Network Reconfiguration for Service Restoration in Shipboard Power Distribution Systems

Transcription

1 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 4, NOVEMBER Network Reconfiguration for Service Restoration in Shipboard Power Distribution Systems Karen L. Butler, Senior Member, IEEE, N. D. R. Sarma, Senior Member, IEEE, and V. Ragendra Prasad Abstract The electric power systems of U.S. Navy ships supply energy to sophisticated systems for weapons, communications, navigation and operation. Circuit breakers (CBs) and fuses are provided at different locations for isolation of faulted loads, generators or distribution system from unfaulted portions of the system. These faults could be due to widespread system fault resulting from battle damage or material casualties of individual loads or cables. After the faults and subsequent isolation of the faults, there will be unfaulted sections that are left without supply. Fast restoration of supply to these unfaulted sections of the SPS is necessary for system survivability. This paper presents a new method to reconfigure the network to restore service to unfaulted sections of the system. The problem is formulated as a variation of fixed charge network flow problem. The method is illustrated using various case studies on a small power system with similar topology to a shipboard power system. Index Terms Network flow method, optimization, reconfiguration, restoration, shipboard power systems. I. INTRODUCTION SHIPBOARD Power Systems (SPS) consist of generators that are connected in ring configuration through generator switchboard [1]. Bus tie circuit breakers interconnect the generator switchboards that allow for the transfer of power from one switchboard to another. Load centers and some loads are supplied from generator switchboards. Load centers in turn supply power to loads directly and supply power to power panels to which some loads are connected. Feeders supplying power to load centers, power panels, loads are radial in nature. Hence the system below the generator switchboards, referred to as the shipboard distribution system by the authors, is radial. Loads are categorized as either vital or nonvital. For vital loads, two sources of power (normal and alternate supply) are provided from separate sources via automatic bus transfers (ABTs) or manual bus transfers (MBTs). Circuit breakers (CBs) and fuses are provided at different locations for isolation of faulted loads, generators or distribution system from unfaulted portions of the system. These faults could be due to widespread system fault resulting from battle damage or material casualties of individual loads or cables. After the faults and subsequent isolation of the faults, there will be unfaulted sections that are left without supply. Fast restoration of Manuscript received April 7, 2000; revised June 7, This work was supported by the Office of Naval Research, USA under Grant N K. L. Butler and N. D. R. Sarma are with the Power System Automation Lab and the Department of Electrical Engineering, Texas A&M University, College Station, TX USA ( klbutler@ee.tamu.edu; ndrsarma@ieee.org). V. R. Prasad is with the Knowledge Based Systems, Inc., College Station, TX USA ( rajvelaga@earthlink.net). Publisher Item Identifier S (01)09424-X. supply to these unfaulted sections of the SPS is necessary for system survivability. Also it is important to maintain the radial nature of the system, for ease of fault location and isolation, and coordination of the protective devices. Additionally, the capacities of the generators and cables must not be violated and voltage magnitudes at each node should be within tolerable limits. The existing shipboard protection systems have several shortcomings in providing continuous supply under battle and certain major failure conditions. The control strategies that are implemented when these types of damage occur are not effective in isolating only the loads affected by the damage, and most significantly are highly dependent on human intervention to manually reconfigure the distribution system to restore power to healthy loads. With the U.S. Navy demands for reduced manning and increased system survivability, new techniques are needed which efficiently and automatically restore service under catastrophic situations. Their goals are to increase survivability, eliminate human mistakes, make intelligent reconfiguration decisions more quickly, reduce the manpower required to perform the functions, and provide optimal electric power service through the surviving system. Shipboard power systems are very similar to isolated utility systems in that the available generators are the only source of supply for the system loads. There are, however, several differences between utility and shipboard power systems, such as ships have large dynamic loads relative to generator size and a larger portion of nonlinear loads relative to power generation capacity, and transmission lines are not nearly as significant as for utilities because of their short lengths. In the literature there are several papers [2] [12] discussing the restoration problem for utility systems. Most of the methods are based on heuristic search techniques. Some of the methods are based on graph theory [9] [11]. Aoki et al. [2] and Lee and Grainger [12] attempt to use network flow approach to solve the problem of service restoration for utility systems. As pointed out by Lee and Grainger [12], the method of Aoki et al. [2] would handle multiple faults as a series of sub-problems and has some limitations. In the method suggested by Lee and Grainger [12], the optimal solution obtained by solving the maximal flow problem is disturbed to meet the radial condition and finally conclude that straightforward application of network flow approach is not suitable for solving the problem for utility systems. The authors for the first time have attempted to present an automatic service restoration method for SPS. This method uses as a basis the restoration techniques developed for utility systems while including in the formulation features that exploit the unique characteristics of SPS topologies. The proposed method comes under the category of network flow methods /01$ IEEE

2 654 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 4, NOVEMBER 2001 Unlike the method of [12], this method does not modify the optimal solution to take care of the radial condition. The proposed formulation is such that it will directly give the optimal solution satisfying the radial constraint. In SPS, the generators are connected in ring configuration through generator switchboards, bus-tie breakers and cable connecting generator switchboards. All components below the generator switchboards are operated in radial configuration and faults on any of these components may interrupt power supply to some loads. If the fault is not on a component in the ring, the network is modified by merging the generator switchboards, bus-tie breakers, and cables connecting these switchboards. This reduced network is used to reconfigure the network to restore the service. On the other hand if the fault is one of the components connected in the ring, then that component is assumed to be isolated from the system and remaining network is used to reconfigure to restore the service. It may be noted that, when a component in a ring is removed, the total network becomes radial. In a previous article by the authors [13], the problem was formulated as a variation of fixed charge network flow problem [14], but did not include voltage constraints which make the formulation more complex. This paper discusses the final version of the method that directly incorporates voltage constraints in the problem formulation and solution. An innovative technique has been developed to formulate the problem as a mixed integer linear optimization problem for which an optimal solution can be easily obtained. The proposed method does not need load flow/power flow analysis to verify the current capacity and voltage constraints. It directly suggests the reconfigured network that restores maximum load satisfying the constraints and also ensuring the radial condition. In this paper, several lemmas and their proofs are given to validate the proposed problem formulation. Further the proposed method has been applied to a simple shipboard power system and various case studies are presented to illustrate the effectiveness of the proposed formulation. The paper is organized as follows. Section II presents the mathematical problem formulation. Lemmas and their proofs are given in Section III. Section IV presents the generalized procedure for reconfiguration for service restoration in shipboard power systems. This is illustrated using various case studies in Section V. Conclusions are given in Section VI. Fig. 1. Example system. II. MATHEMATICAL PROBLEM FORMULATION Consider a simplified shipboard power system (SPS) as shown in Fig. 1. This system consists of three generators connected in a ring configuration. Two generators supply power while the third generator is an emergency generator. Some loads are connected to the load center directly and some via ABT/MBT s. The loads connected via ABT/MBT have an alternate supply. Graphical representation of this system is shown in Fig. 2. Each ABT/MBT is represented with two switches as shown in Fig. 3. Since supply should be from only one source (in radial systems), only one of these switches is in closed position at any given time. The edges in thick lines indicate normal paths and the edges in dotted lines indicate Fig. 2. Fig. 3. Graphical representation of example system. Modeling of ABT/MBT. alternate paths. The switches in open and closed position are depicted as and, respectively.

3 BUTLER et al.: NETWORK RECONFIGURATION FOR SERVICE RESTORATION IN SHIPBOARD POWER DISTRIBUTION SYSTEMS 655 Whenever there is a fault on any of the edges and after it is isolated, there would be no supply to the loads on the paths that are beyond the faulted edge. Supply has to be restored to most of these affected loads, by closing some of the switches that are open. This has to be done while satisfying the capacity and voltage constraints and ensuring the radial condition. Also it is possible that battle damage could initiate several simultaneous faults which affect several loads. In such cases, supply has to be restored to maximum load satisfying the constraints. The mathematical formulation of this problem is described in the following section. A. Problem Formulation The problem is formulated as a variation of the fixed charge network flow problem [14]. Let represent the set of nodes and represent set of edges in the network. The set represents the network under consideration. Let represent the capacity of edge. represents available power with respect to a source or a flow capacity constraint in case of a component like cable or circuit breaker. Let represent the set of load nodes in the network. Let represent the set of edges that are in closed position. Therefore, the set represents the set of open edges in the network. After the fault occurs, there are some edges that are faulty and some that are not faulty. Let represent the set of edges that are faultless. Now the set of edges which are available for the restoration of power to all load points given in is given by and the network would be represented by the set. In this work, DC models of data and electrical behavior have been used. Even though the DC analysis yields approximate results, the optimization algorithm will still tend to determine the optimal configuration among various candidate configurations based on voltage drop and other costs [15]. At a node, let represent the set of edges, for which current flows into the node, and the set of edges for which current flows out of the node where represents the directed edge from node to. Similarly, where represents the directed edge from node to. Let be the flow in edge. Let be defined as follows: if edge is closed otherwise. To restore service through reconfiguration, some of the edges of have to be closed. The mathematical formulation of the problem is shown below with its objective function and constraints. (1) (2) Objective Function: Maximize (3) It may be noted that the above equation represents maximization of the total load supplied at load nodes where represents the load current at node. Constraints: a) At any source node, the sum of the flows going out of the source node should not exceed the total capacity of the respective source node b) At any node, (except source node) sum of flows into the node should be equal to sum of the flows coming out of the node. At a load node At any other intermediate node c) For any load, the load that can be restored is maximum up to its rating for variable type of loads. But for fixed type of loads, it can be restored either to its rating or cannot be restored at all. At a load node which is variable type At a load node which is fixed type where is a 0 1 variable. This will ensure that in the solution is either or 0. d) The flow in an edge must be zero if the edge is open and it must not exceed the capacity of the edge otherwise (4) (5) (6) (7) (8) for (9) e) The system should be radial. This implies that at any node there should be only one edge feeding that node f) Voltages at all nodes should be within tolerable limits: (10) for (11) The expressions for the voltage at any node can be written in terms of voltage of the node that feeds to the node through

4 656 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 4, NOVEMBER 2001 where is a constant which is very large when compared to node voltages. It may be noted that in addition to these constraints, there is also the following constraint on as per (11): (17) Fig. 4. Triplet (i, i, m). an edge. If node is feeding node through an edge, then voltage at node can be written as follows: (12) where is the impedance of edge. Since the system is radial in nature, expressions for voltages can be written starting from the source node whose voltage is assumed to be known. All expressions for voltages in terms of flows and voltages are written as equality constraints in the problem formulation. Since some loads have a normal and an alternate supply path, a node may have more than one edge connected to it. However, only one of the edges is actually feeding the node. For example, consider a node connected to nodes and through edges and whose impedance are and respectively and flows in these edges are and, respectively. Let the set of nodes be referred to as triplet as shown in Fig. 4. Suppose that node is connected from node through edge whose impedance is. Let be the flow in the edge. Depending upon which edge is feeding the node, the voltage at node can be written as follows: when edge or when edge is feeding is feeding In other words, the expression for voltage at node written as (13) (14) can be (15) It may be noted that, due to the radial condition, one of the variables and is equal to one, while the other will be equal to zero. Therefore, the value of will be calculated correctly based on the values of s (status of edges). If the problem is formulated with (15) as one of the constraints, the problem will be nonlinear in nature. Any solution procedure for nonlinear optimization problems usually involves lots of computational effort. Moreover, there is no guarantee that the solution obtained is exactly optimal [16]. Therefore, it is preferable to formulate the problem as a mixed integer linear optimization problem, so that the solution is always optimal. Accordingly, the following expressions are written for the voltage at nodes of type (16.a) (16.b) These constraints will ensure that while optimizing the loads (which affect the flows in the edges), the voltage constraints are also satisfied. The constraints given in (16.a) and (16.b) can also be written as equality constraints as follows: (18.a) (18.b) where and are slack variables. If this problem [presented in (3) (12) and (18)] is solved, it will ensure that the restored network would restore supply power to as much load as possible and also all the capacity and voltage constraints are satisfied. Further, this would also ensure radial condition. It is important to prove that the model represented by (15), the nonlinear model, and the model represented by (18.a) and (18.b), the linear model, give the same optimal solution. This is discussed in the next section. III. PROOF FOR PROPOSED MODELING OF VOLTAGE CONSTRAINTS Let represent the model described by (3) (12) and (15). Similarly let represent the model described by (3) (12), and (18). It is required to prove that the models and give the same optimal objective function value. We shall argue that under the condition for all, any optimal solution (,, )of can be transformed into an optimal solution of. In order to prove this, it is sufficient to prove the following lemmas. Lemma 1: Any feasible solution of is a feasible solution of. Lemma 2: If an optimal solution of is feasible for, then it is also optimal for. Lemma 3: Under the condition for all,any optimal solution of can be transformed into an optimal solution of. The proof of these lemmas is given in the following paragraphs. Lemma 1: Any feasible solution of is a feasible solution of. Proof: Let (,, ) be an arbitrary feasible solution of. Then it satisfies the conditions (4) (12) and satisfies (15) for every triplet (,, ) as described in Fig. 4. Consider a triplet (,, ) of nodes as described in Fig. 4. Then (,, ) satisfies (15) for this triplet. It implies or and (19) and (20)

5 BUTLER et al.: NETWORK RECONFIGURATION FOR SERVICE RESTORATION IN SHIPBOARD POWER DISTRIBUTION SYSTEMS 657 Suppose (19) holds true. Then (16.a) holds as strict equality while (16.b) holds as strict inequality (as is an extremely large positive value). Similarly, when (20) holds true, (16.a) and (16.b) hold as strict inequality and strict equality constraints, respectively. Similarly (16.a) and (16.b) hold for each triplet where a node is connected to two other nodes. It means that the solution (,, ) satisfies (3) (12), (16.a) and (16.b). This implies that (,, ) is a feasible solution of. This Lemma also implies that the set of feasible solutions of is a subset of feasible solutions of. Lemma 2: If an optimal solution of is feasible for, then it is also optimal for. Proof: From Lemma 1, it follows that the set of feasible solutions of is a subset of feasible solutions of. Therefore, if an optimal solution (,, )of is feasible for, then it is also optimal for. Lemma 3: Under the condition for all,any optimal solution of can be transformed into an optimal solution of. Proof: Let (,, ) be an optimal solution of. Let the upper limits on voltages be the same for all nodes, that is, for all. Under these conditions, we shall prove that any optimal solution (, )of can be transformed into an optimal solution of by increasing the values, if necessary, of some of the s. Consider a triplet (,, ) of nodes as shown in Fig. 4. One of the following cases occurs: Case (a):,, Case (b):,, Case (c):,, Case (d):,,. Also, it may be noted that, as per constraint (11), we have and If case (a) occurs, increase and the voltages of nodes downstream of node (in the direction of the flow) by, where is given by (21) If case (b) occurs, increase and the voltages of nodes downstream of node (in the direction of the flow) by, where is given by (22) In either case, the new voltages do not exceed the upper bound as can be seen in Fig. 5. In Fig. 5, and indicate the modified voltages at nodes and for the triplet (,, ) for Case (a). For cases (c) and (d), no changes are made to the voltages because they satisfy (15). Let denote the vector of voltages of all nodes after making the necessary modifications. Then (,, ) satisfies the constraint (15) for the triplet (,, ). Repeating this procedure for every triplet of nodes, we finally get a feasible solution (,, )of that satisfies (15) for all triplets in the network and thus is feasible for. Note that this solution is also feasible and optimal for (as the Fig. 5. Modification of voltages as in proof of Lemma 3: Case (a). vectors and remain unchanged in the transformation). Therefore, (,, ) is optimal for by Lemma 2. It follows from the analysis that an optimal solution of can be obtained by solving the linear model and making an appropriate transformation on voltages as described in Lemma 3. This approach to solve is efficient because we can get an exact optimal solution of the linear system. Moreover, it is relatively easy to solve a linear system such as. As the main objective is to determine an optimal configuration that maximizes the total load satisfying all the constraints, the variables of interest are the optimal values of and for. Since the optimal solution of and the corresponding optimal solution of have the same and, it is sufficient to determine and in the optimal solution for.ifthe voltages in the optimal solution of are required, then the transformation on the optimal solution of is performed as described in Lemma 3. IV. GENERAL PROCEDURE FOR RECONFIGURATION The generalized procedure for reconfiguration for service restoration in shipboard distribution systems based on the theory and problem formulation discussed earlier can be summarized as given below. Step 1) Develop the graphical representation of the given shipboard power system and number the components and nodes in some order. Step 2) If there are any generators connected in ring configuration, then go to step 3. Else go to step 4. Step 3) If there is a fault on any of the component connected in the ring, go to step 3.1. Else go to step 3.2. Step 3.1: The faulted component in the ring is removed from the system. Merge all the nodes corresponding to the generator switchboards connected together in a ring. Establish a fictitious node and a fictitious edge connecting this fictitious node to the merged node. This fictitious node represents a source node whose capacity is equal to the total capacity of the generators connected together and supplying power. The capacity of the fictitious edge is equal to the total capacity of the generators connected together. Go to step 4.

6 658 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 4, NOVEMBER 2001 Step 3.2: Merge all the nodes corresponding to the generator switchboards connected together in a ring. Establish a fictitious node and a fictitious edge connecting this fictitious node to the merged node. This fictitious node represents the source node whose capacity is equal to the total capacity of the generators connected in the ring and supplying power. The capacity of the fictitious edge is equal to the total capacity of the generators connected in the ring. Step 4) Assign variables and to each component (referred to an edge in the graph) in the network. Variable represents flow in an edge and represents status (0 open; 1 close) of the edge. Also assign a variable to each load. Also for each load which cannot be varied discretely assign a 0 1 integer variable. Step 5) Formulate the problem for nonfaulted case as stated in (3) (12), and (18). Simulate the fault conditions by adding/modifying the constraints to the basic problem. Step 6) Solve the optimization problem using a commercial optimization package such as CPLEX [17]. Step 7) The solution gives the reconfigured network. The optimal values of indicate the status of all components (edges) in the reconfigured network. The optimal values of indicate the flows in components. The optimal values of for each load. Step 8) End. indicate the load supplied Fig. 6. Graphical representation of example system 1 after merging components in ring configuration. V. ILLUSTRATION OF THE PROPOSED METHOD To illustrate the restoration method, let s consider an example using the system shown in Fig. 1. Suppose there is a fault on the cable (edge 15) connecting load (at node 17). After it is isolated, there would be no power to the load at node 17 (as can be seen in Fig. 1). In step 1, the graphical representation would be developed as shown in Fig. 2. As explained in step 3 of the procedure, if there are no faults on the components connected in ring, Fig. 2 will be modified as shown in Fig. 6 by merging the nodes corresponding to the generator switchboards connected in ring. In Fig. 6, node 29 represents the new node after merging the generator switchboards and bus-tie-breakers (nodes 2, 21, 22, 11, 26, 25, 24, 27, 28). Node 30 represents the new source node whose capacity is equal to the sum of the capacities of generators supplying power at nodes 1, 10 and 23. Accordingly, the capacity of edge 31 is equal to the capacity of source node 30. If there are faults on components 23 and 30 (bus tie breakers) that are on the ring configuration, Fig. 2 will be modified as shown in Fig. 7. In Fig. 7, node 29 represents the new node after merging two switchboards and bus-tie-breakers (nodes 21, 22, 11, 26, 25, 24, 27, 28). Node 30 represents a new source node whose capacity is the sum of the capacities of generators supplying power at nodes 10 and 23. Accordingly the capacity of the edge 31 is the capacity of the source node 30. In this case generator connected at node 1 will be isolated and will be the source node 1 as shown in Fig. 7. Fig. 7. Graphical representation of example system with faulted generator. It is assumed that the loads, and can be varied from zero amps to 40, 40 and 30 amps, respectively. Such loads represent lump loads consisting of several individual loads. Load is assumed to be a fixed load of 30 amps. To facilitate modeling of the loads as discussed above, a 0 1 integer variable is associated with loads that have fixed values. Accordingly, the loads are modeled as follows. This will ensure that optimal values of loads can be maximum value of 40 amps in case of and ; and maximum value of 30 amps in case of, whereas the optimal value of can be either 0 or 30 amps Also in this example, it is assumed that values of impedance of all edges are 0.01 ohms. The voltage limits are assumed to be

7 BUTLER et al.: NETWORK RECONFIGURATION FOR SERVICE RESTORATION IN SHIPBOARD POWER DISTRIBUTION SYSTEMS (min) and 118 (max) volts at all nodes. Also it is assumed that the capacity of each edge is 80 amps. It is also assumed that the current capacity of each of the generator is 60 amps. Various case studies are presented below to illustrate the effectiveness of the proposed method. For these case studies, the CPLEX program [17] is used to solve each resulting optimization problem. CPLEX is a tool for solving linear optimization problems. It implements optimizers based on the simplex algorithms (both primal and dual simplex) as well as primal dual logarithmic barrier algorithms. CPLEX also solves mixed integer and quadratic problems. Further CPLEX offers a network optimizer aimed at a special class of linear problems with network structures such as a network flow problem. A. Case 1 Case 1.1: Initially, the system without any faults is studied. Based on the explanation given earlier the system is modeled as follows. Objective function: Maximize Subject to: Source node constraint; Initial configuration details; Load details; Load node constraints; Intermediate node constraints; It may be noted that the variables (i.e., flows) should be positive and the variables and should be 0 1 integer variables. In most optimization packages, these need not be indicated explicitly as constraints. The solution obtained from the CPLEX package is as follows: Total load amps Edge capacity constraints; This solution gives the values of flows in the edges satisfying the capacity and voltage constraints and supplying a maximum amount of the loads. Case 1.2: To see the effectiveness of the model, let us put a severe voltage constraint at node 12 as: Radial constraints; Voltage constraints; If the optimization is solved with this modified constraint, the optimal values of loads are:,,,. This indicates that only 20 amps of could be fed under these constraints. Further, since can be either 30 or 0 amps, it was equal to zero amps to satisfy the strict voltage constraints. Case 1.3: Let the capacity of edge 11 which is feeding the load be modified (reduced) to 20 amps when compared to

8 660 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 16, NO. 4, NOVEMBER 2001 Case 1.1. This is done by changing the respective capacity constraint as: The optimization solution to this problem with the above constraint would give the optimal values of the load as,,,. It can be seen that since the load is 30 amps and the capacity of edge 11 is less than or equal to 20 amps, this load cannot be fed. Accordingly the value of. In summary, this case illustrates that the proposed modeling/formulation will supply as much load as possible satisfying the constraints. B. Case 2 In this case, faults on components will be simulated and then the proposed optimization formulation is solved to restore maximum load satisfying the constraints. Simulation of fault: A fault on any component (edge) is simulated in the model by equating the respective flow variable and 0 1 variable (status variable ) to zero. This is done because when there is a fault in a component, this component is not available and there cannot be any flow in it. Because of the fault and after isolating the fault, there will be some loads that will be left without supply. Such loads, referred to as affected loads, need to have their power restored. Also the status of switches at the affected load(s) (which are constrained to be closed in the initial configuration) are now removed from the constraints. This will allow exploration of alternate paths for restoration. Case 2.1: Consider a fault on the circuit breaker (CB) (edge 14) feeding the load (at node 17). Because of this fault, the load will be affected and left without supply. This case is modeled as follows. Since the fault is on component 14, 14 and 14 are now made equal to zero (indicating that this component is not available). Also the affected load due to this fault is (at node 17). The status of the switch at this load,, is now removed from the constraints. It may be noted that initially (unfaulted case). The other constraints are the same as in Case 1.1. Formulating the problem as explained with these modifications, CPLEX generates the following results Total load amps The solution suggests that, indicating that switch 17 was closed to restore service to the affected load (at node 17) and all the loads can be fed satisfying the current and voltage constraints. Case 2.2: Consider a case wherein there is a fault on a component that would affect a load that has no alternate path, referred to as a nonvital load. In such a case, service cannot be restored to that affected load until the fault is repaired. This case is illustrated as follows. Assume that there is a fault on CB (edge 3) supplying load (at node 4) which has no alternate path and a fault on the cable (edge 6) connecting the load (at node 20) which has an alternate path. Faults on components were modeled as explained in the previous case. The optimal solution generated by CPLEX is as follows Total load amps The solution indicates that the optimal value of, meaning that switch 8 was closed to restore supply to load. Further, load cannot be restored since there is no alternate power supply path for it. Also since the capacity limit on edge 10 is 80 amps, only 20 amps of can be supplied. Case 2.3: Assume there is a fault on the components 23 and 30, which represent the bus tie breakers that tie together the generator switchboards in ring configuration. Fig. 2 is modified to Fig. 7 following the procedure explained in step 3. It can be seen that the generator connected at node 1 is isolated. The problem is formulated as explained in the procedure. The optimal solution generated by CPLEX is,, and. Now load is supplied through switch 8 to supply maximum load. The generator at node 1 will supply load. Various cases have been studied using the proposed restoration method by applying it to a small system, similar in topology to a shipboard power system. The cases demonstrate the ability of the restoration method to configure the system within the constraints when there is not a fault, and to reconfigure the system in the presence of a fault. The results obtained matched the expected results exactly. The authors have also tested the proposed method for a large SPS that was based on an actual surface combatant ship. All cases demonstrated that the proposed method provides the optimal solution for service restoration in shipboard distribution systems. VI. CONCLUSION A new and simple method of reconfiguration for service restoration in shipboard power systems was presented. The service restoration problem is formulated as a variation of the fixed charge network flow problem. Since it is in a mixed integer linear form, an optimal result is ensured. The proposed method restores a maximum amount of load while satisfying the capacity and voltage constraints directly. Further, it ensures that the resulting topology is radial. The accuracy of this method has been illustrated through several case studies on a simple shipboard power system. Faults

9 BUTLER et al.: NETWORK RECONFIGURATION FOR SERVICE RESTORATION IN SHIPBOARD POWER DISTRIBUTION SYSTEMS 661 in the network are easily modeled in the formulation. In the studies, the CPLEX optimization package was used to generate solutions for the resulting network. The proposed method does not require any load flow/power flow analysis to verify the current and voltage constraints. It directly suggests the reconfigured network that satisfies the current and voltage constraints. The results from the case studies were very good. Large shipboard power systems have also been studied with similar success. Future work entails adding a procedure for including load priorities in the solution methodology, probably using expert system technology. ACKNOWLEDGMENT The authors would like to thank the reviewers for their valuable comments that improved the quality of the paper. REFERENCES [1] K. L. Butler, N. D. R. Sarma, C. Whitcomb, H. Do Carmo, and H. 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Nash, Nonlinear programming, ORMS Today, vol. 25, no. 3, pp , June [17] ILOG CPLEX, ILOG, Inc., Mountain View, CA, version 6.5, Karen L. Butler (M 90 SM 01) is an associate professor in the department of electrical engineering at Texas A&M University. She received the B.S. degree from Southern University-Baton Rouge in 1985, the M.S. degree from the University of Texas at Austin in 1987, and the Ph.D. degree from Howard University in 1994, all in electrical engineering. In , Dr. Butler was a Member of Technical Staff at Hughes Aircraft Co. in Culver City, CA. She received a 1995 NSF Faculty Career Award and 1999 Office of Naval Research Young Investigator Award. Her research focuses on the areas of computer and intelligent systems applications in power, power distribution automation, and modeling and simulation of vehicles and power systems. Dr. Butler is a Senior Member of IEEE, IEEE Power Engineering Society, and the Louisiana Engineering Society. She is a registered professional engineer in the State of Louisiana, Texas, and Mississippi. N. D. R. Sarma (M 86 SM 01) received the B.Tech (electrical) and M.Tech (power systems) degrees from Regional Engineering College, Warangal, India in 1983 and 1986, respectively, and the Ph.D. degree from Indian Institute of Technology, Delhi, India in From February 1992 to October 2000, he was with the R&D Division of CMC Limited, Hyderabad, India. From October 1997 to June 1999, (on sabbatical leave from CMC) and since November 2000, he is working at Texas A&M University, College Station, TX, USA, as a Post Doctoral Research Associate. His areas of interest include Load Dispatch and Distribution Automation Systems for power utilities. He is a Senior Member of IEEE and IEEE Power Engineering Society. V. Ragendra Prasad received the B.S. (mathematics) and M.S. (statistics) degrees from Andhra University, Waltair, India in 1974 and 1977, respectively, and the Ph.D. degree from Indian Statistical Institute (ISI), Calcutta, India in He served as tenured faculty of SQC and OR Division of ISI during He worked as visiting faculty at Washington State University, Pullman during January August 1996 and at Texas A&M University, College Station, during September 1996 to December Since January 1999, he has been working with Knowledge Based Systems, Inc., College Station. Dr. Prasad provides consulting services to manufacturing industries on mathematical modeling and simulations of problems in engineering, process control and system design. His research interest are system reliability optimization, quality control, stochastic models and mathematical programming. He is a co-author of a book entitled Optimal Reliability Design: Fundamentals and Applications, that was published by Cambridge University Press in January He is a member of INFORMS and life member of Operations Research Society of India and National Institute for Quality and Reliability (India).