OPTIMAL POWER FLOW (OPF) is a very useful tool for

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1 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 1, FEBRUARY A Decentralized Implementation of DC Optimal Power Flow on a Network of Computers Pandelis N. Biskas, Member, IEEE, A. G. Bakirtzis, Senior Member, IEEE, Nikos I. Macheras, and Nikolaos K. Pasialis Abstract This paper presents a decentralized implementation of the DC Optimal Power Flow (OPF) problem on a network of workstations. Each workstation is assigned to a regional Transmission System Operator (TSO), who manages the operation of the transmission system of his own region, as well as cross-border exchanges with neighboring regions. The workstations take part in an iterative process, exchanging information with each other and solving regional OPF sub-problems, until the global OPF solution is reached. Tie-line related information is exchanged between workstations assigned to neighboring regions. A master workstation, assigned to a Super-TSO, checks for the convergence of the algorithm. The parallel processing system is tested on various test systems, including a large real-world system, the Balkan power system. Index Terms Decentralization, optimal power flow, parallel processing. NOMENCLATURE Unit index, identical to bus index; for simplicity of notation it will be assumed that only one unit is connected to every bus. Line from bus to bus index; also used for tie-lines. Fuel cost function of unit, here a quadratic function. Reactance of line. Power flow on line. Power flow on tie-line. Area (region) index. Set of area tie-lines. Area unit active power output vector. Area bus active power demand vector. Area bus voltage phase angle vector. Area network admittance matrix; area tie-lines are ignored. Area modified network admittance matrix; tieline admittances are added to the diagonal entries of the corresponding border buses. Number of area nodes. Number of area tie-lines; identical to number of area border buses. Number of area internal lines. Manuscript received December 29, This work has been supported in part by the National Fellowship Foundation of Greece. Paper no. TPWRS The authors are with the Department of Electrical Engineering, Aristotle University of Thessaloniki, Greece ( pbiskas@egnatia.ee.auth.gr; bakiana@eng.auth.gr). Digital Object Identifier /TPWRS Vector of adjacent area voltage phase angles corresponding to the to buses of area A tie-lines (size ). diagonal tie-line primitive reactance matrix of area. diagonal line primitive reactance matrix of area. node to tie-line incidence matrix of area ( : tie-line from to ). node to branch incidence matrix of area. sensitivity matrix containing the sensitivities of tie-line power flows with respect to nodal active power injections. sensitivity matrix containing the sensitivities of internal-line power flows with respect to nodal active power injections. Decentralized algorithm iteration index. I. INTRODUCTION OPTIMAL POWER FLOW (OPF) is a very useful tool for the management of interconnected transmission systems. Ideally, a single, system-wide Transmission System Operator (TSO), who has access to technical and (limited) market data over the whole interconnection, would solve the OPF of a large, interconnected power system. However, political and organizational issues, such as the difficulty in disclosing utility data and the lack of political intention to devolve powers to a foreign center, large communication requirements between each node of the interconnected system and the central point for real-time data (i.e., network status, participant bids) acquisition hinder the centralized management of the transmission system. Additional obstacles to the central management of interconnected transmission systems are the difficulty to maintain a huge database for all system or utility data, including long-term changes and outages, huge CPU requirements, and concerns about the reliability of the central computer system that implements the centralized operation, and the robustness of the centralized algorithm, especially in highly stressed systems. Therefore, the centralized operation of the transmission grid in multi-area systems is questionable with the current state of the art. For the near future there will be different TSOs in different regions (e.g., countries) within large interconnected power systems /$ IEEE

2 26 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 1, FEBRUARY 2005 The recent trend toward multi-national electricity markets operated by the coordinated actions of regional TSOs imposes the development of efficient methods for the solution of the decentralized OPF problem on real-world distributed computing systems using parallel processing. Parallel processing, the method of having many small tasks to solve one large problem, has emerged during the last decade as a key technology in modern computing. A program is divided into multiple fragments that can execute simultaneously, each on its own processor. Parallel processing has been facilitated by two major developments: Massively Parallel Processors (MPPs) and the widespread use of distributed computing [1]. MPPs are the most powerful computers in the world, combining thousands of CPUs in a single large cabinet, connected to thousands of gigabytes of memory. MPPs offer enormous computational power, but typically cost several million dollars. Distributed computing is a process, in which multiple computers, remote from each other, connected by a network are used collectively to solve a single large problem. Owing to distributed computing, we are able to solve problems several times larger than we could by using a single computer, in a considerably lower cost compared to MPPs. Also, the performance of the distributed system can be optimized by assigning each individual task to the most appropriate architecture. Common between distributed computing and MPPs is the notion of message passing, a model for interactions between processors within a parallel system. In all parallel processing systems, data must be exchanged between cooperative tasks. The Parallel Virtual Machine (PVM) software [2], which supports the message passing model, enables a collection of computers with heterogeneity in architecture types and in computational speeds, to appear as one large virtual machine, namely, to be used as a coherent and flexible concurrent computational resource. PVM handles all message routing, data conversion and task scheduling across a network of incompatible computer architectures. Several algorithms have been proposed for the solution of the decentralized OPF problem [3] [11]. In [12] a distributed implementation is described, in which a network of workstations is used to solve the decentralized OPF algorithm developed in [4]. Several large real-world systems are tested, to show the effectiveness of the distributed implementation as compared to the central one. A new method for the decentralized solution of the DC-OPF of interconnected power systems has been developed by the authors in [13]. The features that differentiate our method from other existing methods are the following: a) The OPF decoupling is performed around tie-lines and not boundary nodes. Therefore, functional inequality constraints, such as tie-line power flow limits, are not coupling constraints of the problem, so they do not hinder the OPF decoupling process. b) In most of the existing literature, the problem decomposition by area is based on the decomposition of the Lagrangian or the Augmented Lagrangian of the entire problem. This paper follows a new decomposition philosophy, [11], [17], based on the decoupling of the first-order (KKT) conditions of the original large-scale problem in such a way that the combination of the KKT of all area sub-problems are identical to the KKT conditions of the original problem. c) No parameter tuning is required. In fact, the only parameter that controls the algorithm convergence behavior is the tie-line power flow tolerance used as termination criterion. This paper extends the work reported in [13] in two ways. First, the DC-OPF sub-problem, to be solved by each individual region in the iterative decentralized DC-OPF algorithm, is transformed to reduce its size. The above transformation provides a physical interpretation of the interaction of the neighboring regions during the iterative solution process. Border node injections and soft grounding terms are introduced to express the effect of neighboring areas on each individual area. In addition, the above transformation provides a guide of the modifications required to convert an autonomous area OPF problem to an interconnected area modified OPF sub-problem. Therefore, the innovative feature of this paper with respect to [13] is not the transformation (which leads to size reduction) itself, but the implications of the transformation regarding the physical interpretation of the interaction of the neighboring regions and the standardization of the process of converting an autonomous area OPF problem to an interconnected area modified OPF subproblem. Second, the decentralized DC-OPF algorithm is implemented on a network of workstations, using the PVM software. Our goal is not to enhance the computational efficiency of the decentralized implementation as compared to the central one, but the implementation itself as a means of preserving dispatching independence of each region. Each workstation is assigned to a regional TSO, who has access to the generation, demand and network data of his own region. Each workstation manages the operation of the transmission system of one region, as well as its cross-border exchanges with neighboring regions. Workstations representing neighboring regions exchange information on tieline flows, tie-line export prices and boundary bus voltage phase angles. Each workstation solves a modified regional DC-OPF sub-problem, and exchanges tie-line information with the workstations of neighboring regions in an iterative manner until the DC-OPF solution of the entire interconnected power system is reached. A master workstation, assigned to a Super-TSO, checks for the convergence of the algorithm. The parallel processing system is evaluated using several test systems, including a large real-world system, the Balkan power system. II. THE DECENTRALIZED DC-OPF PROBLEM This section is divided in three parts. In the first part, the DC-OPF problem decoupling is described, and the regional DC-OPF sub-problem is defined. In the second part, the regional DC-OPF sub-problem is transformed to effectively model interactions with neighboring regions, while in the third part the size of the regional DC-OPF sub-problem is reduced defining a small, dense QP problem. A. DC-OPF Decoupling and Definition of Regional DC-OPF Sub-Problem The OPF decoupling principle is illustrated in Fig. 1. Consider a simple power system with two areas, and, connected by tie-line ( stands for adjacent area ). The tie-

3 BISKAS et al.: DECENTRALIZED IMPLEMENTATION OF DC OPTIMAL POWER FLOW 27 Fig. 1. OPF decoupling principle. line is cut, and new variables, and, are added representing the tie-line flows on the two sides of the tie-line. The terms express the export prices of electricity flowing over tie-line. The same principle is applied in systems with more than two areas. A decentralized solution to the multi-area DC-OPF problem, based on tie-line information exchange between adjacent areas, has been presented by the authors in [13]. The method iteratively solves a modified DC-OPF sub-problem for each area until the DC-OPF solution of the entire power system is reached. At the beginning of each iteration each area,, receives from its neighboring areas three pieces of information per tie-line, namely, (a) tie-line power flow,, (b) electric energy import price,, through the tie-line, and (c) neighboring area border bus phase angle,, as computed by the neighboring areas modified DC-OPF of the previous iteration. If all tie-line power flows, as computed by the neighboring areas,, are equal, within tolerance, to the values,, that area computed during the previous iteration, then the modified OPF problem of area has converged. Otherwise, area uses the other two pieces of information exchange to define its modified regional DC-OPF sub-problem as follows [13]: (1) (2) (3) (4) (5) (6) (7) where (2) represent the area DC power flow equations, (3) defines the system slack bus voltage phase angle since (the slack bus is not omitted in (2)) and is applied to one, reference area only, (4) represent area unit active power output limits, (5) represent area transmission line power flow limits, (6) are area tie-line flow equations, and (7) represent area tie-line power flow limits. Equations (6) invoke variables from area, namely the bus voltage phase angle of the boundary buses that belong to neighboring areas, so they constitute the coupling constraints of area sub-problem. Assuming quadratic unit cost functions, the regional DC-OPF sub-problem (1) (7) is a large, yet highly sparse QP problem, which must be solved for the unknown vectors, and. The bounds on decision variables (4), (7) are treated implicitly by most QP software. The size of the sparse QP constraint matrix is large owing to the power flow and the transmission loading constraints (2), (5). The number of tie-line constraints (6) is small. In the DC-OPF of single-area systems it is common practice [14] to separate the solution of the power flow equations from the optimization by eliminating the bus voltage phase angles from the QP problem constraints. The numerous power flow constraints are thus replaced by a single system power balance constraint resulting in a drastic reduction of the QP constraint matrix size. The size of the QP constraint matrix is further reduced by omitting all branch power flow constraints, which are deemed to be inactive in the optimal solution. This is achieved by including in (5) only the, so called, -effective branch power flow constraints [15], i.e., constraints on the power flow of branches loaded above (e.g., 90% of their rating). The branch loading is computed from a load flow solution in which all units are economically dispatched (according to a classic economic dispatch, without transmission loading constraints). In practice, the number of -effective branch power flow constraints is much smaller than the number of branches. With the elimination of the bus voltage phase angles and the inclusion of the -effective branch power flow constraints only, a small-size, yet dense QP problem must be solved for the singlearea DC-OPF calculation. Appendix 1 discusses the computational requirements of both the sparse and the dense QP formulation of the DC-OPF problem and presents test results showing that the dense QP formulation is more efficient. The application of the above concepts to the decentralized DC-OPF solution of a multi-area power system requires an initial transformation of the regional sub-problem (1) (7). This transformation, in addition to providing a drastic reduction of the regional sub-problem size, also provides a physical interpretation of the interaction of neighboring areas within the framework of the decentralized solution, as well as a cookbook recipe for the modifications that must be made to the standard, autonomous-operation DC-OPF problem of an area, in order to convert it to the modified regional DC-OPF sub-problem that must be solved during each iteration of the decentralized multiarea DC-OPF solution algorithm. B. Transformation of the Regional DC-OPF Sub-Problem Substituting from (6) in the objective function (1) and, also, substituting from (6) in (2) the area sub-problem is

4 28 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 1, FEBRUARY 2005 Fig. 3. Soft grounding of border nodes. Fig. 2. Structure of network admittance matrix. converted to the following form: where (8) (9) (10) (11) (12) (13) (14) (15) The area objective (8) represents area production cost net the revenues from area exports, priced at the import prices of adjacent areas. A similar economic interpretation is presented in [3]. Note that constraints (11) (14) are the vector versions of constraints (4) (7), respectively, and the vector absolute values in (12) and (14) are taken on an element-by-element basis. In (9) the network admittance matrix,, has been replaced by the modified network admittance matrix, (see Nomenclature). Fig. 2 presents the structure of the entire network admittance matrix,, and illustrates the interaction between adjacent areas and the formulation of matrices, and. The contribution to the power flow equations of the off-diagonal entries, corresponding to tie-lines, is taken care of by the first term of the right hand side of (9). The first term of the right hand side of (9) represents active power injection of at each border node of area (see also (15)), as illustrated in Fig. 3. Therefore, the effect of neighboring area,, on area, after the omission of the tie-line, is modeled as an active power injection of at each border bus and as a soft grounding of the border bus through an admittance of. Due to the addition of the soft grounds the modified area admittance matrix,,is no longer singular and the area power flow (9) can be solved for, given, and without the need for selection of an area reference (slack) bus. Only one reference bus is selected for the entire system, in the system reference area (10) by solidly grounding the diagonal entry of the reference area admittance matrix corresponding to the system slack bus through a very large (practically infinite) admittance. Due to the solid grounding of the system slack bus,. In the literature an alternative formulation requiring the selection of a separate reference bus for each area has also been proposed [4] [6], [10]. This formulation requires synchronization of the reference bus voltage phase angles of different areas in each iteration of the decentralized algorithm. Our method uses a single slack bus for the entire interconnected system. With the addition of the border bus nodal injections,, and admittances to ground,, as shown in Fig. 3, the border bus voltage phase angles of each area,, automatically take values which are consistent with the voltage phase angles of the border buses of the neighboring areas,. Thus the need for synchronization of the reference bus voltage phase angles of different areas is avoided in our method. Before proceeding to the model reduction in the next section, the modifications required to convert an autonomous area OPF problem to an interconnected area modified OPF sub-problem, (8) (14), will be summarized: 1) Add new tie-line export variables,, one for each tieline. 2) Modify OPF objective to account for export revenues at constant prices, as in (8). 3) Add fixed power injections of to every boundary node (see Fig. 3). 4) Add admittance to ground at every boundary node, by modifying the corresponding diagonal entry of the area network admittance matrix. 5) Remove area slack bus, except from the slack area. This modification translates to the removal of the solid grounding (large admittance) from the corresponding diagonal entry of the admittance matrix in many modern OPF programs. 6) Add tie-line flow constraints (13), (14) [or (6), (7)]. C. Reduced-Size Regional DC-OPF Sub-Problem The area DC-OPF sub-problem (8) (14) represents a very large, sparse QP problem, which can be solved using a sparse QP routine. This problem can be converted to a very small, dense QP problem by separating the power flow solution from the QP.

5 BISKAS et al.: DECENTRALIZED IMPLEMENTATION OF DC OPTIMAL POWER FLOW 29 This is achieved by eliminating the voltage phase angle vector from the optimization problem using (9): (16) Substituting (16) in (12) and (13), the OPF problem (8) (14) becomes: where (17) (18) (19) (20) (21) (22) (23) (24) which is solved for and. Note that, as discussed above, the numerous branch power flow constraints (12) have been replaced by the much smaller set of the -effective branch power flow constraints (20), as indicated by the subscript eff in (20) and (24). The remaining branch power flow constraints are assumed to be inactive in the optimal solution, and the corresponding Lagrange multipliers zero. This assumption can be verified by a power flow solution and the set of -effective branch power flow constraints appropriately modified. The dense sensitivity matrices and, which contain the sensitivities of tie-line and internal-line power flows with respect to nodal active power injections, are computed one column at a time using a sparse forward and back substitution [16] of a very sparse vector (containing one or two nonzero entries only) with the sparse table of factors of the modified network admittance matrix,. If, and are the multipliers (dual variables) corresponding to constraints (18), (20), and (21), it is proved in Appendix 2 that the nodal prices can be expressed as: (25) (26) If is the multiplier vector corresponding to constraints (22), the price of exporting electricity through tie-line, equals the boundary bus nodal price,, plus the tie-line transmission rent,, which is nonzero (positive or negative) only in the case that the tie-line is congested (27) Note that in the case that the tie-line is not congested (27) becomes:. The regional DC-OPF sub-problem (17) (22) is solved during an iterative algorithm, which under certain conditions [17] converges to the solution of the global DC-OPF problem. The iterative algorithm is discussed in the following section. III. DECENTRALIZED DC-OPF ALGORITHM We distinguish the solution algorithm of the workstation assigned to the Super-TSO, the Control Master Unit called Server, and the workstations assigned to regional TSOs, the Area Slave Units called clients. A. Server Solution Algorithm The Server spawns the several tasks on the clients assigning a unique Task Identifier (TID) to each one. Then, it sends to each client the TIDs of the tasks run on the neighboring area clients, so as to enable communication and information exchange between neighboring area clients. An iterative process follows, in which the Server receives a Local Convergence Flag (LCF) from each client and returns a Total Convergence Flag (TCF) to every client. A LCF declares convergence of the OPF sub-problem of a particular area, while the TCF declares convergence of the OPF of the entire system. The TCF becomes TRUE only when all LCFs become TRUE, in which case the iterative process stops. B. Clients Solution Algorithm First, each client receives the TIDs of the tasks run on the neighboring area clients. An initialization phase follows, in which each client: 1) Initializes all tie-line flows and export prices of area. 2) a) Solves a classic economic dispatch (without network constraints) to compute the active power output,, of all units. b) Solves power flow equations (16) and computes the set of -effective constraints. c) Solves autonomous area OPF problem (17) (20) for. d) Solves power flow equations (16) for. If any branch power flow limits are violated, the client modifies the set of -effective constraints and loops back to 2.c. e) Computes nodal prices,, using (25) or (26). f) Computes export prices,, using (27), where for. 3) Exchanges with neighboring area clients. Next, each client enters an iterative process. At each iteration each client solves the DC-OPF sub-problem (17) (22) of one area. During the solution process each client: 1) Uses the set of area -effective constraints of the previous iteration. 2) Solves modified regional OPF sub-problem (17) (22) for and, in which and are treated as constants taking the values they were assigned during the previous iteration,, i.e., and.

6 30 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 1, FEBRUARY ) Solves power flow (16) for. If any branch power flow limits are violated, the client modifies the set of -effective constraints and loops back to 2. 4) Computes nodal prices,, using (25) or (26). 5) Computes export prices,, using (27). Next, each client exchanges, and of each tie-line with neighboring area clients. The value of the LCF is updated as follows: LCF becomes TRUE only if tie-line flows computed on both sides of each tie-line converge within tolerance (MW): (28) The values of and in (28) are the values assigned during the current iteration,. Each client sends the LCF to the Server and receives the TCF by the Server, which determines if the process will continue. The iterative process continues, until all clients receive TCF equal to TRUE by the Server. A second case is studied, in which no communication of data is required between the clients and the Server [12], but an infinite loop is executed: The Server just spawns the tasks on the clients, assigning a unique TID to each one, and sends the TIDs of the tasks to one another. The clients get into the iterative process, in which, at regular time intervals, they exchange tie-line information only with the clients assigned to neighboring regions. Each area verifies its convergence using (28) and solves its own sub-problem if necessary, i.e., if at least one of its tie-lines violates (28). Fig. 4. Network architecture. TABLE I BALKAN SYSTEM CHARACTERISTICS. * Half total due to double counting of tie-lines. IV. NETWORK AND PARALLEL PROCESSING DESCRIPTION A common method of parallelizing an application, called data parallelism, has been used in our implementation. According to this method, all tasks are the same, but each task only knows and solves a small part of the data, that is, each client manages the data of one region. This method is usually referred to as Single Program Multiple Data (SPMD) model of parallel computing. This fact does not limit the potential for different DC-OPF algorithms to be used by different clients. PVM, a publicly available software tool [2] for simulating parallel processing, has been used to solve the above algorithm on a network of workstations. In our model, a computer running GNU/LINUX was assigned to the Super-TSO (Server), while a number of workstations running WINDOWS 98 are assigned to the regional TSOs (clients). All computers had processing power of a 166 MHz Pentium with 32 MB of RAM. The architecture of our network is shown in Fig. 4. The type of the network connecting the computers is crucial for the efficiency of the decentralized implementation. Under the right circumstances, the network-based approach can exhibit computational efficiency similar to the one of MPPs. To be effective, distributed computing requires high communication speeds. Our implementation used workstations located in three different floors of the Electrical Engineering building, connected via an Ethernet connection with a 10 MBit/s broadcast bus technology. The information exchange is not attained through a LAN connection via a local hub, but through the Internet connections of all computers using their TCP/IPs, Fig. 5. Balkan system topology. TABLE II TEST SYSTEMS CHARACTERISTICS. which further delays the message passing. Slow communication between the different computers has been deliberately chosen, so that the tests are as close as possible to the real-world

7 BISKAS et al.: DECENTRALIZED IMPLEMENTATION OF DC OPTIMAL POWER FLOW 31 TABLE III COMPUTATIONAL TIMES OF CENTRAL AND DISTRIBUTED DC-OPF RUNS (166 MHZ PENTIUM I). implementation of the decentralized OPF algorithm, where the different computers will be located in different countries/states. V. TEST RESULTS The proposed algorithm is evaluated using 8 test systems; the first 5 are derived by repeating the one-area IEEE RTS-96 [18] up to 6 times, while test systems 6 through 8 represent parts of a large, real-world power system, the Balkan power system (2nd UCTE synchronous area). The Balkan power system characteristics are presented in Table I, and its topology is illustrated in Fig. 5. Test system Balkans-3 consists of Bulgaria, Romania and Former Yugoslavia, test system Balkans-4 consists of Bulgaria, Greece, Former Yugoslavia and Albania, and test system Balkans-5 consists of the whole Balkan power system. Table II presents the test systems characteristics, namely, number of regions, buses, units, internal lines and tie-lines. Table III presents the computation times of the central and distributed runs. The communication latency includes the possible waiting time for several processes to be completed before the transfer, plus the time to send and receive the data, whereas the computation time involves the solution of the QP problem of one area. Two cases are examined: Case I involves central coordination by the Server, whereas in Case II no central coordination by the Server takes place, but clients take part in an infinite loop, as explained in Section III. The execution time in this case is the time needed for all clients to reach convergence. As shown, the computing time is the same in both cases, but the latency is reduced in Case II, as expected since no Server-clients exchange takes place during the iterative process. The total execution time of the network implementation is mainly increased by the latency time, which constitutes over 90% of the total time. The high communication latency is due to the following reasons: 1) In the message-passing model the overhead in handling each message is high. There are no restrictions on how much information each message may contain, so message passing is effective only when a large block of data is transmitted from one processor to another. 2) The performance of the network hardware is poor, because of the networking speed and of the nonisolated, from other traffic, Internet connection of the PCs. 3) PVM message-passing routines generally add significant overhead to standard socket operations, which already have high latency. VI. CONCLUSION A method for the decentralized solution of the DC-OPF problem has been presented and implemented on a network of workstations. The decentralized DC-OPF solution is achieved through an iterative algorithm, in which every area solves its own modified DC-OPF sub-problem and exchanges tie-line information with its neighboring areas. The interaction of an area with its neighboring areas is modeled at its border nodes by adding nodal injections proportional to neighboring area voltage phase angles and by adding tie-line admittance terms to ground. The decentralized DC-OPF algorithm is implemented on a network of workstations, using low-speed computer communication. Each workstation is assigned to a regional TSO, while a master workstation is assigned to a Super-TSO checking the convergence of the algorithm. The network implementation and the test results on a large real-world power system demonstrate the applicability of the method to real-life problems. APPENDIX I This Appendix presents a comparison of the computational complexity of the sparse and dense QP formulations of the DC-OPF problem. Consider a system with buses and lines. The sparse QP formulation of the DC-OPF problem requires the solution of a sparse QP problem with: equality (load flow) constraints and double-sided inequality (line loading) constraints. Both sets of constraints are highly sparse. All other constraints represent bounds on decision variables and are implicitly treated by most available QP software. Please note that the fact that the transmission loading inequality constraints are double-sided does not double the number of constraints to be processed by the QP software: a double-sided inequality constraint can be converted to an equality constraint with upper bounded, nonnegative slack variable. The computational requirements of the dense formulation are analyzed as follows:

8 32 IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 20, NO. 1, FEBRUARY 2005 TABLE IV COMPARISON OF SPARSE WITH DENSE DC-OPF PROBLEM FORMULATION (350 MHZ PENTIUM II). A) Pre-processing: A classic economic dispatch. This is computationally equivalent to a QP with one equality (power balance) constraint (unit technical limits are implicitly treated). A DC power flow solution and identification of the -effective line loading constraints. This step requires one sparse matrix factorization and one forward-back substitution. Let be the number of the -effective constraints. Computation of the sensitivities of the -effective line loading constraints with respect to active power injections. This step requires sparse forward-back substitutions. The resulting sensitivity matrix is dense. B) Dense QP solution: The resulting small QP problem has: one equality (power balance) constraint, and inequality ( -effective line loading) constraints. Both sets of constraints are dense. However, is very small, of the order of 10 20, even in large power systems with several thousands of buses, owing to the high degree of redundancy existing in the transmission system. C) Post-processing (nodal price calculation): The computation of the nodal prices requires a dense matrix vector multiplication as indicated by (25) (with tie-line related term omitted). Table IV gives the QP size and the execution time of both formulations on the two and six area IEEE RTS-96, and the Balkan-5 power system. In all cases the loading limits of certain lines have been reduced to impose congestion. All cases were run on a 350 MHz Pentium II/256 MB RAM PC using a state of the art commercial sparse QP package [19]. It is observed that the improvement of the dense over the sparse QP formulation increases as the system size increases. APPENDIX II The computation of nodal prices of electricity is presented in this Appendix. Adding multipliers, for constraints (18) (22) the Lagrangian function,, is formed and the firstorder (f/o) conditions associated to vector, are written, as follows: (29) The vector of nodal prices,, cannot be directly derived from the f/o conditions of the sub-problem (17) (22), owing to the fact that the nodal power balance (load flow) equations have been eliminated from the optimization sub-problem. The f/o conditions of the equivalent sub-problem (8) (14) are used to link the values of the nodal prices, with the values of the multipliers,, present in (29). Adding multipliers, for constraints (9) (14), the Lagrangian function is formed and the f/o conditions associated to vector, are written: (30) Introducing the vector of nodal prices,, from (30) to (29), (29) can be written as: (31) for reference area. Similarly, the nodal price vector for all other areas except the reference area is computed as follows: (32) since in this case the system power balance (18) is missing. REFERENCES [1] PVM Parallel virtual machine: A users guide and tutorial for networked parallel computing, A. Geist, A. Beguelin, J. Dongarra, W. Jiang, R. Manchek, and V. Sunderam. [Online]. Available: [2] PVM Parallel Virtual Machine Software [Online]. Available: [3] A. J. Conejo and J. A. Aguado, Multi-area coordinated decentralized dc optimal power flow, IEEE Trans. Power Syst., vol. 13, pp , Nov [4] B. H. Kim and R. Baldick, Coarse-grained distributed optimal power flow, IEEE Trans. Power Syst., vol. 12, pp , May [5], A comparison of distributed optimal power flow algorithms, IEEE Trans. Power Syst., vol. 15, pp , May [6] J. H. Kim, J. K. Park, B. H. Kim, J. B. Park, and D. Hur, A method of inclusion of security constraints with distributed optimal power flow, Elect. Power Energy Syst., vol. 23, pp , [7] J. A. Aguado, V. H. Quintana, and A. J. Conejo, Optimal power flows of interconnected power systems, in Proc. IEEE Power Engineering Society Summer Meeting, vol. 2, 1999, pp [8] J. A. Aguado and V. H. Quintana, Inter-utilities power-exchange coordination: A market-oriented approach, IEEE Trans. Power Syst., vol. 16, pp , Aug [9] J. A. Aguado, V. H. Quintana, and A. J. Conejo, A computational comparison of two different approaches to solve the multi-area optimal power flow problem, in IEEE Canadian Conf. Electrical and Computer Engineering, vol. 2, 1998, pp [10] J. Contreras, A. Losi, M. Russo, and F. F. Wu, Simulation and evaluation of optimization problem solutions in distributed energy management systems, IEEE Trans. Power Syst., vol. 17, no. 1, pp , Feb [11] F. J. Nogales, F. J. Prieto, and A. J. Conejo, Multi-area AC optimal power flow: A new decomposition approach, in Proc. PSCC 99 Conf., Trondheim, Norway, June July 1999, pp

9 BISKAS et al.: DECENTRALIZED IMPLEMENTATION OF DC OPTIMAL POWER FLOW 33 [12] R. Baldick, B. H. Kim, C. Chase, and Y. Luo, A fast distributed implementation of optimal power flow, IEEE Trans. Power Syst., vol. 14, pp , Aug [13] A. G. Bakirtzis and P. N. Biskas, A decentralized solution to the DC-OPF of interconnected power systems, IEEE Trans. Power Syst., vol. 18, pp , Aug [14] B. Stott and E. Hobson, Power security control calculations using linear programming, parts I & II, IEEE Trans. Power Apparat. Syst., vol. PAS-97, no. 5, pp , Sept./Oct [15] A. P. Meliopoulos and A. G. Bakirtzis, Corrective control computations for large power systems, IEEE Trans. Power Apparat. Syst., vol. PAS- 102, no. 11, pp , Nov [16] W. F. Tinney, V. Brandwajn, and S. M. Chan, Sparse vector methods, IEEE Trans. Power Syst., vol. PAS-104, no. 3, pp , [17] A. J. Conejo, F. J. Nogales, and F. J. Prieto, A decomposition procedure based on approximate Newton directions, in Mathematical Programming, ser. A. New York: Springer-Verlag, Sept [18] C. Grigg, P. Wong, P. Albrecht, R. Allan, M. Bhavaraju, R. Billinton, Q. Chen, C. Fong, S. Haddad, S. Kuruganty, W. Li, R. Mukerji, D. Patton, N. Rau, D. Reppen, A. Schneider, M. Shahidehpour, and C. Singh, The IEEE reliability test system 1996, IEEE Trans. Power Syst., vol. 14, pp , Aug [19] LINDO API 2.0 User s Manual [Online]. Available: Anastasios G. Bakirtzis (S 77 M 79 SM 95) was born in Serres, Greece, in February He received the Dipl. Eng. degree from the Department of Electrical Engineering, National Technical University, Athens, Greece, in 1979 and the M.S.E.E. and Ph.D. degrees from Georgia Institute of Technology, Atlanta, in 1981 and 1984, respectively. Since 1986, he has been with the Electrical Engineering Department, Aristotle University of Thessaloniki, Greece, where he is currently a Professor. His research interests are in power system operation and control, reliability analysis and in alternative energy sources. Nikos I. Macheras was born in Ierapetra, Creta, Greece in November He received the Dipl. Eng. Degree from the Department of Electrical and Computer Engineering, Aristotle University Thessaloniki, Greece in His interests are in computer networks, parallel processing and GNU/Linux. Pandelis N. Biskas (S 01 M 04) received the Dipl. Eng. and Ph.D. degrees from the Department of Electrical Engineering, Aristotle University, Thessaloniki, Greece, in 1999 and 2003, respectively. His research interests are in power system operation and control and transmission pricing. Dr. Biskas is a member of the society of Professional Engineers of Greece. Nikolaos K. Passialis was born in Thessaloniki, Greece, in October He is currently a student at the Department of Electrical and Computer Engineering, Aristotle University Thessaloniki. His interests are in computer networks, parallel processing, Database development, and GNU/Linux.