Abstract
A comprehensive scheme based on decentralized control, partitioning, multi-agent systems, and fuzzy logic is presented in this paper for the voltage control of power systems. In our proposed smart self-healing method, two types of control agents are defined, namely master and local, which are applied in two steps. In the first step, the power system returns to the normal state after fault occurrence. Immediately after a fault detection in a power system, the system is divided into three subsystems using spectral graph partitioning. Partitioning is conducted based on reactive power flow in transmission lines. For each subsystem, a local control agent and a performance index (PI) are defined. Whenever the PI of a subsystem exceeds its threshold limit, the local control agent uses the Sugeno fuzzy system to intelligently select and apply control actions. In the second step as performed by the master control agent, the power system is transformed to an optimal state by solving the optimization problem. Simulations on a 39-bus New England reveal the effective performance of the proposed method.
ACCORDING to the definition of Institute of Electrical and Electronics Engineers (IEEE) and International Council on Large Electric systems (CIGRE), voltage stability is the ability of power systems to retain stable and acceptable voltages in all buses following a fault occurrence [
Recently, smart grids have quickly developed in power systems. One of the fundamental features of smart grids is self-healing. A self-healing grid is capable of power system monitoring and can identify disturbances and take required actions to restore the grid. Therefore, a smart grid can manage those issues which are too complicated for human operators. This safety structure in power systems can prevent many faults.

Fig. 1 State and control of self-healing grid.
1) Preventive control: return the grid from a fragile state to a normal state. This control is a preventive measure and is activated prior to fault occurrence.
2) Emergency control: immediately return the grid to a normal state after fault occurrence. The process should occur quickly.
3) Restoration: restore the grid to its normal state after a fault (blackout).
4) Optimal control: return the grid to a state with a greater safety margin.
The present study concentrates on the emergency and optimal voltage control in self-healing grids. Emergency voltage control problems have been variously presented in many studies. These studies can be classified into centralized and decentralized control approaches. In centralized control approach, the model predictive control (MPC) is generally used to solve emergency voltage control problems. MPC contains three main blocks: control actions, trajectory prediction and an objective function. In this control approach, within a prediction horizon, the most appropriate control actions are applied to the power system [
Today, fuzzy control theory has been used in many electrical engineering issues [17]-[21]. A new strategy based on the Takagi-Sugeno fuzzy controller is proposed in [
Recently, multi-agent systems (MASs) have been heavily developed. In the research of artificial intelligence, agent-based system technology has been hailed as a new paradigm for conceptualizing, designing, and implementing software systems. Agents are sophisticated computer programs that act autonomously on behalf of their users across open and distributed environments to solve an increasing number of complex problems. Increasingly, however, applications require multiple agents that can work together. An MAS is a loosely coupled network of software agents that interact to solve problems which are beyond the individual abilities or the knowledge of the problem solvers.
An MAS has the following two main advantages over a single agent or centralized approach:
1) An MAS distributes computation resources and capabilities across a network of interconnected agents. Whereas a centralized system may be plagued by resource limitations, performance bottlenecks, or critical failures, an MAS is decentralized and thus does not suffer from the “single point of failure” problem associated with centralized systems.
2) An MAS enhances the overall system performance, specifically in terms of computation efficiency, reliability, extensibility, robustness, maintainability, responsiveness, flexibility, and reuse.
In [
To create a coordinated decentralized control, the power system should be partitioned to form a local control system in each subsystem. In [
A coordinated decentralized control for emergency voltage control in power systems is proposed in [
1) The system is partitioned before the fault occurs. If the fault in the power system is a line outage, the electrical distance matrix of the system will vary. Consequently, pre-fault partitioning does not render precise partitioning for these types of faults. In [
2) The system is controlled only by switching on the capacitors or load shedding.
3) The control variables are applied unintelligently and step by step until the PI value becomes zero. Therefore, the number of required control steps will increase for restoring the grid to its normal state.
This study presents a smart self-healing grid control for emergency voltage control in two steps. In the first step, following fault detection, the system is immediately partitioned based on the reactive power flow in transmission lines. The system is partitioned so that each subsystem exhibits the lowest exchange of reactive power with its neighboring subsystems. Spectral graph partitioning is used here for power system partitioning. Control actions in this paper include altering the reference voltage of generators as well as switching the capacitors and load shedding. Load shedding is applied if no other options exist to prevent a voltage collapse in the power system. The exact values of the control variables are determined using a fuzzy control system. Membership functions (MFs) in fuzzy control are regulated based on the pre-assessment of the power system in offline mode. In the case of obligatory load shedding, load buses with the shortest electrical distances from fault locations are selected. It should be mentioned that control actions in each subsystem are coordinated using MASs when no connection exists between the subsystems. With the use of control actions determined by FISs, the power system is transferred to its normal state in the first step. In the second step, the teaching-learning-based optimization (TLBO) is used to return the power system to its optimal state.
The remainder of this paper is organized as follows. Section II addresses the problem of coordinated and emergency voltage control. Section III introduces the structure of the proposed self-healing grid control based on a decentralized control. Section IV evaluates the proposed method using several simulations on a 39-bus New England power system. Finally, Section V discusses the obtained results.
A model of a power system for emergency voltage control including a combination of algebraic and dynamic equations is expressed as:
(1) |
(2) |
where and are dynamic and algebraic equations, respectively; is the dynamic state variable; is the algebraic output variable; is the system control variables; and is the time.
Since we intend to study the long-term voltage instability problem, the quasi-steady state (QSS) of generators suffices to investigate the behavior of the system [
(3) |
(4) |
(5) |
(6) |
where
is the voltage of the
In this paper, control variables include susceptances of capacitors, reference voltages of generators, and the amount of load shedding. Load shedding is applied only if no other options exist to prevent a voltage collapse in the power system.
The structure of a self-healing grid based on decentralized control is presented in

Fig. 2 Proposed voltage control scheme in a self-healing grid.
In the proposed emergency control method, numerous preliminary simulations are conducted for each power system. Then, after the system status with different faults is identified, a preliminary knowledgebase of the system is collected in offline mode. This preliminary knowledge is the basis for defining MFs of the fuzzy system. Then, a fuzzy control system is created for each control variable. Following a fault occurrence, the power system is immediately divided into three subsystems using spectral graph partitioning. For each subsystem, a local control system is defined. Based on the PI value of each subsystem, the control system smartly determines and applies the values of control variables to the system based on FIS.

Fig. 3 Proposed algorithm for a decentralized emergency voltage control in a self-healing grid.
For transferring to the optimal state, data from all power systems are required. These data are transmitted from LCAs to MCA, and the optimal control variables are determined by MCA by solving the optimization problem with TLBO algorithm. Reducing the voltage deviations is the objective function in this step. To solve this optimization problem, TLBO is proposed although it is possible to apply other optimization algorithms such as the imperialist competitive or particle swarm optimization. The control variables in the optimal control step include alterations in the susceptance of capacitors and reference voltage of generators.
Following a fault, the power system is divided into three subsystems using spectral graph partitioning. In our recent study [
To determine the vulnerability of a grid to a voltage collapse, the PI introduced in [
(7) |
where
is the minimum voltage limit;
is the maximum reactive power limit of the
(8) |
where is the output voltage of the generator; is the maximum voltage of the generator excitation system; is the rotor angel; is the terminal voltage angel; and and are the reactances of the direct and quadrature axes of the stator, respectively.
Similar to [
(9) |
(10) |
where is power system impedance matrix; is the number of bus loads in the subsystem ; and is the distance between the fault location i and load number . Then these distances are ordered from the smallest to the largest.
Whenever the PI value equals 0 in each subsystem, control actions are terminated. In addition, after 20 s, the emergency control system is deactivated. At this moment, the power system is restored from the fault state to the normal state. To restore the system to its optimal state, an optimization problem must be solved. Solving this optimization problem takes 240 s (four times the time constant) after the emergency control system is deactivated. In this stage, the power system is in the steady-state mode. Therefore, substituting the load model with the steady-state model is possible, and the QSS model can also be employed for generators. In this paper, the optimization problem is expressed as:
(11) |
where is the voltage deviation of the system which should be minimized; is the desirable voltage of the bus which is assumed to be 1 p.u.; and is the number of load buses. Control variables of the optimal state are susceptances of capacitors and reference voltages of generators. Finally, refers to the equality equations such as the load flow and refers to inequality equations such as limits of reactive power, voltage of generators, voltages of load buses, and limits of control variables. The variables and are fully described in Appendix A.
The fuzzy control utilized in this paper is a Sugeno type. Regulating MFs is the most difficult part in the process of forming a fuzzy control system. Two general methods can be used to define the rules in a fuzzy system.
1) Extracting the rules out of the collection of input and output data such as the stock market forecasting for tomorrow’s prediction of an index status, the fuzzy system uses a set of input and output data for the past year to determine its fuzzy rules.
2) If the input and output datasets are not available, the knowledge of experts can be used. This requires a person with sufficient knowledge of the system who knows how influential any change in a control variable is on other system parameters. To this end, for the 39-bus system, offline calculations are employed to obtain the best knowledge empirically based on trial and error. Thus, for each fault (outage of a line or a generator), an offline simulation is conducted on the 39-bus system to obtain the rules empirically. The values for MFs are presented in

Fig. 4 MFs in a fuzzy system.
Fuzzy system rules for control actions are listed in Table I. It should be mentioned that the susceptances of capacitors and load shedding can both accept discrete values. Consequently, the output of fuzzy system is rounded to the closest discrete values for these control variables.
The smart selection of control actions based on the Sugeno fuzzy system boosts the solving speed and reduces the number of control actions compared to the unsophisticated utilization of control actions. This reduces the costs of the power system.
The proposed method is implemented on a 39-bus New England power system. All features and settings are based on this power system. Three subsystems in the proposed method are also considered [
To evaluate the performance of the proposed self-healing control system, a 39-bus New England power system is studied, which is fully described in [
Control variables are applied on the system every 5 s. Simulations are performed using two methods: ① unsophisticated application of control variables (non-fuzzy); ② application of control variables using the Sugeno fuzzy method (fuzzy). The results are then compared. In Table II, the step changes in control actions are listed with the non-fuzzy method. However, with the fuzzy method, the control actions are determined based on IPI. For example, assume IPI equals 1.5 in Subsystem 2. According to
In the optimal control step, MCA receives the required data (
) from all LCAs. Then, MCA uses TLBO to determine the optimal control variables to transfer the power system to the optimal state. The initial population and number of iterations are assumed to be 40 and 400, respectively. TLBO algorithm is completely described in [
In this section, two scenarios are investigated. Tripping the generator 32 (G32) is the first scenario. For the second scenario, lines 5-6 and 6-7 are simultaneously removed from the grid. In both cases, it is assumed that the fault occurs at 30 s.
In this scenario, G32 is tripped at 30 s.

Fig. 5 System response after outage of G32 without any control actions.
When G32 is tripped and the control system detects the fault, spectral graph partitioning immediately divides the power system into three subsystems as shown in

Fig. 6 Partitioning of power system after outage of G32.
The IPI values for Subsystem 1 (IPI1) are presented in

Fig. 7 IPI1 with fuzzy and non-fuzzy methods in Scenario 1.

Fig. 8 PI1 with fuzzy and non-fuzzy methods in Scenario 1.
For Subsystem 2, according to

Fig. 9 IPI2 with fuzzy and non-fuzzy methods in Scenario 1.

Fig. 10 PI2 with fuzzy and non-fuzzy methods in Scenario 1.
It should be noted that after G32 is tripped, the IPI value of Subsystem 3 does not reach 1 with either methods. Therefore, LCA3 is not activated with either method. In this scenario, the control system restores the grid from an emergency to a normal state without load shedding.
Figures 11 and 12 show the voltages of buses 11 and 12 in both methods, respectively. As can be seen in these figures, the voltage reaches nearly the same value in both methods. However, according to Figs. 7-12, two significant results are obtained: ① voltage control is faster with the proposed method (fuzzy method); ② fewer control actions are required with the proposed fuzzy method, which is economically significant. The number of control actions is 7 with the fuzzy method (2 and 5 steps in Subsystems 1 and 2, respectively). This number is 13 times with the non-fuzzy method (5 and 8 steps in Subsystems 1 and 2, respectively).
Now, after restoring the system from a fault state to a normal one, the proposed self-healing grid control should restore the system to an optimal state. This is accomplished using TLBO algorithm at 240 s after the emergency control system is deactivated.

Fig. 11 Voltage at bus 11 with fuzzy and non-fuzzy methods in Scenario 1.

Fig. 12 Voltage at bus 12 with fuzzy and non-fuzzy methods in Scenario 1.

Fig. 13 Voltage profile in both normal and optimal states for Scenario 1.
In this scenario, lines 5-6 and 6-7 are simultaneously outed from the power system. Based on

Fig. 14 System response after outages of lines 5-6 and 6-7 without any control actions.

Fig. 15 Power system partitioning after simultaneous outages of lines 5-6 and 6-7.
According to

Fig. 16 IPI1 with fuzzy and non-fuzzy methods in Scenario 2.

Fig. 17 PI1 with fuzzy and non-fuzzy methods in Scenario 2.
According to

Fig. 18 IPI2 with fuzzy and non-fuzzy methods in Scenario 2.

Fig. 19 PI2 with fuzzy and non-fuzzy methods in Scenario 2.
It should be noted that with neither method does this fault shift IPI to 1 in Subsystem 3. Consequently, LCA3 will not be activated with either method. In this scenario, the control system returns the power system from an emergency to a normal state with two steps of load shedding at bus 7 with both methods.
Figures 20 and 21 show the voltage of buses 5 and 7 with both methods, respectively. Both methods nearly reach the same amount of voltage. With the fuzzy method, however, the retrieval speed is higher, and fewer control actions are employed. This is economically significant. The number of control actions with the proposed method is 9 (3 and 6 steps in Subsystems 1 and 2, respectively). The non-fuzzy method requires 11 steps of control actions (4 and 7 steps in Subsystems 1 and 2, respectively).
After restoring a fault state to the normal one, MCA should shift the power system to the optimal state. As in the previous scenario, the voltage profile in both normal and optimal states is shown in

Fig. 20 Voltage at bus 5 with fuzzy and non-fuzzy methods in Scenario 2.

Fig. 21 Voltage at bus 7 with fuzzy and non-fuzzy methods in Scenario 2.

Fig. 22 Voltage profile in normal and optimal states of Scenario 2.
In this paper, a self-healing system is proposed for a coordinated voltage control based on spectral graph partitioning, MASs, FISs, and TLBO algorithm. The proposed method is based on the smart selection of control actions in each subsystem according to IPI values with an FIS. Results indicate that this method reduces the number of control actions in the system compared to the non-fuzzy method, which is an economically significant result. This paper shows that the proposed control method is decentralized and faster than centralized methods. This paper uses spectral graph partitioning based on a reactive power flow in the transmission lines after fault detection to determine the subsystems. The advantage of partitioning based on a reactive power flow in the transmission lines is that each control variable is allocated to the subsystem. It is most influential on reducing the number of control actions required in the subsystem.
Appendix
Each graph is composed of several vertices and edges. For example, graph
is composed of a set of vertices
, and
is the weight of the edge between vertices
and
. The adjacency matrix for each graph is represented as
. In the graph partitioning problem, the goal is to partition the graph into K subgraphs so that the overall weight of edges between subgraphs can be minimized as [
(A1) |
where and are the sets of vertices in each subgraph; is an n-vector with the sum of all entries equal to 1; and is an matrix in which is:
(A2) |
In addition, must be orthogonal ( is a diagonal matrix) and . The notation denotes the Frobenius norm. In fact, the aforementioned conditions guarantee that each row of matrix has only one non-zero array, which is equal to 1. After (A1) is solved, if equals 1, it implies that bus is located in zone .
Since in (A1), is always a constant value, it can be rewritten as:
(A3) |
In graph partitioning, eigenvalues and eigenvectors are used to solve (A3). If is normalized so that the sum of each row is 1 and , the largest eigenvalue for will be equal to 1.
Let be the largest eigenvalues of and be the corresponding orthonormal eigenvectors. It is convenient to define:
(A4) |
where and . is an orthogonal matrix whereby , and since the largest eigenvalue of is equal to 1, we can obtain:
(A5) |
With the assumption that in which is a matrix, we can write:
(A6) |
(A7) |
Based on (A6) and (A7), if
is equal to the identity matrix, the upper limit of (A5) and consequently the maximum of (A3) are obtained. The condition for holding this assumption is [
(A8) |
The optimization problem is defined as:
(A9) |
(A10) |
(A11) |
(A12) |
(A13) |
(A14) |
(A15) |
(A16) |
(A17) |
(A18) |
(A19) |
(A20) |
(A21) |
(A22) |
(A23) |
(A24) |
(A25) |
where
and
are the numbers of buses and generator buses, respectively. Equations (A10) and (A11) are equilibrium equations in the load buses, (A12) and (A13) are equilibrium equations in the generator buses, and (A14) and (A15) are algebraic equations in stators. The QSS equations are described in (A16)-(A21). Equation (A22) indicates thermal limits in any branch, (A23) describes the susceptance limitation, and (A24) and (A25) express the limitations of the reference voltages of the excitation system and voltage buses, respectively. Equations (A10)-(A21) are fully described in [
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