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Static Switching Security in Multi-Period Transmission Switching Cong Liu, Member, IEEE, Jianhui Wang, Member, IEEE, and James Ostrowski, Member, IEEE
Abstract—Transmission switching can improve the economic benefits of a power system through changing its topology during operations. However, the switching operation itself represents a step change in power systems, which is, to some extent, similar to a contingency that can bring disturbances into systems. This paper proposes a new model for multi-period, static-security-constrained transmission switching. Because the power flow on the network will be redistributed instantaneously after the switching operations, the new model involves using disjunctive programming, which considers two sets of power flow equations under possibly different topologies before/after switching. Each switchable transmission element is modeled into four actions or disjunctions. An action transition diagram coupling of four actions in different hours is used to represent feasible paths of instantaneous changes in the element status. Disjunctive formulations are transformed into mixed-integer programming problems. We compare the difference of the previous transmission switching model and the proposed one by using several numerical tests and verify the effectiveness of our solution methodology in the six-bus and RTS-96 systems. Index Terms—Disjunctive programming, mixed-integer linear programming, static security analysis, transmission switching.
Variables and Functions: On/off status indicator of transmission branch. Action indicator of a transmission branch (four actions). Boolean variable associated with each disjunction in disjunctive programming. Power flow through a transmission branch. Generation dispatch of a unit (MW). Phase angle of bus. Generation cost curve. Constant and Set: Lower and upper bounds of generator output (MW).
NOMENCLATURE
Lower and upper bounds of flow through a transmission branch (MW).
Index:
Susceptance of a transmission line between bus and .
Index of generating unit. Index of transmission branch. Index of bus node.
Maximal ramp up/down rates (MW/time interval).
Index of time interval.
Load (MW).
Index of system status ( denotes normal operating status, denotes index of potential contingency).
Big-M positive parameter.
Index of disjunctions.
Set of branches.
Maximal allowable angle difference of the switching line (radian). Set of bus nodes.
Manuscript received June 06, 2011; revised October 28, 2011 and March 09, 2012; March 09, 2012; accepted March 21, 2012. Date of publication April 26, 2012; date of current version October 17, 2012. The submitted manuscript has been created by UChicago Argonne, LLC, Operator of Argonne National Laboratory (“Argonne”). Argonne, a U.S. Department of Energy Office of Science laboratory, is operated under Contract No. DE-AC02-06CH11357. This work is supported by the U.S. Department of Energy, Office of Electricity Delivery and Energy. Paper no. TPWRS-00535-2011. The authors are with the Decision and Information Sciences Division,Argonne National Laboratory, Argonne, IL 60439 USA, and also with the University of Tennessee, Knoxville, TN 37996 USA (e-mail:
[email protected]; jianhui.
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPWRS.2012.2192486 0885-8950/$31.00 © 2012 IEEE
Number of components in power system (generator and branch). Set of time intervals. Set of system status. Set of single-generator contingencies and set of single-branch contingencies. Contingency indicator of generator or transmission line .
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I. INTRODUCTION
T
HE electric power system can be considered as a real-time balanced supply chain in which the commodity (power/ energy) is transferred from the supplier (power plants) to customers (electricity users) through a transmission network. In power system operations, the topology of the transmission network is usually fixed, unless a transmission element is incapable of providing function due to forced outages or maintenance. In practice, researchers in both industry and academia have continually tried to improve voltage profiles, correct contingencies, or reduce power losses through changing topology and adjusting electrical parameters by using flexible AC transmission systems (FACTS) devices [1]–[6]. The concept that transmission switching can increase economic benefits and improve social welfare was first introduced in an innovative study by O’Neill et al. in a market environment [7]. Fisher et al. [8] presented and analyzed the optimal transmission switching problem and showed that optimizing both generation dispatch and network topology can result in substantial savings. The Fisher et al. study used a mixed-integer programming (MIP) model that employed binary variables to represent the transmission line action. Hedman et al. [9] used a similar model, but their model contains sensitivity analysis for the system uncertainties that they encounter when employing transmission switching. The paper concludes that topology changes could affect locational marginal prices, load payments, generation revenues, and congestion costs. Hedman et al. [10] and Khodaei et al. [11] presented a model that makes decisions regarding both unit commitment and transmission switching. Different decomposition strategies are used to solve the model. The studies described above found that transmission switching could achieve large reductions in operating costs. Following the publication of these studies, impacts on the security and reliability of power system operations became major concerns for practical implementation of transmission switching. Hedman et al. [12] considered the N-1 contingency reliability criterion in transmission switching. Hedman et al. [10] further developed a decomposition strategy to address the computational challenges with the security-constrained unit commitment and transmission switching with N-1 reliability. The proposed decomposition method can find a feasible solution faster by solving two smaller MIP problems instead of the original large problem. However the optimality of the solution cannot be guaranteed. While these transmission switching studies improve power system modeling, researchers often ignore an important fact: the switching operation itself represents a step change in power systems, which is, to some extent, similar to a contingency. Although transmission switching is a type of scheduled, deterministic, discrete event, while the forced outage of a transmission line is a stochastic, uncontrollable, discrete event, both actually introduce step disturbances into power systems. Security and reliability analyses are therefore needed to help prevent the adverse impacts on power system operations that can be brought about by transmission switching. Under different time scales and assumptions, people apply different models to analyze the behavior of power systems.
Fig. 1. Security-based optimal transmission switching.
Likewise, researchers analyze the security and reliability effects of implementing transmission switching in different hierarchies, as shown in Fig. 1. In the operations planning stage, on a longer time scale, static security-constrained optimal transmission switching is the first layer of defense for power system operations. A mathematic model is usually related to algebraic equations. On a shorter time scale, changing the network topologies or operating conditions, as occurs in switching operations, can cause transient phenomena in power systems. Because of the nature of the physical phenomena, power system transients can be divided into electromechanical and electromagnetic processes [1], [13]. Transmission switching may change the angle stability limits of transmission lines or cause interaction between the (electric) energy stored in capacitors and the (magnetic) energy stored in inductors. Such an interaction can, for example, motivate surge over-voltage during a switching operation. To analyze these phenomena, researchers must study the characteristics of dynamic systems or develop numeric calculations of differential equations. This paper focuses on the development of a new model for static security-constrained transmission switching by disjunctive programming. Scheduled switching operations can change the topology of a network in much the same way as line outages. In static security analysis, it is assumed that the outputs of generating units remain the same just before and after line outages [1], [14]. Likewise, switching a transmission line would not instantly change the outputs of generating units. However, the power flow will change immediately after the network topology changes. After switching, the power flow will be redistributed instantaneously. Therefore, two groups of power flow equations under possibly two different topologies in each time interval are applied for restricting generation dispatches in our proposed model. The action of each switchable transmission element is modeled into four categories or disjunctions: (0) “stay offline,” (1) “stay online,” (2) “switch on,” and (3) “switch off.” Transitions among the four actions of all transmission elements represent the instantaneous changes in topology. In addition, we propose an action transition diagram to define feasible transition paths among the four actions during two adjacent time intervals. The new model not only considers the robustness of the
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system under contingencies, but it also ensures the system static security at the instant of switching operations. This paper also describes the process used to transform the disjunctive formulations into an MIP problem. The remainder of this paper contains the following sections: Section II presents the existing model and a new model for transmission switching that considers static security based on disjunctive programming; Section III describes the solution of the proposed model; and Sections IV presents case studies. Discussions and conclusions are drawn in Sections V and VI, respectively. II. MODELING OF TRANSMISSION SWITCHING A. Disjunctive Programming Formulation for Optimal Transmission Switching Disjunctive programming is one type of optimization problem aiming to find the minimum or maximum of a real objective function by systematically choosing the values of decision variables from related feasible sets while satisfying a logical system of conjunctive and disjunctive statements. In a linear case, the feasible set of a disjunctive linear programming is the union of individual polyhedral. This approach facilitates the development of the models by the underlying logic structure of the problem as well as the intuitive formulation process. The objective function of the optimal transmission switching problem—like the economic dispatch (ED) problem—is to minimize operating costs during the normal operating state as shown in objective function (1). The cost curve of a generating unit in (1) is usually a convex quadratic function, which is linearized by using a piecewise function for this study. The constraints (2) represent the ramping characteristic of a generator in the normal operating state . The upper bound and lower bound of a generator’s output are given in constraint (3). The constraint (4) represents Kirchhoff’s laws governing power flow: the flow withdrawn from a node should be equal to the flow injected into the node. The relationship of the phase angle difference between nodes and the power flow through the branch are governed by constraint (5), in which and are Boolean variables corresponding to two exclusive disjunctions. Each disjunction, containing a set of different linear equations and/or inequalities, is connected with the others by the logical OR operator that enforces that only one set of equations is active. If is true, transmission branch is providing service and transmitting energy, and the power flow through the branch must be less than a given maximal limit . In the other is true. The selection of disjunction indicates that case, the transmission branch is not included in the system, and the power through the branch is zero: (1) (2) (3) (4)
are two end nodes of branch
(5) (6) (7) (8)
The N-1 reliability criterion is a deterministic reliability standard that obliges the system to remain secure despite the loss of any single transmission element or generator. The selected critical contingencies should be satisfied while system operators adopt transmission switching to improve system operating cost during operation. As defined in the nomenclature, denotes index of system state. represents supposed generator or transmission element forced outage in the system. We introduce binary parameters (not decision variable) and to indicate whether the predefined contingency is related to generating unit or transmission branch . If all and are equal to 1, there are no contingencies and the system is in a normal operating status. in constraint (3) ensures that the output of a generating unit is zero if the generator is the considered contingency. Likewise, parameter indicates the status of transmission branch if it is not switched off intentionally. However, if transmission branch is switched off intentionally, the first disjunction in (5) becomes active and the branch is offline no matter what the value of is. In this case, the corresponding contingency constraints become redundant. When a generator contingency occurs, other committed generating units are re-dispatched to cover imbalance of generation and load caused by the loss of the generator. The generation re-dispatch has to satisfy the constraints in constraints (7), which govern the adjustment distance of generation outputs to account for the change from a normal operating state to a contingency state. When a forced transmission outage occurs, the short-term emergency rate of a transmission line is used as a parameter in (5), and the ramping distances of generating units are set as zero in (6). The short-term emergency rate is higher than the normal emergency rate of a transmission line. For example, in the RTS96 test system, the short-term emergency rate is around 125% of the normal emergency rate for all transmission branches. B. Proposed Model on Optimal Transmission Switching Considering Static Switching Security According to the formulations above, the outputs of generating units at the current time only satisfy the power flow equations and line flow bounds under the current network topology. All generators are considered to maintain approximately the same outputs at the moment of alternating network topology if there is no islanding in the grid. However, as far as the power flow goes, we can deduce that the flow that passes through the branches will redistribute instantaneously after switching. The instant changes in power flow in the transmission branches may lead to violations of their normal flow rate capacity or possibly to other violations, such as voltage over-bounding, if AC power
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TABLE I ACTIONS OF TRANSMISSION BRANCHES
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same as those in the previous model. However, compared with constraint (3) and (4), there are two generating capacity constraints and two nodal flow balance constraints (9)–(12) during each time period in the proposed model. and are two different points in time that occur before and after the intentional switching in topology. However, net power injections (generation minus load) in each node are the same at and as indicated in (15) and (16). The constraint (13) consists of four disjunctions instead of the two shown in (5). The action transitions diagram can be mapped into logic expressions (14). It should be noted that the fault may happen in either or , at which the corresponding network topologies are different: (9) (10) (11) (12)
Fig. 2. Action transition path diagram.
flow equations are used in the transmission switching problem. The model in Section II-A ignores the fact that the transmission switching changes the topology as well as the power flows through the network instantly. The dispatch before switching will not be able to adjust the system to another secure operating point immediately given generation re-dispatch is a relatively slow process. To prevent security risks to the system during switching operations, we propose a new model that can address multi-period optimal transmission switching while ensuring the static security of the entire system. In the proposed model, each switchable branch has four operating actions, as shown in Table I. In the first action (Action 0), the branch stays offline (circuits breaker is always open). In the second action (Action 1), the branch is always online and the circuits breaker is always closed. In the third and fourth actions (Actions 2 and 3), the branch changes its status from offline to online or online to offline, respectively, during the same time interval. Note that the four actions are exclusive from each other: one, and only one, action for each switchable branch is active during each time interval. In addition, action transitions paths between two adjacent time intervals have to obey the defined rules. For example, Action 0 has to transfer to Action 2 before its transition to Action 1, which means that line has to be switched on before it can be switched off. The action transitions diagram, as depicted in Fig. 2, shows four actions for each switchable branch and their feasible transition paths. We describe the proposed model in (1), (2), and (9)–(18). The objective function, lower and upper bounds of generation outputs, and ramp rate limits of generating units are exactly the
are two end nodes of branch
(13) (14) (15) (16) (17) (18)
The proposed model can be used in both day-ahead and realtime scheduling. Unit commitment (UC) can replace ED in the optimal transmission switching problem [10], [11]. UC, to some extent, can be considered a more general form of economic dispatch. Binary variables are introduced to represent the online/offline status of the generating units in the UC problem. In addition, unlike ED, some prevailing constraints (e.g., minimum on/off times, start-up and shut-down costs, generation capacity limits, operating reserve requirement) are included in the UC problem. Co-optimization of transmission switching and UC will not impact the proposed constraints for the transmission switching part, i.e., (11)–(14). For simplicity and saving space
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here, we do not present the formulations of UC in this paper. Complete descriptions of the UC problem are provided in references [14]–[17]. III. SOLUTION METHODOLOGY A. Reformulation of Disjunctive Programming Into MIP The disjunctive programming model can be solved in two different ways. The first is to use the branch-and-bound method, in which branching is implemented in term of logic conditions and propositions [19]. The second is to transform linear disjunctive programming equations to MIP-based equations. Sawaya et al. [20] and Raman et al. [21] showed that linear disjunctive programming also can be systematically transformed into an equivalent MIP whose continuous relaxation is often tight, leading to more efficient solution methods. Furthermore, some commercial solvers, such as CPLEX or Gurobi, which are based on branch-and-cut algorithms, can be used to solve the problem conveniently. In this study, we use the second method and transform the disjunctive programming model into MIP formulations. The equivalent MIP formulations for constraint (5) are shown as follows: (19) (20) is a big positive number that can be determined where by finding the shortest path between two terminal buses of opened branch [12]. A similar method is used in the transmission planning problem [25]. However, it is not easy to find the shortest distance in the transmission switching problem because the topology is still unfixed until the solution to the transmission switching problem is obtained. Here, we set as a conservative value . The variable is the integer decision variable to indicate whether the transmission branch is on or off. By using binary variables to replace Boolean variables in (13), (11)–(14) becomes (21)–(27), and logic expression (14) can be replaced by (25) and (26): (21) (22) (23) (24) (25)
Fig. 3. Rolling runs for optimal transmission switching.
B. Decomposition for the Multi-Period Transmission Switching The optimal transmission switching problem is a non-deterministic polynomial-time hard problem that may have exponential time complexity regardless of whether an MIP [27] or a disjunctive programming formulation is used. If the scheduling horizon is relatively long (e.g., 24 h), it is very difficult to obtain the optimal solution within a reasonable time. In our proposed model, the coupling constraints between adjacent hours include the ramping constraints and the static switching security constraints as shown in (21)–(25). In order to obtain a feasible solution more quickly, we divide the scheduling horizon into several sub-periods. The transmission switching problems are then run in a rolling fashion, as shown in Fig. 3. The dispatches and the transmission line statuses obtained from the previous run will serve as the initial point of the next run. By this way, it is ensured that the coupling constraints are satisfied and the obtained result is a feasible solution. We should clarify that this method still cannot guaranty the optimality because the entire scheduling period is not solved at one time. However, in a real operational setting, obtaining the best feasible solution within a given time limit is more practical than proving the optimality. An alternative method is to introduce Lagrangian multipliers to relax the bounded or violated coupling constraints between two adjacent sub-periods. Then the dual problem of the original problem will be decomposed into several sub-problems associated with each sub-period. With the updated Lagrangian multipliers, the dual problem is solved iteratively. The solution of the dual problem will provide a lower bound for the solution. After solving the dual problem in each loop, we fix the topology of the network and solve the security-constrained economic dispatch problem which can result in an upper bound for the solution. The program can stop until the gap between the upper bound and lower bound satisfies a small predefined value.
(26) (27) Although there are more binary variables in the proposed model compared with the previous model (19), (20), the constraint in (25) is a set of constraints where at most one variable may be nonzero. This special type of constraints can be used branching strategy and accelerate the calculation in CPLEX [26].
IV. CASE STUDIES We use a six-bus power system and a RTS96 test system to compare our proposed model with the previous transmission switching model. The program is coded in AMPL, which formulates the problem and sends it to the MIP solver CPLEX 12.2. All case studies are solved on a computer with an Intel i7 core and 8-GB memories.
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TABLE II UNIT COMMITMENT OF SIX-BUS SYSTEM
Fig. 4. Hourly generation of unit 1 and 2 in cases.
TABLE III STATUSES OF TRANSMISSION BRANCHES OF SIX-BUS SYSTEM
A. Six-Bus Power System We first calculate the minimal cost of transmission-constrained unit commitment and generation dispatch with a deterministic hourly load. The characteristics of the generators, lines, and hourly load distribution over the 24-h horizon are the same as those in Khodaei et al. [11], except that we change the flow limit of transmission line 4–5 to 30 MW. The initial statuses of all transmission lines are set to be online. The global solution can be obtained by setting the MIP gap to zero. The details of the three cases conducted are shown as follows: Case 1) Unit commitment without transmission switching Case 2) Existing transmission switching model Case 3) Proposed transmission switching model considering static switching security The unit commitment solution for Case 1 is shown in Table II, in which unit 1 and unit 2 are on all 24 h while unit 3 is committed from hour 9 because of a load increase and congestion of transmission line 4–5. The unit commitment status at hour 0 represents the initial state of a unit before the operating day. Because there is no transmission switching in this case, the statuses of all transmission lines are always on, as shown in Table III. Transmission congestion leads to lower dispatch levels of unit 1, even if it is the least expensive. The expensive units, 2 and 3, are turned on to help mitigate transmission flow violations. The total operating cost is $128 862. In Case 2, we consider the co-optimization of unit commitment and transmission switching and use the previous transmission switching constraints (5) or (19) and (20), which are similar to those used in Hedman et al. [9] and Khodaei et al. [11]. Transmission switching makes system scheduling and operations more flexible and economical. As shown in Table II,
the expensive unit 2 in Case 2 is no longer online during hours 1–10 and hours 23–24 in comparison with Case 1. Moreover, unit 3 does not need to be committed at all in the 24 h. Table III shows the statuses of transmission lines 2–4 and 4–5. Switching off some transmission lines can mitigate the congestion, as well as reduce the generation cost. The operating cost in Case 2 is $121 123, which is lower than that in Case 1. In Case 3, we co-optimize unit commitment and transmission switching by adopting our proposed transmission switching model. No contingencies are included in this test. The results of unit commitment listed in Table II indicate that, different from Case 2, unit 2 in Case 3 is committed between hours 1 and 2. In Case 3, transmission lines 2–4 and 4–5 can operate in four actions that more accurately specify their statuses, as shown in Table III. For example, the transmission line 2–4 is switched from online to offline at hour 15 and switched from online to offline at hour 18. Fig. 4 compares dispatches of unit 1 and unit 2 in Case 2 and Case 3, respectively. The green line at hours 1, 2, 15, and 18 is lower than the blue line, meaning that unit 1 generates less electricity in Case 3 than in Case 2 at those hours. On the other hand, unit 2 in Case 3 generates more power than in Case 2 because the generation dispatch in a single hour in Case 3 have to satisfy two sets of DC power flow equations before and after switching under two possibly distinct topologies. When the switching actions of lines 2–4 and 4–5 alter the network topology at hours 1, 15, and 18, the power flows through lines 2–4 and 5–6 at and of hour 1, 15, and 18 are different as shown in Fig. 5. The total operating cost in Case 3 is $121 557. Although the operating cost in Case 3, based on our proposed model, is higher than that in Case 2, the schedule resulting from Case 3 is more reasonable and secure because the model in Case 3 considers the static switching security constraints just before and after the switching operation. The model used in Case 2 ignores the fact that scheduled transmission switching instantly changes the topology, as well as the power flows throughout the network. The instant changes in network power flow may lead to violations of the normal transmission line capacity, subjecting the system to security risks. B. RTS96 Power System We use the modified RTS 96 system [12], [22]. The modified RTS 96 system has three identical zones, 117 transmission branches, 111 generators, and 73 buses. The peak load is set at
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TABLE V COMPUTATIONAL RESULTS OF DIFFERENT MODELS
TABLE VI TRANSMISSION BRANCH STATUS IN CASE 1 AND CASE 2
Fig. 5. Power flow through branches before and after switching operation.
TABLE IV HOURLY LOADS AND NUMBER OF COMMITTED UNITS
8547 MW. The initial statuses of all transmission lines at time interval 0 are online. The spinning reserve and operating reserve requirement are set as 4% and 7.5% of the hourly load. The MIP gap is set as 0.8%. However, we record the best upper bound (feasible solution) CPLEX finds as the solution if CPLEX runs out of memory. 1) Comparison Between the Previous and Proposed Models: Two cases are considered to compare transmission switching with and without static switching security constraints. Case 0) No transmission switching Case 1) Previous transmission switching model Case 2) Proposed transmission switching model considering static switching security Four periods are considered in this case and the load at each period is shown in Table IV. The results in Table V show the operating cost in both Case 2 and Case 3 decreases considerably compared to Case 1. The proposed model results in a higher operating cost in Case 3 compared to the previous model in Case 2, primarily because the new formulation has additional static switching security constraints. However, the operating cost of the more reliable schedule in Case 3 only increases by a small amount. The third and fourth columns in Table V show the branch-and-bound nodes and the CPU times in different cases. It should be of note that the computation time in Case 3 does not have to be higher than that in Case 2 since the additional variables and constraints introduced in the new formulation reduce the solution space of the previous formulation. The unit commitment and transmission branch status are shown in Table VI and Table VII. The unit commitment and transmission line status that remain the same in Case 2 and Case 3 are not listed in the tables. As can be seen, the new model in Case 3 leads to the different commitment as well as switching actions compared to the previous model in Case 2.
TABLE VII UNIT COMMITMENT STATUS IN CASE 1 AND CASE 2
Fig. 6. Violations on transmission branches.
To show that the new model considering the static switching security constraints will result in a more reliable generation and switching schedule, we illustrate in Fig. 6 the network violations that result from lack of modeling the static security in the previous model. In the figure, it can be seen that a number of transmission branches are overloaded beyond their normal transmission capacity limits when the static security constraints are absent. 2) N-1 Reliability: Inclusion of all N-1 contingency constraints into transmission switching makes the MIP problem even harder to solve. The huge number of N-1 contingency
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constraints increase the computation time significantly for solving linear programming (LP) relaxations of the MIP problem in CPLEX. The computational mechanism behind the branch-and-cut algorithm is to generate a branch tree and solve LP at some nodes of the tree to obtain a lower bound while using some heuristic methods to find the upper bound (feasible solution) so that the gap between the lower and upper bounds can decrease iteratively. LP is a very important component of the branch-and-cut algorithm and is called hundreds and thousands of times in solving the problem. Consequently, the slower LP solution times mean the increase in overall solution time. In our case, even though we only select 30 switchable branches and solve a 2-h transmission switching problem based on our proposed model, the wall clock time for solving the problem is about 17 h. In practice, only a few critical contingency cases are included in the unit commitment formulation or the contingency analysis is separated from the unit commitment problem [11], [14]–[16]. Similarly, contingency analysis can be implemented after transmission switching. In other words, if the generation schedule from transmission switching without contingency constraints lead to any violations in contingency cases, these contingency cases or critical constraints can be included into the optimal transmission switching problem and the updated optimal transmission switching problem is solved again. By using this method, the computational time can be improved. However, there is a need to develop appropriate contingency screening techniques to select critical contingencies under a variable topology caused by transmission switching. Another heuristic method to deal with the N-1 reliability criterion was presented in [10]. The status of the network topology (integer variables) can be fixed after optimal transmission switching and then the N-1 contingency constraints can be checked in the UC or ED problem. 3) Multi-Period (24 H) Transmission Switching: As a 24-h co-optimization of unit commitment and transmission switching for the RTS96 system is very difficult to solve even with the most advanced commercial solvers, heuristic decomposition methods are used to separate security-constrained unit commitment from transmission switching [10], [11]. The optimal transmission switching problem can be further decomposed by hour in [10], because ramping constraints are not active and bounded. This heuristic decomposition can find a feasible solution in a reasonable time but cannot guaranty the optimality of the solution. Here we take an approach similar to that of [10]. We solve the unit commitment problem first without transmission switching. Then, given the unit commitment status, the optimal transmission switching is solved. However, as our proposed model has more coupling constraints related to static security in addition to the ramping constraints, we use the rolling method as shown in Fig. 3. The results from the previous sub-period are used as the initial conditions in the next sub-period. Our results show the final gaps before CPLEX stops calculations in each run are 0.38%, 0.25%, 0.61%, 0.35%, 0.26%, and 0.67%. The total wall clock time for running the six sub-problems is 132.5 s with at most 5 open branches. By
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TABLE VIII SOLUTION TIMES WITH HEURISTIC METHOD
setting different maximum numbers of open branches, the solution times are shown in Table VIII. The results show this decomposition method makes the 24-time interval transmission switching problem solvable. But it still cannot guarantee the optimality as we mentioned before. V. DISCUSSIONS In traditional power system scheduling problems, the real continuous load profile is approximated by discretization, which is essentially one type of difference approximation method. This discretization leads to the discrete generation output levels between the hours in Fig. 4. In practice, hourly scheduling models cannot exactly reflect the intra-hour cost and real-time generation outputs. In practical physical systems, generation outputs of committed units change continuously within an hour and between adjacent hours. In this paper, we focus on the modeling of the step change in power flow before and after switching actions while capturing the fact that the generation output cannot be changed instantly along with the switching actions. The assumption of a step change in generation output between adjacent hours is still used. Although we acknowledge that modeling of continuous changes in generation output between adjecent hours is very meaningful and practical, such as modeling task is complicated and deserves another full paper on its own. In addition, the model proposed in this paper is not the only method to improve the static switching security during operations. An alternative method is to adopt the existing transmission switching models [8]–[12] with very short time intervals. Since ramp rates of generators are constant, the load and dispatch of generating units at adjacent time points are almost the same if the time interval is short enough. In this case, violations of static security constraints caused by switching transmission lines may be limited. VI. CONCLUSIONS This paper proposes a new static, security-constrained, optimal transmission switching model that is based on disjunctive programming. The proposed model considers the power flow equations under different topologies before and after switching actions in each time interval. Our proposed model improves the static security of the power system operations when implementing transmission switching. The numerical results show that the new model can still lead to considerable economic benefits by applying transmission switching. REFERENCES [1] X. Wang, Y. Song, and M. Irving, Modern Power System Analysis. New York: Springer, 2008.
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[2] M. Noroozian, L. Angquist, M. Ghandhari, and G. Andersson, “Use of UPFC for optimal power flow control,” IEEE Trans. Power Del., vol. 12, no. 4, pp. 1629–1634, Oct. 1997. [3] G. Schnyder and H. Glavitsch, “Integrated security control using an optimal power flow and switching concepts,” IEEE Trans. Power Syst., vol. 3, no. 2, pp. 782–790, May 1988. [4] J. G. Rolim and L. J. B. Machado, “A study of the use of corrective switching in transmission systems,” IEEE Trans. Power Syst., vol. 14, no. 1, pp. 336–341, Feb. 1999. [5] W. Shao and V. Vittal, “Corrective switching algorithm for relieving overloads and voltage violations,” IEEE Trans. Power Syst., vol. 20, no. 4, pp. 1877–1885, Nov. 2005. [6] G. Granelli, M. Montagna, F. Zanellini, P. Bresesti, R. Vailati, and M. Innorta, “Optimal network reconfiguration for congestion management by deterministic and genetic algorithms,” Elect. Power Syst. Res., vol. 76, pp. 549–556, Apr. 2006. [7] R. P. O’Neill, R. Baldick, U. Helman, M. H. Rothkopf, and W. Stewart, “Dispatchable transmission in RTO markets,” IEEE Trans. Power Syst., vol. 20, no. 1, pp. 171–179, Feb. 2005. [8] E. B. Fisher, R. P. O’Neill, and M. C. Ferris, “Optimal transmission switching,” IEEE Trans Power Syst., vol. 23, no. 3, pp. 1346–1355, Aug. 2008. [9] K. W. Hedman, R. P. O’Neill, E. B. Fisher, and S. S. Oren, “Transmission switching—Sensitivity analysis and extensions,” IEEE Trans. Power Syst., vol. 23, no. 3, pp. 1469–1479, Aug. 2008. [10] K. M. Hedman, M. C. Ferris, R. P. O’ Neill, E. B. Fisher, and S. S. Oren, “Optimization of generation unit commitment and transmission switching with N-1 reliability,” IEEE Trans. Power Syst., vol. 25, no. 2, pp. 1052–1061, May 2010. [11] A. Khodaei and M. Shahidehpour, “Transmission switching in security-constrained unit commitment,” IEEE Trans. Power Syst., vol. 25, no. 4, pp. 1937–1945, Nov. 2010. [12] K. W. Hedman, R. P. Oren, E. B. Fisher, and S. S. Oren, “Transmission switching with contingency analysis,” IEEE Trans. Power Syst., vol. 24, no. 3, pp. 1577–1586, Aug. 2009. [13] A. Gomez-Exposito, A. J. Conejo, and C. Canizares, Electric Energy Systems-Analysis and Operation. Boca Raton, FL: CRC/Taylor & Francis, 2008. [14] A. Wood and B. Wollenberg, Power Generation, Operation, and Control, 2nd ed. New York: Wiley, 1996. [15] M. Shahidehpour, H. Yamin, and Z. Li, Market Operations in Electric Power Systems. New York: Wiley, 2002. [16] F. Bouffard, F. D. Galiana, and A. J. Conejo, “Market-clearing with stochastic security-part I: Formulation,” IEEE Trans. Power Syst., vol. 20, no. 4, pp. 1818–1826, Nov. 2005. [17] D. N. Simopoulos, S. D. Kavatza, and C. D. Vournas, “Reliability constrained unit commitment using simulated annealing,” IEEE Trans. Power Syst., vol. 21, no. 4, pp. 1699–1706, Nov. 2006. [18] C. Liu, M. Shahidehpour, and L. Wu, “Extended benders decomposition for two stage SCUC,” IEEE Trans. Power Syst., vol. 25, no. 2, pp. 1192–1194, May 2010. [19] E. Balas, 50 Years of Integer Programming-Disjunctive Programming. New York: Springer, 2010. [20] N. W. Sawaya and I. E. Grossmann, “A cutting plane method for solving linear generalized disjunctive programming,” Comput. Chem. Eng., vol. 29, no. 9, pp. 1891–1913, 2005. [21] R. Raman and I. E. Grossmann, “Modeling and computational techniques for logic based integer programming,” Comput. Chem. Eng., vol. 18, no. 7, pp. 563–578, 1994. [22] C. Grigg and P. Wong et al., “The IEEE reliability test system—1996, a report prepared by the reliability test system task force of the application of probability methods subcommittee,” IEEE Trans. Power Syst., vol. 14, no. 3, pp. 1010–1020, Aug. 1999.
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[23] J. McCalley and S. Asgarpoor et al., “Probabilistic security assessment for power system operations,” in Proc. IEEE PES General Meeting, Jun. 2004, vol. 1, pp. 212–220, 2004. [24] A. L. Motto, F. D. Galiana, A. J. Conejo, and J. M. Arroyo, “Network constrained multi-period auction for a pool-based electricity market,” EEE Trans. Power Syst., vol. 17, no. 3, pp. 646–653, Aug. 2002. [25] S. Binato, M. Veiga, F. Pereira, and S. Granville, “A new benders decomposition approach to solve power transmission network design problems,” EEE Trans. Power Syst., vol. 16, no. 2, pp. 235–240, May 2001. [26] IBM ILOG CPLEX V12.1-User’s Manual for CPLEX. [Online]. Available: ftp://ftp.software.ibm.com/software/websphere/ilog/docs/optimization/cplex/ps_usrmancplex.pdf. [27] M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness. New York: Freeman, 1979.
Cong Liu (S’08–M’10) received the B.S. and M.S. degrees in electrical engineering from Xi’an Jiaotong University, Xi’an, China, in 2003 and 2006, respectively, and the Ph.D. degree from the Illinois Institute of Technology, Chicago, in 2010. Currently, he is working as an energy systems computational engineer in the Decision and Information Sciences Division of Argonne National Laboratory, Argonne, IL. His research interests include numerical computation, optimization, and control of power systems.
Jianhui Wang (M’07) received the Ph.D. degree in electrical engineering from Illinois Institute of Technology, Chicago, in 2007. Presently, he is a Computational Engineer with the Decision and Information Sciences Division at Argonne National Laboratory, Argonne, IL. Dr. Wang is the chair of the IEEE Power & Energy Society (PES) power system operation methods subcommittee and co-chair of an IEEE task force on the integration of wind and solar power into power system operations. He is an editor of the IEEE TRANSACTIONS ON SMART GRID, a guest editor of a special issue on Electrification of Transportation of the IEEE POWER AND ENERGY MAGAINE, and an editorial board member of APPLIED ENERGY. He is the technical program chair of the IEEE Innovative Smart Grid Technologies conference 2012.
James Ostrowski (M’10) received the Ph.D. degree in the Industrial and Systems Engineering Department at Lehigh University, Lehigh, PA. While at Lehigh, he was awarded second place in the George Nicholson Student Paper competition for his work dealing with symmetry in integer programming. He was working in the Decision and Information Sciences Division of Argonne National Laboratory, Argonne, IL. Currently, he is an Assistant Professor at the University of Tennessee, Knoxville. His research interests include integer and stochastic programming.
