The Maximum Flow Algorithm Applied to the Placement ... - CiteSeerX

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The Maximum Flow Algorithm Applied to the Placement and Distributed Steady-State Control of UPFCs Austin Armbruster, Student Member, IEEE, Michael Gosnell, Student Member, IEEE, Bruce McMillin, Member, IEEE, Mariesa Crow, Senior Member, IEEE

Abstract— The bulk power system is one of the largest manmade networks and its size makes control an extremely difficult task. This paper presents a method to control a power network using UPFCs set to levels determined by a maximum flow (maxflow) algorithm. The graph-theory-based max-flow is applied to the power system for UPFC placement and scheduling. A distributed version of max-flow is described to coordinate the actions of the UPFCs distributed in a power network. Two sample power systems were tested using max-flow for UPFC placement and settings. The resulting system characteristics are examined over all single-line contingencies and the appropriateness of the maximum flow algorithm for power flow control is discussed. Index Terms— load flow control, operations research, power transmission control, power system control, unified power flow control

I. I NTRODUCTION

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HE bulk power system grid is arguably the largest manmade interconnected network in existence. The sheer size of the power network makes control an extremely difficult task. Cascading failures are the most severe form of contingency that can occur in a power system. A typical cascading failure involves transmission line overloads. In a system where there is a large directional power flow from one region to another, the loss of a single transmission line can instigate a cascading failure. If the first transmission line is lost, the power it carried must be shunted across remaining transmission lines. The redirected power may cause one or more of these transmission lines to overload. If not immediately mitigated, an initial contingency may lead to a second, a third, and so on, until load must be shed to stabilize the system. The family of Flexible AC Transmission System (FACTS) devices holds considerable promise as network-embedded controllers. Distributed Unified Power Flow Controller (UPFC) devices in the network can help alleviate the overload problem by directing extraneous power flow away from highly loaded lines. Although one of the most promising applications of UPFCs is to use them to regulate active and reactive power flow across critical transmission corridors, transmission

The National Science Foundation has supported this work under contracts DGE-9972752 and ECS-0085666. The authors are with the University of Missouri–Rolla Intelligent Systems Center, 1870 Miner Circle, Rolla, MO 65409-0350 USA. A. Armbruster, M. Gosnell, and B. McMillin ({aearmbru, mrghx4, ff}@umr.edu) are members of the Department of Computer Science. M. Crow ([email protected]) is a member of the Department of Electrical and Computer Engineering.

providers have been reluctant to install them due to their cost and lack of systematic control paradigms. Grid control has historically been decentralized due mainly to geographic and regulatory constraints. UPFCs are naturally decentralized due to their wide geographic distribution throughout the transmission system, and are good candidates for distributed power grid control methodologies. In addition, the rapid expansion of communication technologies, including optical and wireless communication, is making it possible to incorporate real-time, distributed, decision-making processes over a widespread area using a variety of information. As such, distributed FACTS control can be designed to work cooperatively to maintain long-term power system security. Before UPFCs can be implemented on a wide-scale basis, they must also coordinate their actions with each other dynamically and rapidly in the event of a contingency; therefore, a static setting is not sufficient. In addition to the development of distributed control procedures, the placement of the UPFCs within the transmission system is another key issue. For the UPFCs to achieve maximum control and resiliency to all possible contingencies, the UPFCs must be placed along critical pathways. The placement of the devices must consider the network topology and resultant power flows after any possible system contingency. One promising technique that addresses control, communication and coordination is the graph theory-based maxflow technique, originally used for transportation control and allocation. This approach was first proposed for power systems in [1]. The max-flow algorithm finds a set of flows through a network that approximate the power system flows which are then used as a basis to set the controls of the UPFCs. The power system then responds to these settings to stabilize the system. This paper describes a graph-theory-based distributed maxflow algorithm used to help identify critical transmission corridors, determine UPFC placement, and adjust power flow to avoid cascading failures. Contributions of this paper include: 1) the use of the max-flow algorithm to adaptively control the UPFCs in the event of contingencies to maintain system security, 2) the use of the max-flow algorithm to determine the placement of UPFCs in the power system, 3) the introduction of a distributed max-flow algorithm to coordinate the actions of the UPFCs in a decentralized network, and

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II. M AXIMUM F LOW AND THE P OWER S YSTEM N ETWORK This section will give an introduction to the maximum flow (max-flow) algorithm [2]. The algorithm will be applied to power system flow control and also as a basis in which to place and coordinate the UPFCs. In the max-flow algorithm, the power network is modeled as a directed graph G(N, A) where power flow is represented as flow in the graph. The set of nodes, N , corresponds to the buses of the power network. The power line between buses ni , nj ∈ N is represented by an arc aij ∈ A. Each arc is assigned a tuple of the amount of remaining flow and the maximum flow, uij , for that particular line. Initially, the maximum flows are set as the steady state power flow values of the system so that the max-flow algorithm approximates the actual power flows. For the basic max-flow algorithm there are two special nodes, the virtual source (s) and the virtual sink (t), representing the combination of the generator(s) and load(s), respectively. Each line out of the virtual source has a maximum flow that matches the generation of the connected node, and each line into the virtual sink represents the load demanded by the connected node. Constrained by power flow equations, each node in the graph must also satisfy ΣPin = ΣPout , except for the virtual source and sink nodes. The virtual sink may be considered the network “ground” node. The nodes s and t, together with G, form the graph G (N  , A ). Fig. 1 is an example power system which has an equivalent directed graph representation in Fig. 2. Modeling the power network as a directed graph is useful when a line is outaged due to a contingency, and the resulting power flow stresses the network. If too much power is drawn over lines with inadequate capacity, the lines may successively overload and trip off-line, causing a cascading failure. In the case of a contingency, a method is needed to rapidly rebalance the power by either redirecting the power flow across transmission corridors with greater capacity, or possibly by shedding load to maintain adequate power to the remaining loads. By modeling this problem as the maximum flow in a directed graph [3], the UPFC settings can be determined. The formal max-flow algorithm is given in Fig. 3. The algorithm

Fig. 2. Power network shown as a directed flow graph with virtual nodes s and t. Edges are labeled with (flow, capacity). The capacity over all edges is fully utilized.

works by successively computing the maximum flow along a directed path from s to t until no more power can be added. The appropriateness of this approach is addressed in Section IV. Fig. 4 shows the system from Fig. 2 with flow settings to compensate for the loss of a line. Without increasing capacity of the remaining lines, the max-flow tells us some loads cannot be satisfied. The power system does not actually (nor is it expected to) behave precisely as the max-flow algorithm predicts. Line losses, reactive power flow, and system nonlinearities cause the power to flow differently; however the max-flow scheduling can be used to determine the setting of the UPFCs. The premise of this approach is that if the UPFCs are appropriately placed throughout the system, then, acting in a coordinated fashion via the max-flow algorithm, they can control the power flow through critical transmission corridors. The UPFCs will subsequently be able to have significant effect on the power flow across the non-controlled corridors such that the impact of contingencies on overloads can be substantially mitigated. III. A D ISTRIBUTED V ERSION OF THE M AX -F LOW A LGORITHM In practice, the max-flow control algorithm cannot be run from a single centralized site, but must be decentrally run at each UPFC site. The UPFCs are physically distributed throughout the system; therefore, the original max-flow algorithm must be modified to reflect the distributed nature of the UPFCs. A modified max-flow algorithm appropriate for decentralized control is presented in this section. To provide real-time power flow scheduling, the proposed graph theory-based max-flow strategy is implemented such that it can respond to contingencies in real-time over a wide geographic area. The only way to accomplish this is to devise a

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Forward Labeling of aij : If ni is labeled and nj is not, and uij > f (aij ), nj gets labeled, and Δij = uij − f (aij ) Backward Labeling of aij : If ni is labeled and nj is not, and f (aij ) > 0, nj gets labeled, and Δij = f (aij )

Algorithm: 1) Assign an initial flow (f (aij ) = 0 for all arcs in A). 2) Mark s labeled and all other nodes unlabeled. 3) Search for a node that can be labeled by either a forward or backward edge. If none found, flow is maximum, stop. If nj = t go to step 4, otherwise, repeat step 3. 4) Backtrack the path computing the minimum Δij used. If aij used a forward labeling, f (aij ) ← f (aij ) + Δij . If aij used a backward labeling, f (aij ) ← f (aij )− Δij . Go to step 2. Fig. 3.

Ford and Fulkerson sequential algorithm for max-flow [2]

distributed algorithm in which the UPFCs cooperatively reach an agreement from their individual settings. Using the communication and computational processes of the UPFCs along with a network interconnection, the max-flow is implemented as a distributed long-term control algorithm. Much of the previous work in UPFC power flow control has concentrated on determining a priori how power is distributed in the network from each generator. The proposed distributed max-flow algorithm computes the actual flow balance needed to prevent cascading failures in a distributed real-time manner. With distributed max-flow, the algorithm simultaneously explores augmenting flow paths from each generator by passing probe and response messages over the communications links. Winning paths are selected for the resulting power flow on a “first discovered” basis. Each UPFC runs the distributed maximum flow algorithm shown in Fig. 5 which is a modification of the one from [4]. The primary change is that flow paths are selected from all generators concurrently. To use the distributed max-flow algorithm for the control algorithm, the power flow problem must be described in appropriate terms. The arc capacities are initially set to the steady-state power flow values. By starting from the steadystate values, the maximum flow algorithm attempts to keep the network in a physically feasible configuration. If maxflow cannot satisfy the loads with these settings, changes are made to try to satisfy the load. First, all of the settings are increased to the corresponding line capacities. The max-flow algorithm is then restarted with the unsatisfied sink nodes as source nodes and source nodes as sink nodes. The max-flow algorithm changes slightly to compensate for the change in source and sink nodes. When the arc flows are modified, the negative value of the Δ is added to or subtracted from the arc flow. The final change to the algorithm is in the calculation

of the Δ values to be Δij ← uji − fji . The resulting flow is a solution that satisfies all loads and line capacities, if one exists. The lines to which the UPFCs are attached are set to the max-flow settings to control power flow in the UPFCs. The UPFCs remain coordinated through passing messages over a communication network. To meet real-time constraints, the number of messages must be minimized. Tests were conducted with the distributed max-flow algorithm with the IEEE 118-bus system to determine the control settings for UPFCs in the presence of contingencies. The program was executed on a cluster consisting of 16 Sun Ultra 10’s and 16 Sun Ultra 5’s each with 100 Mega-bit Ethernet connection. The workstations are connected through a Catalyst 3200 switching network with a 2.1 Gb/s backplane. Using 5 workstations to represent the UPFCs and another 19 to represent generators, the program requires 20 seconds to execute after a contingency occurs. Although execution in under a minute may be adequate to mitigate many cascading failures, other max-flow algorithms, such as [5] can be applied to improve this speed for distributed UPFC control. IV. UPFC P LACEMENT AND A PPROPRIATENESS OF C ONTROL S ETTINGS A. UPFC Placement for Contingencies The max-flow algorithm provides an excellent mechanism for determining the placement of UPFCs in the power system. The placement of series devices in a power network is considerably different than the placement of shunt devices, since series devices more significantly impact active power. Fairly comprehensive studies have been performed to determine the optimal placement of STATCOMs due to their similarity to SVCs and static capacitor banks [6], [7], but these are not appropriate for the placement of series devices (SSSC and UPFC) in the bulk power system. A variety of approaches have been proposed for placing series devices (usually considering the TCSC) in the system, but little comprehensive work exists that determines the device controls with respect to changes in system topology or loading. Many utilities considering investment in UPFCs would like assurances that the devices can be placed and controlled to mitigate any foreseeable contingency. To meet this objective, a usable placement algorithm must: • minimize the number of required UPFCs, • minimize the number of overloaded lines, and • provide the UPFC settings (controls) to minimize an economic objective such as fuel costs, line losses, etc. and this minimization must occur across the set of all possible system contingencies. In other words, the placement of a minimum number of UPFCs and their respective settings must be chosen such that for any given contingency, no system lines are overloaded and any economic constraints are met. Several authors ([8]–[19]) have achieved good results on portions of this problem, but to date, no one has attempted to perform this optimization over the entire set of possibilities. This distinction is important. Most of the placement algorithms for series devices typically consider only one set of system

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flows and/or topology. In a real system, however, it is necessary to consider the placement of the devices such that they will provide power flow control regardless of system topology (i.e. loss of line, load, or generator). To accomplish this, the setting of the devices must be determined at the same time as the placement of the devices is determined. This is a large and very complex optimization problem that cannot easily be solved with traditional tools. Previous work in determining the optimal location of UPFCs can be primarily categorized into sensitivity-based [8]–[12] or stochastic (genetic algorithm-based) [13]–[19]. Most of the UPFC placement algorithms have been applied to a system only at steady-state, therefore the control settings of the devices are chosen for a particular topology, load, and generation profile and may also be computationally time-consuming. With the exception of [12], these methods do not consider the placement and setting of the devices over the entire set of possible contingencies, ensuring that no lines are overloaded for any loss of line. But this is exactly the application that is of interest to transmission service providers. The max-flow algorithm provides a cohesive approach for placing and controlling the devices to accommodate any possible contingency. The application of the max-flow algorithm on the power grid is accomplished by using the UPFCs to enforce the desired flow across each UPFC-controlled line. The UPFCs control the active power flow across the lines. The objective is to find the optimum placement of the minimum number of UPFCs to maximize the benefits of the max-flow algorithm. To properly control the network, the UPFCs need to be arranged in such a way that the grid remains stable under all possible single line contingencies. A simplistic assumption is that if a UPFC can be placed on every line in the system, then the maxflow algorithm provides the control setting for each device regardless of contingency to help ensure no line overloads.

This assumption is not realistic since UPFCs are too expensive to place on every line and max-flow does not account for line losses. However, the max-flow algorithm can be used to identify critical corridors across contingencies on which the UPFCs will provide the greatest impact on power flow. The most comprehensive power system examination involves an exhaustive search that calculates the power flow under every combination of UPFC placements over each possible single line contingency, thereby testing every possible scenario. The so-called brute force placement approach grows as nk where n is the number of lines in the system and k is the number of UPFCs. The brute-force placement of one UPFC is computationally tractable since the computational effort scales linearly with the number of lines in the system. For each potential single UPFC placement, the line flows are calculated across the set of all single line contingencies and compared to the line capacities, and the overloaded lines are counted. Collecting the data from all single-UPFC placements yields an ordering of placements that provides a crude estimate of the lines that are candidates for UPFC placement. This grouping is called the “top candidate lines.” The brute force method for one UPFC placement will yield not only the best placement for a single UPFC, but also a set of top candidate lines. To determine the best placement for two UPFCs, the top fifty candidate lines provide a set of UPFC pairs to test: (line 1, line 2), (line 1, line 3), . . ., (line 1, line 50), (line 2, line 3), . . . for a total of 1225 combination pairs. Similarly, for three UPFCs, there are combinations of triples: (line 1, line 2, line 3), (line 1, line 2, line 4), . . .. To minimize the number of possible triplet combinations, only the top 20 candidate lines are used. Therefore, there are 1140 combinations to analyze. This process can be carried out to include any number of UPFCs or top candidate lines. The top candidate lines used for testing up to 10 UPFC devices

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Distributed Max-Flow Algorithm: Assign an initial flow (f (aij ) = 0 for all arcs in A) for each power source si , do in parallel 1) 2) 3) 4) 5)

Mark si labeled. Select an unlabeled outgoing arc nj . If there are no more unlabeled outgoing arcs, stop. Label node si and the arc with (si , Δij ). Explore the arc asi j with a probe(si , Δij ). Wait for messages to return. If the return message is • path(si , Δij ), set f (asi j ) ← f (asi j ) + Δij • blocked(si): All paths to t are blocked by other searches. Unlabel the arc asi j . Go to step 2. • echo(si ): All paths to t along asi j are filled to capacity. Go to step 2.

for each node, nl 1) Wait for a probe(si, Δkl ) 2) If the node is already labeled by si , return echo(si ) to the sender of probe. Go to step 1 3) If the node is already labeled (by some sj ), return blocked(si) to the sender of the probe. Go to step 1 4) If the node is a sink node, • If alm used a forward labeling, f (alm ) ← f (alm ) + Δlm . • If alm used a backward labeling, f (alm ) ← f (alm ) − Δlm . clear labels for si , send path(si , Δlm ) to sender of probe, and go to 1. 5) Select an unlabeled arc, alm . 6) If there are no more unlabeled arcs that are not already at capacity, send echo(si ) to the sender of the probe, nk , and clear the label for si and go to 1. 7) Δlm ← ulm − flm . 8) Label the arc and send a probe of (si , Δ=min(Δkl , Δlm )). 9) Wait for messages to return. If the return message from node nm is • path(si , Δlm ) – If alm used a forward labeling, f (alm ) ← f (alm ) + Δlm . – If alm used a backward labeling, f (alm ) ← f (alm ) − Δlm . clear labels for si , set Δkl ← Δlm , and send path(si , Δkl ) to sender of probe. • blocked(si), all paths to t are blocked by other searches, clear labels for si and send blocked(si) to sender of probe. • echo(si ): All paths to t are filled to capacity. Go to 5. 10) Go to 1. Fig. 5.

Distributed algorithm using multiple sources for max-flow. Uses the same definitions as Fig. 3.

are shown in Table I. The number of combinations tested for each set of UPFC devices is user-defined and can be chosen according to the computational effort desired. The values given in Table I were chosen to maintain comparable computational effort regardless of the number of UPFCs. B. Effectiveness of Max-Flow for Power Networks It is theorized that once the critical corridor line flows are enforced by UPFCs according to the maxflow algorithm, the remaining power flow in the system will not overload most of the lines, while still satisfying the load. To test this theory, brute-force and combination-based placement strategies were used to find placements of UPFCs on two power systems. The first system was the IEEE 118 bus test system containing 118 busses, 186 lines, and 20 generators. The second test system was a scaled down version of the Western Electricity

Coordinating Council (WECC) power system containing 179 busses, 263 lines, and 29 generators. The best solutions found for UPFC placements up to 10 devices are shown in Tables II and III. The aggregate overloads over all single-line contingencies and the aggregate amount of overload is shown in relation to the number of UPFCs placed. The relationship between the number of UPFCs and number of overloads is shown in Fig. 7. Additionally, placements from Tables II and III were weighted with a single Performance Index (PI), calculated as PI =

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TABLE I

TABLE II

M ULTIPLE -UPFC PLACEMENTS GENERATED FROM COMBINATIONS FROM

IEEE 118- BUS UPFC PLACEMENT RESULTS

THE ORDERED LIST OF SINGLE -UPFC PLACEMENTS .

Number of UPFCs Placed 2 3 4 5 6 7 8 9 10

Number of Top Lines Used 50 20 15 10 10 10 11 12 13

Number of Combinations Tested 1225 1140 1365 252 210 120 165 220 286

of lines in the system, and  1 when line n is overloaded w(n) = 0 when line n is not overloaded Function w(n) ensures that only overloaded lines (where Sn > Smax ) are included in the calculation of PI. Thus, the performance index sums the squares of the ratio of overload for all overloaded lines. The PI can then be used as an optimization criteria for evolutionary algorithms where lower PI values indicate a less overloaded system. These results are achieved through proper placement of the UPFC and the application of the maximum flow algorithm to determine the active power flow control setting of each UPFC. There is, however, a point at which the benefit of applying more UPFCs saturates. This can clearly be seen in the IEEE 118-bus system from Fig. 7. After placing around five devices,

Number of UPFCs 0 1 2 3 4 5 6 7 8 9 10

Number of Overloads 116 107 95 86 81 75 73 71 70 69 88

Amount (MW) 25.7 16.3 13.8 12.6 11.5 10.2 9.65 9.46 9.19 8.90 8.68

Performance Index (PI) 0.734 0.510 0.460 0.417 0.390 0.359 0.348 0.341 0.337 0.328 0.320

the incremental decrease in line overloads abate to roughly one overload per additional UPFC. V. C ONCLUSIONS This paper presents a novel method for controlling the active power flow through the power transmission system using UPFCs. This method is based upon the graph-theoretic maximum flow algorithm to determine the scheduling of active power flow on each line in the system. The maximum flow algorithm can actively compensate for line outages by redirecting power flow to avoid cascading failures. The maximum flow algorithm is implemented using UPFCs to enforce active power flow constraints on critical corridors within a power system. The IEEE 118-bus test system and the 179-bus WECC system were used to show the effectiveness of the proposed

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Aggregate overloads over all contingencies in two test systems with placement of up to 10 UPFCs.

TABLE III WECC 179- BUS UPFC PLACEMENT RESULTS Number of UPFCs 0 1 2 3 4 5 6 7 8 9 10

Number of Overloads 154 128 113 98 83 75 67 61 56 52 49

Amount (MW) 905 851 832 815 797 792 789 777 770 767 759

Performance Index (PI) 5.08 4.81 4.76 4.71 4.66 4.62 4.60 4.58 4.56 4.55 4.54

[2] [3] [4]

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[7]

placement and scheduling algorithms. The number of line overloads resulting from line outages was reduced by using UPFCs set according to the maximum flow algorithm. Work in this area continues in developing a fault-tolerant distributed maximum flow algorithm that can be implemented in practice.

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Austin Armbruster (S’01) received his BS in Mechanical Engineering in 2000 and MS in Computer Science in 2001 from the University of Missouri–Rolla. He is currently pursuing a Ph.D. in Computer Science at UMR and is employed at Boeing’s Phantom Works in St. Louis, Missouri. His research interests include constraint-based modeling and hardwaresoftware co-design.

Michael Gosnell (S’03) received his BS in Computer Science/Mathematics and Chemistry in 2002 from Buena Vista University, Storm Lake, IA. He received his MS in Computer Science from the University of Missouri–Rolla in 2003 and is currently pursuing a Ph.D. at UMR. He is a student member of IEEE and ACM. His research interests include wireless broadcasting, parallel and distributed computing, and fault tolerance in wireless airborne systems. Bruce McMillin (S’86–M’88) received the BS degree in electrical and computer engineering and the MS degree in computer science from Michigan Technological University, Houghton, Michigan, respectively, and the PhD in computer science from Michigan State University. Dr. McMillin is currently a professor of computer science at the University of Missouri–Rolla. He directs interdisciplinary teams in formal methods for fault tolerance and in computational engineering on multiprocessing systems. His research interests include fault tolerance and security, parallel algorithms, software engineering, and distributed systems theory. Mariesa Crow (SM’94) received the B.S.E. degree from the University of Michigan at Ann Arbor, and the Ph.D. degree from the University of Illinois, at Urbana–Champaign. Currently, she is the Dean for the School of Materials, Energy & Earth Resources and a Professor of Electrical and Computer Engineering at the University of Missouri–Rolla, Rolla. Her research interests include developing computational methods for dynamic security assessment and the application of power electronics in bulk power systems.