Title: Network partitioning strategy for parallel power system restoration
Abstract: IET Generation, Transmission & DistributionVolume 10, Issue 8 p. 1883-1892 Research ArticleFree Access Network partitioning strategy for parallel power system restoration Lei Sun, Lei Sun School of Electrical Engineering, Zhejiang University, Hangzhou, 310027 People's Republic of ChinaSearch for more papers by this authorCan Zhang, Can Zhang School of Electrical Engineering, Zhejiang University, Hangzhou, 310027 People's Republic of ChinaSearch for more papers by this authorZhenzhi Lin, Zhenzhi Lin School of Electrical Engineering, Zhejiang University, Hangzhou, 310027 People's Republic of ChinaSearch for more papers by this authorFushuan Wen, Corresponding Author Fushuan Wen [email protected] School of Electrical Engineering, Zhejiang University, Hangzhou, 310027 People's Republic of China Department of Electrical and Electronic Engineering, Universiti Teknologi Brunei, Bandar Seri Begawan, BE1410 BruneiSearch for more papers by this authorYusheng Xue, Yusheng Xue State Grid Electric Power Research Institute, Nanjing, 210003 People's Republic of ChinaSearch for more papers by this authorMd. Abdus Salam, Md. Abdus Salam Department of Electrical and Electronic Engineering, Universiti Teknologi Brunei, Bandar Seri Begawan, BE1410 BruneiSearch for more papers by this authorSwee Peng Ang, Swee Peng Ang Department of Electrical and Electronic Engineering, Universiti Teknologi Brunei, Bandar Seri Begawan, BE1410 BruneiSearch for more papers by this author Lei Sun, Lei Sun School of Electrical Engineering, Zhejiang University, Hangzhou, 310027 People's Republic of ChinaSearch for more papers by this authorCan Zhang, Can Zhang School of Electrical Engineering, Zhejiang University, Hangzhou, 310027 People's Republic of ChinaSearch for more papers by this authorZhenzhi Lin, Zhenzhi Lin School of Electrical Engineering, Zhejiang University, Hangzhou, 310027 People's Republic of ChinaSearch for more papers by this authorFushuan Wen, Corresponding Author Fushuan Wen [email protected] School of Electrical Engineering, Zhejiang University, Hangzhou, 310027 People's Republic of China Department of Electrical and Electronic Engineering, Universiti Teknologi Brunei, Bandar Seri Begawan, BE1410 BruneiSearch for more papers by this authorYusheng Xue, Yusheng Xue State Grid Electric Power Research Institute, Nanjing, 210003 People's Republic of ChinaSearch for more papers by this authorMd. Abdus Salam, Md. Abdus Salam Department of Electrical and Electronic Engineering, Universiti Teknologi Brunei, Bandar Seri Begawan, BE1410 BruneiSearch for more papers by this authorSwee Peng Ang, Swee Peng Ang Department of Electrical and Electronic Engineering, Universiti Teknologi Brunei, Bandar Seri Begawan, BE1410 BruneiSearch for more papers by this author First published: 01 May 2016 https://doi.org/10.1049/iet-gtd.2015.1082Citations: 30AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract Parallel restoration is an efficient way to speed up the restoration process after a wide-area outage or a complete blackout of a power system. To implement efficient parallel restoration, an appropriate network partitioning method is necessary. Given this background, a two-step network partitioning strategy for parallel power system restoration is presented. In order to minimise the restoration time of generating units, a grouping model for non-black-start units is developed first. Based on the graph theory, a power network partitioning model is presented with the objectives of minimising the number of the interconnected lines and maximising the electrical distance of the interconnected lines among different restoration subsystems. The well-established AMPL/CPLEX is employed to solve the developed optimisation models. Finally, the IEEE New England 10-unit 39-bus system and a part of Zhejiang provincial power system in China are employed to illustrate the feasibility and effectiveness of the developed models. 1 Introduction Although well-developed protection and automatic control devices can effectively enhance the robustness of modern power systems, it is almost impossible to completely prevent power outages or blackouts from occurrence. For example, 130 power outages were recorded in USA from 1999 to 2004 [1]. The blackouts could lead to significant economic losses and even affect social security. Hence, much attention has been given to efficient and stable restoration of power systems [2–4]. Power system restoration strategies can generally be classified into ‘build-down’ and the ‘build-up’ ones [5, 6]. The ‘build-down’ strategy refers to restarting individual generators one by one, while in the ‘build-up strategy’ a power system is first partitioned into several restoration subsystems and subsequently all restoration subsystems are restored simultaneously. Generally, the ‘build-up’ strategy could reduce the complexity of the restoration problem and restore the power system quickly. The partitioning strategies for power system restoration have been well studied. A sectionalising strategy based on wide area measurement system is proposed in [7], which makes each power plant or substation in the restoration subsystems observable so as to guarantee the power system security in the restoration procedure. The ordered binary decision diagram-based method is used to partition a power system in [8, 9], and the partitioning result can then be optimised in real time after unpredictable changes in system conditions and operators’ choices. A strategy based on division and interconnection of the restoration subsystems is presented in [10], and does not require any heuristics and experts’ experience and can determine the sequence of interconnecting the restoration subsystems. A black-start partitioning strategy employing the characteristics of scale-free networks is proposed in [11], and can provide consistent restoration paths under given scenarios and be applied in unexpected situations. In [12], a black-start partitioning strategy is presented for maximising the number of the units to be restarted and accelerating the whole power system restoration procedure, taking into account the optimal startup sequence of the units and the critical maximum or minimum startup time constraints. A sectionalising methodology for parallel power system restoration is proposed based on the cut-set matrix in the graph theory in [13], and some constraints, such as black-start availability, load-generation balance, voltage stability and the ability to monitor synchronisation between adjacent subsystems, are respected for maintaining the steady-state stability of each subsystem. The effect of the number of the subsystems on the restoration process is studied in [14] and the optimal subsystem boundaries are determined to minimise the energy not supplied by using the genetic algorithm. In [15], the subsystems with strong internal connections (not weak external connections) are determined based on the spectral clustering, and the constraints presented in [13] are also considered. It can be concluded that only the network topology is taken into account in existing partitioning strategies for power system restoration, while the capacities of the units, the amount of the loads and the electrical characteristics of transmission lines are not comprehensively considered. Furthermore, the issues regarding restoration of the units in the restoration subsystems are rarely examined. The objective of partitioning a power system is for restoring the power system rapidly, so it is necessary to take the restoration of the units in each subsystem into account. Given this background, a two-step partitioning strategy is proposed in this work for quickly restoring power systems. The major contributions of this paper could be summarised as follows. First, in the existing publications the blackout area is sectionalised into several subsystems without taking the restoration of separated subsystems into consideration, while in this paper two issues, i.e. the partitioning of a power system and the grouping of units, are carried out simultaneously by employing the two-step network partitioning strategy. Second, a unit grouping model is proposed to divide all the units into several groups, which is beneficial to the rapid restoration of each separated subsystem. Finally, the unit grouping model and partitioning model of a power system are described as a constrained linear Boolean programming problem and a constrained quadratic Boolean programming problem respectively, and can be efficiently solved by the AMPL/CPLEX solver. The rest of the paper is organised as follows: Section 2 introduces a partitioning model from the well-established graph theory and describes some graph simplification principles for power systems. In Section 3, a two-step partitioning strategy, which consists of a unit grouping model with all the units assigned to several groups and a partitioning model for power system restoration subsystems, is proposed. In Section 4, the proposed partitioning strategy is demonstrated on the IEEE New England 10-unit 39-bus system and a part of Zhejiang provincial power system in China. Concluding remarks are made in Section 5. 2 Graph partitioning Before the graph theory is applied to partition a power system, the network topology representing the characteristics of an actual power system could first be simplified. The characteristics of actual power systems should be taken into consideration in power network partitioning. 2.1 Abstraction principles for actual power systems Abstraction principles for forming a topology graph of an actual power system are stated as follows: (1) Only transmission networks with voltages at 220 kV and above are considered. (2) Power plants and substations are equivalent to buses, in which the wiring structure is ignored. High-voltage transmission lines and transformer lines are equivalent to edges. (3) If there are two or more transmission lines between two different buses in an actual power system, only one line is considered in the abstracted topology graph. (4) The differences among bus voltage magnitudes and physical properties of transmission lines are ignored [16]. It should be pointed out that the proposed approach based on the bus branch model can be easily implemented in an actual restoration process. There is at least one circuit breaker in each side of an actual transmission line. In the presented bus branch model, if a branch is to be restored, the circuit breakers on a transmission line are to be restored in sequence. Otherwise, if a branch is not restored, the circuit breakers equipped in the transmission line remain open. 2.2 Mathematical model of graph partitioning problem In the graph partitioning problem, buses are assigned to two or more partitions for approaching the given objective with specified constraints. So far, graph partitioning methods have been widely applied to many research domains, such as parallel computation, sparse matrix sorting, and VLSI (very large scale integration) [17–19]. One of the commonly used objectives in graph partitioning is to minimise the number of the interconnection lines among the partitions. For example, each vertex in a graph usually represents a computational task in parallel computation, while each edge represents the communication requirements between two tasks concerned. The weight of a vertex represents the time required to complete the task of the vertex. Thus, the objective of the graph partitioning in parallel computation is to minimise the number of the connected edges and the communication, taking into consideration the balance between processors and computing in each partition [20]. For a directed graph with n buses and m lines, a line/bus incidence matrix A is defined to describe the relationship between the lines and buses. First, a partition index column vector X is introduced, and its dimension is equal to the number of buses. When the topology graph is divided into two subsystems (denoted as Zq,1 and Zq,2, which represent the partitioned subsystems respectively after the qth time in solving the graph partitioning model), the value of xj in vector X is equal to −1 if bus j is in subsystem Zq,1 or 1 if bus j is in subsystem Zq,2. Hence, once X is known, it can be determined whether each bus belongs to subsystem Zq,1 or not. A m-dimension column vector Y is defined as the multiplication of the incidence matrix A and the index vector X, and its element yl is equal to 0 if line l is within one subsystem, and yl is equal to ±2 if line l is an interconnection line between two subsystems. It can be concluded that the multiplication between the vector Y and its transpose vector is four times of the number of the interconnection lines among the subsystems [21]. Thus, the optimal partitioning problem for a topology graph with n buses and m lines, with an optimisation objective of minimising the number of the interconnection lines among the subsystems, can be formulated as (1)where xi is the ith element of X, and the value of xi can be −1 or 1. Thus, the optimal partitioning problem can be transformed into a typical quadratic Boolean programming one by linear transformation. 2.3 Graph simplification Generally, there are many buses and lines in actual power systems. In order to reduce the computational burden and improve computational efficiency, it is desirable to simplify the topology graph. Three kinds of graph simplifications including removal of degree-one-nodes, removal of degree-two-nodes, and removal of closed loops are introduced in [22], and employed in this work. A node degree represents the number of the lines directly connected with the node concerned. A node with degree one means that the node is directly connected by one line only. A node with degree two means that the node is directly connected by two lines. These three graph simplifications can be briefly introduced as follows: Removal of degree-one-nodes. If node j is connected with the system through one branch, node j cannot be assigned to an independent subsystem. So any degree-one-node and its adjacent node can be merged as a new node. Removal of degree-two-nodes. Assuming that node j is a degree-two-node and only directly connected with nodes i and k through lines. If branch i–j (or j–k) is the interconnected line between two subsystems, node j becomes a node with degree one when the branch i–j (or j–k) is disconnected. According to the simplification principle (1) in Section 2.3, node j and node k (or i) can be merged as a new node. It is assumed that there is a virtual line which connects node i with node k directly. The disconnection of any line connecting with node j is equivalent to that of the virtual line. Hence node j can be removed. Removal of closed loops. An independent circle is defined as two or more nodes connected with each other; each node connects with two different nodes except for the special one that connects with more than two nodes. The intersection of the independent circle and other buses in a power system is the special node, which can be considered as a new generating bus after the removal of the independent circle. All nodes in the circle can be replaced by a new node. Due to the characteristic of the models to be developed in the following section, one more simplification is presented, i.e. merging of grouped buses. Before partitioning the whole system, partial buses, including the buses with non-black-start units and the buses in the restoration paths for supplying the cranking power to those buses with non-black-start units from the black-start unit in the same group, should be assigned to one group. The buses in the same group are equivalent to a new bus in the graph partitioning model for guaranteeing that they are in the same subsystem. Generators to be restored in each group should be merged with the black-start generator in the same group into a new black-start generator bus. The active power of the new black-start generator bus is equal to the sum of the active power of all the generators in the same group. The load of the new black-start generator bus is equal to the sum of the loads in the buses of the generators in the group and the buses in the restoration paths. Merging the grouped buses into a new bus is to assign the buses in the same group into the same subsystem. 3 Partitioning model In this section, a partitioning strategy for parallel power system restoration is presented. First, we describe the principles and constraints concerned in partitioned subsystems. The number of subsystems is determined and the constraints must be accommodated for secure restoration. Then a unit grouping model and a partitioning model for a given power system are presented to identify the groups with non-black-start generators and the tie lines connected with the subsystems, respectively. Besides, the steps of the proposed partitioning strategy for power system restoration are shown in the last part of this section. 3.1 Partitioning principles and constraints of each subsystem In order to restore a power system, there should be at least one black-start unit for supplying startup power and one load for balancing power in the power system to be restored. Thus, the number of subsystems should equal to the smaller one between the number of the black-start generators and the number of load buses [7] (2)where s is the maximum number of the subsystems; n is the number of buses; Nbi is a Boolean parameter representing whether there is a black-start generator in bus i. The value of Nbi is equal to 1 if there is a black-start generator, or Nbi is equal to 0; Nli is a Boolean parameter representing whether there is a load in bus i. The value of Nli is equal to 1 if there is a load, or Nli is equal to 0. After determining the number of the subsystems, several constraints have to be respected in each subsystem to ensure that the power system be restored securely and effectively. These constraints are as follows. 3.1.1 Constraint of black-start generators In each restoration subsystem, at least there exists one black-start generator supplying startup power to non-black-start generators, and certain important loads should be involved in the subsystem restoration of the power system. The black-start generator in each subsystem can be restarted by itself. The cranking power generated by the black-start generator will be delivered to the non-black-start generators and some important loads in the same subsystem by the restored transmission lines. Important loads mainly include military bases, governmental organisations, hospitals, and data centres. Loss of electric power supply to these loads can lead to significant economic losses and very negative social impacts. These so-called important loads should be restored as soon as possible in the system restoration process. It should be pointed out that during the early restoration stage, the lines are lightly loaded, which may result in very high harmonic resonance voltage. If a transformer is overexcited because of the power frequency voltage, the harmonic resonance voltage will increase [23]. Hence, some load, even not important, should be restored in each subsystem to maintain the voltage profile within an acceptable range. This constraint can be mathematically described as (3)where Vz is the bus sets of restoration subsystem z; PGj is the active power of generator located in bus j, and PGj is equal to 0 if there is no generator located in bus j. 3.1.2 Power balance constraint As shown in (3), at least one generator exists in each subsystem. What is more, the active output power of generators and loads should be balanced approximately, and the absolute value of their difference should be within a given threshold value. This constraint is mainly employed to guarantee that the frequency is maintained within an acceptable range. In an actual power system, the reactive power imbalance is related to the sequence of the restoration paths, and is not examined in this paper. In the early stage of the restoration, in selecting the loads to be restored, resistance-type loads should be given some priority, and hence the reactive power issue can be mitigated. With the restoration procedure going on, reactive power compensators could be used to maintain the reactive power balance and hence improve the voltage profile in an actual restoration procedure. Comparatively, the balance of active power plays a key role in the restoration procedure [9]. Thus, only the balance of active power is considered in this work. The following constraint should be accommodated for any restoration subsystem z (4)where d is the allowable threshold of unbalanced active power in each restoration subsystem. In [8], the permitted power balance error limit is set as 0.11Psubsys (Psubsys is the overall active power output of units in subsystems) when the lowest system average frequency is 57 Hz. However, according to [24] it is beneficial to maintain the frequency greater than 58.5 Hz for the sake of secure restoration. Therefore, d should be smaller than 0.065Psubsys. A lower d could improve the frequency response ability of a power system, and 0.05Psubsys is selected as the allowable threshold. 3.1.3 Minimum active power output constraint for each generating unit In each restoration subsystem, there should be sufficient loads for balancing the increasing generation outputs. Each unit has its minimum output power, so the loads should not be less than the minimum output power of all the units in each restoration subsystem [7]. For any restoration subsystem z, this constraint can be mathematically described as (5)where is the minimum active power of the generator located at bus j. 3.1.4 Frequency constraint The power balance constraint shown in (4) limits the absolute value of the difference between the generation output and the load demand in the subsystems within a given threshold. In the restoration procedure, the active power output of generators should be more than the load demand and some margin can be maintained by controlling the restored load in each stage. Thus, the frequency could be regulated by the generators in each restoration subsystem [24, 25]. 3.1.5 Voltage constraint During the parallel restoration process, the available voltage regulation methods could be used to maintain the voltage profile within the acceptable operating range in the subsystems, such as adjusting the terminal voltages of generators, the transformer ratios, and the outputs of reactive compensation equipment's. In the restoration procedure, the system dispatchers could maintain the voltage profile in each restoration subsystem within the permitted limits through voltage regulation [2, 23, 26]. 3.2 Unit grouping model In order to determine which black-start unit should supply startup power to the non-black-start unit, the unit grouping model representing the first step of the proposed two-step partitioning strategy is first introduced. It takes the shortest time for the non-black-start units to get startup power from the black-start units in each group. Subsequently, the units in the same group and the connection lines among them are merged as a new equivalent unit in the next power system partitioning model. After the partitioning process of the whole power system is completed, the equivalent units can be restored by the units and lines prior to grouping. This can effectively prevent the units to be restored in a restoration subsystem from being assigned to another restoration subsystem. The optimisation strategy of unit restoration is a multi-stage optimisation problem with various constraints. The restarting of the unit depends on the active power supplied by the system, the power required to restart the unit and its critical maximum/minimum interval. In the proposed unit grouping model, the non-black-start units, which should be supplied with startup power by black-start units as soon as possible, are grouped. Thus, the restoration time of a transmission line can be set as their weight. The well-known Dijkstra algorithm, which is a classical shortest path algorithm, is employed to determine the shortest path from black-start units to the non-black-start units. The transmission path for a non-black-start unit consists of the transmission lines delivering the startup power from a black-start unit to the non-black-start unit. The time of recovering the concerned transmission path is taken as the restoration time of the non-black-start unit. In [27], the problem of estimating the restoration duration is addressed, and the ‘most likely’ restoration time of a transmission line is employed as the restoration time. Before the unit grouping model is introduced, a grouping identification matrix V with b rows and r columns is defined, where b is the number of the black-start units in the whole power system, and r is the number of units to be restored. In this paper, the black-start units are named as source buses. Matrix V is composed by r b-dimensional grouping identification column vectors Vg = (v1g, v2g…vbg)T, where g is the serial number of units to be restored (1 ≤ g ≤ r); v1g, v2g, …, vbg are Boolean variables representing the grouping situation of unit g. If only veg is 1 in column g and all other elements are 0, then unit g belongs to group e. Thus, Vg represents the restoration relationship between unit g and all the black-start units. In the unit grouping model, the number of groups should be equal to that of the black-start units, and the objective is selected as the minimum time consumed by the non-black-start units for obtaining the startup power from the black-start unit located in the same group. Thus, the optimal grouping model can be described as (6) (7) (8)where v1g, v2g,…,vbg are non-negative Boolean variables; (w1g, w2g, …, wbg)T is the restoration time matrix of unit g, and its element wtg is the time consumed by supplying the startup power from black-start unit t to unit g. β is a proportional factor. Equation (8) implies that the number of the non-black-start units in any group e is not more than β times the total number of the non-black-start units in the power system, so most of the units to be started can avoid being assigned to the same group. AMPL, an acronym for ‘A Mathematical Programming Language’, is an algebraic modelling language for describing and solving high-complexity problems for large-scale mathematical computation. AMPL supports dozens of solvers, both open source and commercial products, including CPLEX, Gurobi, MINOS, IPOPT, SNOPT, and KNITRO. One particular advantage of AMPL is the similarity of its syntax to the mathematical notation of optimisation problems. This allows for a very concise and readable definition of problems in the domain of optimisation. CPLEX can solve linear, quadratic, second-order cone and mixed-integer programming problems. The optimisation model described in (6)–(8) is a typical linear Boolean programming problem with constraints, which can be solved efficiently by AMPL/CPLEX. It should be pointed out that if there are two or more optimal solutions for the unit grouping model, the solution with the minimum charging capacity in transmission lines will be selected as the final optimal one. 3.3 Partitioning model of a power system Parallel restoration strategy can improve the overall restoration speed of the power systems. One of the most important factors affecting the efficiency of power system restoration is the synchronisation problem among the partitioned restoration subsystems. The restoration operation may fail during the synchronisation of the subsystems because it is difficult to take many factors, such as the transient overvoltage and the reduction of standing angles, into consideration. The more interconnected lines between restoration subsystems, the greater the risks involved in accomplishing the restoration operations securely. Thus, the number of the interconnected lines among the subsystems should be minimised when the power system is partitioned, so as to improve the synchronous efficiency of parallel restoration. The other objective function in the proposed power system partitioning model is to maximise the electrical distance of the interconnected lines among different restoration subsystems. The power system is a complex network with community structure characteristics. The partitioning results should reflect the community structure to some extent, which means that the buses within a subsystem are strongly connected while buses within different subsystems are weakly connected. The strength of connections is measured by the electrical distance that represents the electrical characteristics of the network. Line admittances can be regarded as electrical distances and used to measure the strength of connections of a power network [15, 28]. The line resistance can be ignored because its value is much less than that of the line reactance. In this work the normalised line susceptance is employed as the line weight. As the second step of the proposed two-step partitioning strategy, the power system partitioning model which aims to minimise the number of the interconnection lines and to maximise the line weight in the inte
Publication Year: 2016
Publication Date: 2016-03-24
Language: en
Type: article
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