency and/or improve system reliability. However, there is an underlying question as to why a line that is taken out of service would have been built in the first place. Applying transmission switching to reduce costs may seem counter-intuitive as it seems to contradict the purpose of transmission planning. Transmission lines are built because they are needed to maintain reliability and they are built to facilitate additional trading of energy, i.e., increase the transfer capability between areas in order to access cheaper energy. As a result, there is the common misconception that optimal transmission switching can only benefit system operations when there was previously poor transmission planning. If transmission switching were never beneficial and never feasible, such a result would mean that a single topology is the optimal topology for every possible network condition over a long planning horizon; to have one perfect topology out of a vast number of possible network configurations over such varying operational situations is unlikely. Optimal transmission switching and transmission planning are two distinct problems with different goals. Even if the network is optimally planned, optimal transmission switching can still be beneficial. First, the basis for optimality of transmission expansion planning is the aggregate of benefits due to building a transmission element over a long time horizon. This is distinctly different than optimal transmission switching, which is a short-term optimization problem that determines the optimal topology for very specific operating conditions over a short time horizon. There is no guarantee that the optimal transmission expansion project is necessary to meet reliability standards during every period throughout its lifecycle, e.g., the line may only be required during peak-hours. Furthermore, there is also no guarantee that the line provides an economic benefit to the system for each period over its lifecycle. Based on optimization theory, it is known that the optimal investment over a long planning horizon, i.e., choosing one investment for all periods, need not be the same as the optimal investment for each individual period. In fact, the optimal transmission expansion project could propose the building of a line that is a detriment, regarding system reliability and economic efficiency, to the system during a few specific hours but is overall the best investment choice over a long planning horizon. Consequently, the fact that transmission switching may be beneficial and feasible does not guarantee inefficient transmission planning since they are two distinct problems. Moreover, it is well known that the redundancies built into the grid in order to handle a multitude of contingencies over a long planning horizon causes dispatch inefficiency. The purpose of optimal transmission switching is to remedy this issue by solving for the best topology to have for specific short-term operating conditions. The concept of short-term network reconfiguration is further supported by the fact that transmission expansion planning is a very granular process. Due to the high

12 534 K.W. Hedman et al. level of uncertainty regarding future network conditions, it is next to impossible to determine the optimal topology over such a long planning horizon. As system conditions change, it should be expected that the optimal topology may change from one period to the next and the choice regarding which topology is best for a specific period is better answered just prior to the period since there is less operational uncertainty. Finally, transmission expansion planning is a very difficult optimization problem, which limits the modeling complexity and further decreases the validity of the solution. These factors further argue in support of short-term network reconfiguration. 4 Optimal Transmission Switching 4.1 Optimal Power Flow The flow of electric energy follows Kirchhoff s laws. The Alternating Current Optimal Power Flow (ACOPF) problem is the network flow problem for the AC electric transmission grid and it is used to dispatch generation optimally subject to the network flow constraints and reliability constraints. The ACOPF optimization problem is a non-convex optimization problem involving trigonometric functions and, thus, it is a difficult problem to solve. Equations 4 and 5 represent the equations for the flow of electric power into bus n from transmission line k (line k is connected from bus m to bus n), see [26]. P k ¼ V 2 m G k V m V n ðg k cosðy m y n ÞþB k sinðy m y n ÞÞ; 8k (4) Q k ¼ V m V n ðg k sinðy m y n Þ B k cosðy m y n ÞÞ B k V 2 m ; 8k (5) Due to the difficulty with solving the ACOPF problem, a linear approximation is commonly used in its place, both by academia and by the industry. This problem is referred to as the Direct Current Optimal Power Flow (DCOPF) problem, which contains all linear constraints. Many assumptions are made to go from the equations listed by (4) and (5) to produce the DCOPF line flow constraint, (6). The voltage variables are assumed to take on a per unit value of one, the angle difference between buses m and n is assumed to be small so that the cosine terms are assumed to be one and the sine terms are assumed to be the angle difference itself, the reactive power flow constraint, (5), is ignored, and the resistance is assumed to be negligible. Note that the basic DCOPF model does not account for reactive power flow or losses; however, there are ways to account for reactive power and losses in the DCOPF. These assumptions produce the crude approximation that is listed by (6) below. P k ¼ B k ðy n y m Þ; 8k (6)

13 Flexible Transmission in the Smart Grid: Optimal Transmission Switching Direct Current Optimal Power Flow Problem For the purpose of this research, it is assumed that generators cost functions are linear, i.e. constant marginal cost 3, and, hence, the DCOPF problem is a Linear Programming (LP) problem since all of the constraints are linear. The objective is to minimize the total generation cost, (7); note that since the load throughout the network is assumed to be fixed, i.e., perfectly inelastic, minimizing the total cost is the same as maximizing the total market surplus. Constraint (8) restricts the difference of the bus voltage angles for any two buses that are connected by a transmission element. Constraint (9) specifies the operational constraints for generator g; for the basic DCOPF formulation, it is assumed that the minimum operating level for the generator is zero even though most generators do not have a zero minimum operating level. In order to enforce the true minimum operating levels of generators, i.e., if their minimum operating level is not zero, requires the use of a binary unit commitment variable thereby changing the linear program into a mixed integer linear program. Constraint (10) is the node balance constraint that specifies that the power flow into a bus must equal the power flow out of a bus. Generator supplies at a bus are injections while the load is a withdrawal. Constraint (11) represents the thermal capacity constraint on transmission line k; it is generally the case that P max k ¼ P min k. Finally, constraint (6) represents the linear approximation of the real power flow on transmission asset k. OPF formulations generally include lower and upper bound constraints on the voltage angle difference, y n y m, for any two buses that are connected by a transmission asset, see (8). In the DCOPF formulation, P k is equal to the susceptance times the angle difference thereby allowing (8) to be subsumed by (11), i.e., placing lower and upper bounds on the angle difference for a line places bounds on the power flow for that line. Therefore, the thermal capacity lower and upper bounds, P min k and P max k, can be replaced by B k y max and B k y min if those bounds are tighter than the thermal capacity constraints for the lines. Therefore, constraint (8) is not included in the optimal transmission switching DCOPF formulations that are presented in the following section; instead, we update P min k and P max k accordingly. Minimize: X c g P g (7) s.t. g y min y n y m y max ; 8k (8) 3 In reality, generator cost functions are quadratic in output (aside from startup and no load costs); however, in practice, such cost functions are approximated by piecewise linear functions represented as block offers at different marginal prices. The DCOPF formulation with piecewise linear cost functions is also a linear programming problem.

14 536 K.W. Hedman et al. 0 P g P max g ; 8g (9) X P k X P g ¼ d n ; 8n (10) 8kðn;:Þ 8kð:;nÞ P k þ X 8gðnÞ P min k P k P max k ; 8k (11) P k ¼ B k ðy n y m Þ; 8k (6) 4.3 Mathematical Modeling of Optimal Transmission Switching In order to introduce optimal transmission switching into the DCOPF, a binary variable is first needed to reflect the state of the transmission line. z k is used as a binary variable for transmission asset k; it takes on a value of one when the asset is in service (circuit breakers are closed) and it takes on a value of zero when the asset is out of service (circuit breakers are open). The first and easiest modification to make is to multiply the lower and upper bounds in (11) by the binary variable. Thus, when z k equals zero, the line flow variable will be forced to be zero; otherwise, if z k is one, the constraint (11) will appear in the OPF formulation in its original form. This modification is shown in the later formulations as (11.1). The modification to (11), however, is not sufficient. If zk equals zero and Pk equals zero, then (6) will force the bus angles to equal each other. This is not the desired outcome; if the line is taken out of service by the opening of an electrical switch, there should be no constraint on the angle difference for two buses that are not directly connected (unless there will be a breaker reclosing procedure to bring the line back into service). This creates a situation where (6) is modified into what is known as an indicator constraint. There are different ways that this relationship can be modeled. A simple way to implement this relationship is to break (6) into two inequality constraints, (6.1) and (6.2), and use what is known as a big M value. As a result, when zk equals one, the Mk in each inequality is multiplied by zero and these inequalities will then enforce (6) as desired. When zk equals zero, the value of Mk is large enough such that it allows yn and ym to take on different values as desired. The value of Mk does place an indirect bound on the difference between these two angles; however, this is the desired outcome. When there is a breaker reclosing procedure to bring a line back into service, the operator must limit the angle difference between the two buses that are about to be directly reconnected. The use of Mk provides a way to model this relationship. Furthermore, it is important to have Mk be as small as possible as it is well known that the use of a big M value to create such relationships substantially impacts the computational performance of the MIP. This reclosing rule provides that needed minimum value on Mk and for this formulation, Mk ¼ Bkyrec.

15 Flexible Transmission in the Smart Grid: Optimal Transmission Switching 537 Formulation 1: Minimize: X c g P g (7) g s.t. 0 P g P max g ; 8g (9) X P k X P g ¼ d n ; 8n (10) 8kðn;:Þ 8kð:;nÞ P k þ X 8gðnÞ P min k z k P k P max k z k ; 8k (11.1) B k ðy n y m Þ P k þð1 z k ÞM k 0; 8k (6.1) B k ðy n y m Þ P k ð1 z k ÞM k 0; 8k (6.2) z k 2f0; 1g; 8k (12) There are additional ways to introduce the state of the transmission asset into the DCOPF formulation. The second formulation introduces a new variable to the formulation to allow P k to be replaced by B k (g k y m ). Note that in (10.1), the y m is the voltage angle corresponding to the from bus for line k. Then, (6) is replaced with constraints (6.3) and (6.4) to force g k to equal y n if the line is in service; if the line is out of service, g k is not forced to equal y n by (6.3) and (6.4). Due to (11.2), g k will equal y m when the line is out of service. Thus, the new variable, g k, equals the to bus voltage angle value, y n, when the line is in service and it equals the from bus voltage angle value, y m, when the line is out of service. Once again, there is a reclosing rule that places a restriction on y n minus y m through (6.3) and (6.4) since g k equals y m when the line is out of service. The big M value in this formulation is represented by y rec to enforce this reclosing rule. Formulation 2: Minimize: X c g P g (7) s.t. X 8kðn;:Þ g 0 P g P max g ; 8g (9) B k ðg k y m Þ X P g ¼ d n ; 8n (10.1) 8kð:;nÞ B k ðg k y m ÞþX 8gðnÞ

16 538 K.W. Hedman et al. P min k z k B k ðg k y m ÞP max k z k ; 8k (11.2) y n g k þð1 z k Þy rec 0; 8k (6.3) y n g k ð1 z k Þy rec 0; 8k (6.4) z k 2f0; 1g; 8k (12) 5 Economic and Market Implications of Optimal Transmission Switching 5.1 Economic Savings Resulting from Optimal Transmission Switching Optimal transmission switching has been researched for various test cases and formulations, [19 25]; it has been studied with a DCOPF, an N-1 DCOPF, and a unit commitment N-1 DCOPF formulation with the IEEE 73-bus test case (RTS96 system), the IEEE 118-bus test case, and two large scale, 5000-bus test cases provided to the authors by the Independent System Operator of New England (ISO-NE). Table 1 presents the best found economic savings for these various formulations and test cases. Additional sensitivity studies were done with these test cases as well, see [19 25], with all studies showing that optimal transmission switching can provide substantial economic benefits. Since these test cases vary in size and generator costs, the best indicator of the potential of optimal transmission switching is the percent savings instead of the dollar savings. Of the solutions in Table 1, the only solution proven to be the optimal solution is the IEEE 118-bus DCOPF result. Consequently, the true optimal solutions for the rest of the test cases may provide even more economic savings. If optimal transmission switching can be practically implemented and save even a fraction of the savings that are shown here, such would be a remarkable result for the three-hundred billion dollar electric industry in the USA. Table 1 Economic savings from transmission switching for various test cases and formulations Formulation IEEE 73-Bus (RTS 96) IEEE 118-Bus ISONE 5000-Bus 1HR DCOPF % Savings 25% [19, 20] 13% [23] $ Savings $512 [19, 20] $62,000 [23] 1HR % Savings 8% [21] 16% [21] N-1 DCOPF $ Savings $8,480 [21] $530 [21] 24HR UC N-1 DCOPF % Savings 3.7% [22] $ Savings $120,000 [22]

17 Flexible Transmission in the Smart Grid: Optimal Transmission Switching 539 For the N-1 results, it was assumed that if a line is temporarily taken out of service for a given steady-state (no contingency) period and if a contingency occurs during that period, then the operator would not have the choice to reclose the line, i.e., place the line back in service immediately after the contingency. Thus, all of the N-1 solutions in Table 1 enforce N-1 without acknowledging the possibility to implement post-contingency transmission switching actions. This is a conservative approach since it is well known that there are SPSs that exist today involving postcontingency switching actions, [16]. The DCOPF results in Table 1 do not enforce N-1. However, these results still provide very useful information. First, these results demonstrate that the redundancy built into the grid, in order to handle a multitude of contingencies over a long planning horizon, does cause dispatch inefficiency. Second, the results estimate the potential savings for the concept of just-in-time transmission, [23]. The concept basically states that we should be able to co-optimize the topology with generation while accounting for the ability to implement actions similar to the SPSs that involve post-contingency switching actions. Transmission assets that are a detriment to dispatch efficiency can be kept offline during steady-state operating periods but they can be switched back into service, if needed, just-in-time once there is a contingency in order to bring the grid back to its redundant, reliable state. This concept is discussed in more detail in Sect In [24], the authors examined the potential yearly savings for the IEEE 118-bus test case with the DCOPF optimal transmission switching formulation. The unconstrained economic dispatch solution 4, which is a lower bound to the optimal transmission switching problem, for the IEEE 118-bus test case was 3.07% below the DCOPF solution. This is likely a unique result corresponding to this particular IEEE test case as most systems have a much larger gap between the unconstrained economic dispatch solution and the DCOPF solution. This gap defines the maximum potential savings for the optimal transmission switching DCOPF problem; for this yearly case study in [24], optimal transmission switching saved 3.05% out of this 3.07% gap. 5.2 Generation Cost, Generation Rent, Congestion Rent, and Load Payment Harnessing the control over transmission can be used for a variety of operational benefits; optimal transmission switching suggests that operators should co-optimize the generation with the network topology, while meeting reliability requirements, in order to reduce the overall system operating costs. This approach is not 4 The unconstrained economic dispatch problem is a dispatch problem without transmission network constraints.

18 540 K.W. Hedman et al. controversial for a vertically integrated utility that takes the role of serving its region at least cost as the savings would be passed on to the consumers. In standard Independent System Operator (ISO) markets that are based on a nodal pricing system, i.e., they use Locational Marginal Prices (LMPs), the goal of the operator is to maximize the market surplus while ensuring a reliable system (note that when load is perfectly inelastic minimizing the total cost achieves the same objective as maximizing the market surplus). LMPs are the dual variables (shadow prices) on the node-balance equations in the OPF formulation, the dual variable on Eq. 10 in Sect. 4.2; it reflects the marginal cost to deliver another unit of energy to that location in the network. With an LMP pricing system, generators are paid their LMPs and the load pays their LMP to consume. Even though optimal transmission switching increases the surplus in the market, there is no guarantee that, with the implementation of this new technology, all market participants will be better off than before. Figure 3 demonstrates the unpredictable impact optimal transmission switching can have on groups of market participants, [20]. The generation rent is the short term generation profit for all generators and, thus, the generation revenue is equal to the generation rent plus the generation cost, which can also be determined by summing each generator s production times its LMP (note that we are not including other payments made to generators, i.e., uplift payments). The load payment is defined as the sum of each load times its LMP. The congestion rent is defined as the sum of each line s flow times the dual variable on its capacity constraint, (11); this dual variable is often referred to as the flowgate marginal price. Since the DCOPF is an LP, it has a well defined dual. Based on duality theory, complementary 180% 160% 140% Generation Cost Generation Revenue Generation Rent Congestion Rent Load Payment 120% 100% 80% 60% 40% J=0 J=1 J=2 J=3 J=4 J=5 J=6 J=7 J=8 J=9 J=10 Case 1 Case 2 Best Fig. 3 Generation cost, generation rent, congestion rent, and load payment for various transmission switching solutions IEEE 118-bus test case, [20]

19 Flexible Transmission in the Smart Grid: Optimal Transmission Switching 541 slackness, strong duality, and by identifying the parts of the dual problem that reflect these four terms, it can be shown that the following identity holds: load payment generation rent congestion rent (objective of dual) ¼ generation cost (objective of primal). The congestion rent is often also labeled as the cost to send energy from a source to a sink location, which translates into the difference in LMP between these two locations times the quantity. The J index on the x-axis is a reflection of how many lines were opened; J ¼ 0 reflects the base case where all lines are in service; all values for J ¼ 0 are normalized to 100% in the graph to reflect each term s value from the optimal DCOPF solution for the original topology. The plots reflect how each term varies from one transmission switching solution to the next as compared to that term s value in the base case when all lines are in service. For instance, the load payment for the J ¼ 0 DCOPF solution is $7,757/h and the load payment is 79% of that value for the J ¼ 4 optimal transmission switching solution (with the restriction that only four lines can be opened). Case 1 and case 2 reflect solutions found by a heuristic technique and the best solution represents the optimal transmission switching solution for this IEEE 118-bus test case when there is no restriction on how many transmission assets can be temporarily taken out of service. This figure identifies the following interesting results: first, the majority of the savings are first obtained by only opening a few lines; this is an important result in term of computational complexity because it states that good solutions can be found by only searching for a few lines to open instead of considering all possible topology configurations (the other studies we have conducted agree with this statement). Next, there is a plateau affect in the sense that many transmission switching solutions are extremely close to the optimal solution. There are many solutions that are very close in objective and yet the results show that there can be drastically different outcomes for the market participants with these solutions. This is an interesting result as it is highly unlikely that this concept will be implemented and that optimality will be proven. It is also interesting to note that each term, except for the objective of the primal, the generation cost, is at some point below 100% as well as above 100%. Finally, the optimal solution ends up providing the generators with the highest generation rent out of all solutions and every category outside of the generation cost is at least 20% higher than the corresponding DCOPF solution, J ¼ 0, for the original topology. 5.3 LMPs MIPs do not have well defined duals; the LMPs from the optimal transmission switching MIP problem come from the node, which is an LP, in the branch and bound tree where the optimal solution was found or they can be reproduced by fixing the integer variables to their optimal values and then by solving the