یک طراحی چند هدفه جدید برای جایابی بهینه واحدهای توربین گازی به منظور بهبود قابلیت خودراه اندازی

1. Introduction [1]

The modern power systems are being controlled in a high level of reliability and security. However, different unpredicted events may occur and, in turn, cause to have a wide area of cascading failures and then a blackout for the entire system [1-2]. This can impose huge economic losses and negative social impacts. Therefore, the quick restoration of a power system after a blackout is one of the most important concerns for power system dispatchers, operators and planners [3].

The process of power system restoration includes three stages: preparation, transmission paths reconfiguration and load restoration [4]. In the preparation stage, the black-start units are commissioned, and then transmission paths are connected to crank non-black-start units (NBSU) in order to be started up [5]. The objective of this stage is mainly to maximize the generation capability. In this regard, many researchers have formulated this objective as an optimization problem [3-6]. In [3] and [5] authors propose an algorithm that formulates generator start-up sequence as a MILP problem. In [3] the firefly algorithm (FA) is used to search the final optimal sequence of NBSUs, transmission paths and load pick up sequence by the aid of renewable energy sources (RES) already installed in power system. In [5] an optimal generator start-up strategy is addressed to find a sequence of starting time for all NBSUs and also to update the generation capability during system restoration. In [6], an initial restoration sequence of generators is obtained by using a mixed integer linear problem (MILP) method. Then, the final restoration sequence is manually acquired by using this initial guess after many trials. A decision making support system is addressed in [7-8] to search the optimal black-start strategy by using a concept of the vague-valued fuzzy measure. In [9] a steady state model is developed to optimize the total generator power output through searching the optimal start-up sequence of NBSUs by using the shortest path decision. In [10], the optimal start-up sequences of generators are designed by applying an algorithm based on Dijkstra method for finding the shortest path during power system restoration. In [11], the optimal network reconfiguration was designed into two separately steps. In the first step, the optimal sequence design of generating units was done to maximize the restored generation capability, and in the second step, the optimal transmission path is obtained to crank the generating units.  In [12] a combined design of the generating start-up sequence and the transmission path search is presented as an optimization problem and solved by using firefly algorithm.

In all researches published so far, to the best of our knowledge, the selection of black-start unit (BSU) as a starting point of restoration process has been based on the existed units within the power system without performing any design procedure to locate these units.  In this regard, although the hydro and Gas Turbine (GT) power plants that are generally considered as the BSUs, due to the high speed of commissioning, it will be possible that these plants do not have enough capabilities to crank NBS units [13]. Some of the hydro power plants are not available because of the insufficient flow of the water or the low level of the stored water behind the dam [14]. Moreover, when the already-installed gas turbine power plants continuously work in the system, there is a high risk that after blackout, the technical problem in fuel system causes to trip the unit or it may not permit to start the unit for restoration preparation [15]. Furthermore, in both gas turbine and hydro power plants, the failure of control systems may stop the process of starting BSU [15-16]. Therefore, considering redundant BSUs is necessary to crank and to increase the speed of restoration [17-19].

There are many situations during the normal condition of a power system that the system should be able to respond to the sudden load demands very quickly. In these cases the new installed GTs that can be started very fast could be one of the best options to help the system increase the generations in the shortest possible time [20-21]. Therefore, it will be more economical to place these units in the locations where the minimum power losses are achieved.  They must also justify another important constraint which is the voltage violation during the normal condition. Therefore, the problem of the optimal placement of GTs, as the good candidates for redundant BSUs, will be solved by increasing the capability of black-start generation and also improving the power system performance in normal conditions.  In doing so, a multi-objective function is used to obtain the best locations of GTs, the optimal transmission path and the optimal generators start-up sequences by considering the black-start capability criterion.

There are several techniques for finding the optimal transmission path in a power system including expert system [22], artificial neural networks [23], power transfer distribution factor (PTDF)-based approach [24], linear programming-based approach [25], heuristic approaches [26] and Dijkstra algorithm [27].  Among all these techniques, Dijkstra algorithm has a simpler structure and, therefore, the computational complexity of any proposed design is decreased by using this algorithm [28].  Also, the CPU time of running the proposed optimization model during the optimization process is decreased.

The multi-objective function consists of minimizing the Unavailable Energy Capability (UEC) of the system during the restoration process and also minimizing the voltage deviations of buses and active power losses in transmission system during normal condition to achieve the best Pareto optimal set. Then, a new population-based meta-heuristic technique, called crow search algorithm (CSA) based on fuzzy decision making approach is used to obtain the Pareto optimal solutions. The idea of using CSA is initiated from one of the real and intelligent behavior of storing, hiding and retrieving process of crow’s food by themselves [29]. The high level of efficiency, accuracy and robustness to search the Pareto optimal solution and the finding capability of the proposed algorithm make it be a desirable approach for optimizing the MOP addressed in this paper. The main contributions of this paper are as follows:

• · Optimal placement of the Gas Turbine (GT) as the redundant energy source to improve the black-start capability,
• · Considering all objective functions related to the both restoration and normal conditions simultaneously, and
• · Presenting a multi-objective crow search algorithm (MOCSA) for searching the optimal decision variables.

The 39-bus New England power system and a 32-bus power system are employed as two test systems. The simulation results show that not only the proposed method is capable of finding the best locations of the redundant GTs, but also it finds the optimal start-up sequences of NBS units with the optimal transmission paths in order to speed up the restoration process.  It also improves the performance of the power system in normal conditions.

The rest of the paper is organized as follows: Section 2 describes the problem formulation. In Sections 3 and 4, the solution approach (MOCSA) is given to solve the proposed multi-objective problem. Simulation results are given in Section 5 and finally, Section 6 concludes the paper.

2. Problem Formulation

The formulation of the multi-objective model for searching the location of a GT including three objective functions and their constraints is given in this section.

2.1. Objective Functions

The employed objective functions for the proposed multi-objective model are explained in this subsection.

2.1.1. Unavailable Energy Capability Objective Function

The first objective function is used to minimize the total generation capacity (MWh) which is not available after blackout [3] and is given by the following equation [3]:

The optimal start-up sequence of NBS units and the optimal locations of GTs are obtained by the optimization of the first objective function. The optimal transmission paths are also determined when the objective function f₁is calculated. This will be described in the next subsection.

2.1.1.1. Optimal Path Search Algorithm

A minimum spanning tree method is used here to find the shortest paths between two nodes in network and can be shown as a graph [30]. This algorithm, which is an iterative process, has been introduced by W. Dijkstra [30]. A simple example of this algorithm is shown in Fig. 1. In this figure, a shortest path is found between node A (source node) and node E (destination node). The start point of this algorithm is shown by coloured node A (step 1). In the first iteration, the algorithm searches the adjacent node from the source node, which must be the nearest node to the source node (step 2).  In the second iteration, the algorithm searches the second nearest node from the source node
(step 3).  This node must be the nearest node from the first node which was the closest to the source node. This procedure iteratively continues so that the minimum spanning tree for the source node is found (step 4). This tree gives an optimal path which includes all nodes.

Fig (1): Dijkstra algorithm [30]

2.1.2. Minimization of Active Power Loss

Minimizing the active power loss is formulated as follows [31]:

2.1.3. Voltage Violation

The following objective function is given to reduce the voltage violation in all PQ buses [31]:

2.2. Decision Variables

The decision variables are optimally obtained after completing the optimization process in order to achieve the best optimal solution. In the proposed design method, a constrained optimization problem will be described as a multi-objective function and minimized subject to  and . The and are defined as the decision variables and imply all start-up times of NBS units and the location of a GT (redundant BSU) respectively.

2.3. Multi-objective Design

In multi-objective design it is impossible to obtain an optimal solution when all objective functions are simultaneously being optimized, because the objective functions are different and each function is not compromised with other functions. In this condition, a decision maker chooses a “most effective” answer as an optimal solution [32]. Generally, the equation of a MOP can be formulated as follows;

Optimize

The vector is defined as decision variables and obtained by solving Eqn. (4). Also, this MOP is executed subject to regard the total number of all constraints. These constraints consist of inequality () and equality () constraints as given in Eqns. (5) and (6), respectively.

In this study, Eqns. (5) and (6) include all equality and inequality constraints given in Eqns. (1), (2) and (3) which are the starting times of NBSUs and the generation capability of BSUs, active and reactive power balance and also the voltages of buses.

Generally, the proposed MOP is optimized by storing a group of most desirable answers in a repository and renewing the group at any iteration [33]. In this technique, the best acquired answers are defined as non-dominated answers or Pareto optimal set. An answer can be determined as a Pareto optimal set if and only if Eqn. (7) is satisfied.

In Eqn. (7)  is defined as the Pareto optimal set of the possible search space of. It also implies that enhancing one objective function by an allowable answer will be gained by degrading the optimal solution of at least one of the other objective functions.

2.4. Best Acceptable Solution

In the optimization process, it is required that a solution be considered between all non-dominated answers of the Pareto optimal set as the best allowable answer [33]. In this paper, a fuzzy mechanism is used for three objective functions (f₁, f₂, f₃) because their values are not precise. In this regard, a fuzzy membership function is defined for achieving the most desirable answer. For all individual recorded in repository, the following equation is formulated [33]:

where and  are described as the maximum and minimum limits of each objective function respectively. These amounts are calculated by comparing with the values acquired by the single optimization of each objective function.

The normalization of the membership of the recorded individuals could be done according to the following equation [34]:

where ,,  are the total number of objective functions, the number of Pareto optimal answers in repository and the weight of K_th objective function respectively. Finally, the best compromise solution is considered as the solution which has the maximum .

2.5. Repository Size Control

One of the main factors in the design of any evolutionary optimization approach is to control its repository’s size [35-36]. In this paper, a fuzzy clustering technique is used to control the size of the repository without changing its characteristics [35]. In doing so, each solution of the repository is defined as a cluster with definite radius. Neighbor clusters are combined until the proper repository size is achieved. In the combining stage, the member with the higher membership value of combined clusters is chosen to be recorded in the repository. The flowchart of the whole mechanism using fuzzy based clustering is given in Appendix.

3. Crow Search Algorithm

The idea of crow search algorithm (CSA) was firstly created from one of the real and intelligent behavior of crows by A. Askarzadeh in 2016 [29]. This algorithm formulates the storing, hiding and retrieving process of crow’s food by themselves.  The full details of CSA are given in [29].

One of the most important specifications of meta-heuristic algorithms is satisfying a good balance between diversification and intensification [37]. In CSA, intensification and diversification are principally balanced by the parameter of awareness probability (AP). By decreasing the value of AP, the searching process will go into a local area in CSA where a current good solution is obtained in this area.  On the other hand, increasing the value of AP conducts searching process in CSA into the global area (randomization). Therefore, diversification is increased by applying the large values of AP.

4. Implementation of CSA in Multi-objective Design for Optimal Placement of GTs

Now, the design procedure for implementation of CSA in multi-objective design is described as follows:

Step 1: Determine the required data which are flock size of crows (N), flight length (fl), awareness probability (AP), maximum iteration, and Pareto archive dimension.

Step 2: Generate the initial random population of crow’s position and their memories.

Step 3: Compute the value of objective functions for each crow by solving Equation 4.

Step 4: Investigate the constraints of the problem by using Eqns. (1), (2) and (3). If the constraints are not satisfied, consider a high value of penalty for objective functions related to the current crow, otherwise, ignore it.

Step 5: Search the non-dominated solutions among the possible solutions and record them in the Pareto repository according to Eqns. (8) and (9).

Step 6: Check the repository size. If the size of repository is more than the maximum predefined size, remove the extra member with fuzzy based clustering mechanism.

Step 7: Generate the new population of crow’s position by using Eqns. (10) and (11).

Step 8: Check the feasibility of the new population of crow’s position.  If the new position is possible, go to step 9.  Otherwise, go to step 7.

Step 9: Compute the value of objective functions for each crow by solving Equation 4.

Step 10: Investigate the constraints of the problem by using Eqns. (1), (2) and (3). If the constraints are not satisfied, consider a high value of penalty for objective functions related to the current crow, otherwise, ignore it.

Step 11: Update the repository by replacing the new non-dominated solutions according to Eqns. (8) and (9).

Step 12: Check the repository size. If the size of repository is more than maximum predefined size, remove the extra member with fuzzy based clustering mechanism.

Step 13: Check the convergence criterion.  If the maximum iteration number is reached, stop the execution of the algorithm.  Otherwise go back to step 7.

The flowchart of the above-mentioned steps is given in Fig. 2.

5. Simulation Results

In this section, the proposed method is performed on two different power systems to show its efficient performance during restoration and normal conditions.

5.1.  The 39-bus New England Power System

The single line diagram of the 39-bus New England power system is depicted in Fig. 3 and its information can be found in [38]. This system is segmented into two subsystems only at restoration condition according to [3], and each subsystem must have at least one black-start unit [39]. Therefore, the number of already installed BSUs is assumed to be 2, which are G6 and G10 [3, 5]. As mentioned in section 1, it is essential that the redundant BSUs (GTs) be considered for each power system, and their optimal locations are obtained.

Fig (2): Flowchart of application of CSA on multi-objective design

Fig (3): IEEE 39-bus power system [37]

Now, the proposed design procedure is performed in two stages. First, the single objective design is done to specify and to use the limit values of three objective functions f₁, f₂and f₃ ((Eqns. (1), (2) and (3)) for the second stage (multi-objective design). Then, the multi-objective design is carried out to determine the optimal solution (or optimal decision variables), which are the start-up times of NBS units and the locations of GTs. The required data of generators for black-start process are given in Table 1 according to [5]. In this table, refers to the cranking time for generator to start to ramp up for connecting with system, and  is the rate of generator ramping.

Table (1): Characteristics of generating units in 39-bus system [4]

5.1.1. Single Objective Design

In this part, single objective design is performed by using CSA algorithm. The extracted data of this part are used as the limits for the multi-objective approach.

5.1.1.1. Restoration Condition

In this subsection, the objective function f1 (Eqn. (1)) is minimized by using CSA so that all optimal start-up times of NBS units and the best locations of GTs (redundant BSUs) are obtained and the results are given in Table 2. The convergence curve of f1 is also shown in Fig. 4. The last row of Table 2 shows the best value of the objective function f1 used for the limits considered in multi-objective method. By using the optimal search path algorithm explained in section 2.1.1.1, the optimal start-up sequences of NBSU units are obtained and given in Table 2. In this regard, each subsystem includes one redundant BSU and four NBSUs. . It should be mentioned that the capacity of each gas turbine is selected to be 50 MW because the total amount of power needed for starting all NBS units (Pstart) in each subsystem is 50 MW.

Fig (4): CSA-based convergence curve of UEC

Table (2): Optimal design of generators start-up process using CSA-based design considering redundant BSU

The colored optimal path of cranking power for all NBSUs is depicted in Fig. 5. In this figure, the sequence of start-up for all NBSUs is determined by a specific color. For more clarification, the details of these paths are given in Table 3.

Fig (5): Optimal generator start-up of IEEE 39 bus power system using redundant BSU

As mentioned before, G6 and G10 have been considered as main BSUs. A comparison is performed between two different cases where the system is equipped with new GTs as redundant BSUs and the case where new GTs are not installed. The results are presented in Table 4. The results given in this table show that the value of UEC for all cases is desirably decreased when new GTs are used when either G6 or G10 fails to operate. For example, in case 1 when G6 fails to operate, the value of UEC by using new GTs (2309.9 min. p.u.) is lower than the ones without new GTs (2870.60 min. p.u.). For the worst situation (case 3) when both G6 and G10 fail to operate, by using new GTs as redundant BS units, the value of UEC is 2334.90 min. p.u., which means that the black-start process is successfully performed.

Table (3): The optimal cranking power paths of NBSUs in New England power system

Table (4): Effect of GTs as the redundant BSUs in black-start process during different cases