Multi-Objective Coordination of Local and Centralized Volt/Var Control with Optimal Switch and Distributed Generations Placement

Abstract

Distribution network operators have been generally engaged over the last few decades toward distribution system automation to enhance reliability and efficiency of power systems. From the viewpoint of reliability, optimal sectionalizing switch placement is an effective strategy since it mitigates both power network asset cost and number of customers not supplied by outages. From efficiency perspective, introducing capacitors and distributed generations (DGs) and DGs optimal sitting in power distribution systems with the goal of volt/var control can reduce system loss and voltage deviation. In this regard, this paper proposes a methodology to determine the optimal location of sectionalizing switches and DG units while the concept of Volt/Var control problem is also considered. The objective functions are formulated to enhance the network condition during normal and contingency conditions. A new enhanced sine cosine algorithm is proposed for providing the best compromise solution. A self-adaptive probabilistic mutation strategy is appended into the original algorithm to balance exploitation and exploration in order to achieve the promising search space and ultimately converge to the global or near-global optimum. A 33-bus IEEE test system is deployed to assess the superiority of the proposed framework.

Keywords

Distributed generator (DG) placement multi-objective optimization sine cosine algorithm switch placement volt/var control

1. Introduction

The necessity of enhancing power distribution network reliability and efficiency besides integration of distributed generations (DGs) and restricted financial budgets for network automation force the electric utilities to deploy innovative alternatives. The most cost effective technique for network reinforcement and its efficiency enhancement is smart grid paradigm which can obtain affordable and sustainable solutions [1, 2]. In this regard, distribution network automation is one of the most capable toolsets for network reliability and efficiency development [2] utilizing sectionalizing switch placement and actions and thus decreasing capital and operational expenditure as well as outage-experienced regions following a fault. It is to be noticed that the efficiency improvement in terms of system loss and voltage deviation reduction obtained by distributed generation increases as they located appropriately in the network. Thus, optimization technique is needed to find an optimal solution for switch placement and DG sitting to simultaneously optimize the aforementioned cost, voltage deviation and loss. Another concern of power distribution network operator is to enhance the restoration capability. Therefore, the main aim is to restore the maximum amount of load within low rate of outage of capacitors and DGs by a sequence of switching operations following the isolation of a fault [3], since disconnecting DGs and/or capacitors decreases the advantages related to theirs high penetration. Further, the scenario of DGs lost is ineffective and leads to needless loss of power. Therefore, the proposed strategy should have this ability to preserve connections to the DGs and capacitors following the disturbance and at the same time balance the active and reactive power generated and consumed in the network in order to potentially enhance the efficiency and reliability of the system.

Considerable efforts have been devoted to optimal switch placement problem considering functional requirements and cost advantage in single or multi-objective optimization approach [4 –9]. In the switch sitting as an investment decision, the algebraic sum of investment cost as well as customer interruption cost is deployed as the objective function [4, 5]. In [4] an immune algorithm is used to solve the optimal switch placement problem while sum of customer service outage cost and switches investment cost is considered as the objective function. The reliability index and the customer type are two major parameters which affect the objective function. In [5] two-stage decomposition approach is used to find the optimal number of switches and their optimal locations. In this regard the solution space is divided into independent sub spaces then the optimization problem is solved in each subspace. In [6] the optimal switch placement problem is formulated while improving the reliability indices is considered as the objective function and the analytic hierarchical process decision-making algorithm is offered to determine the optimal location of switches. In [7] the particle swarm optimization (PSO) method is proposed to reduce the number of affected customers by power outages using the optimal switch placement problem. In this paper, the number of customers not supplied is considered as the objective function. In [8] a three-state approach is presented to define the number and locations of two types of switches in radial distribution feeders. The authors state that the proposed method provides a global optimal solution for switch placement problem. In [9] a reliability index based on energy not supplied is considered as the objective function in optimal switch placement problem while different types of switches are also considered.

In other hand, the system restoration strategies are also proposed to open the naturally closed sectionalizing switches and at the same time close the naturally open tie switch in order to raise the interrupted load demand. In this context, numerous techniques are suggested to restore the distribution network after a blackout. In [10] the authors proposed a method using a restoration plan based on the status of each feeder at each time interval. A Subgradient-Based Lagrangian Relaxation method is used to solve the problem while restoration index achieved from the dual formulation of the problem. In [11] the service restoration methods are investigated from the implementation time point of view and a method is proposed which can restore the customers in about 1 minute after a blackout. In [12] a reduced state dynamic programming method is proposed to solve the distribution system restoration problem. In this regard, the problem states i.e. timing and selection of feeders to be energized are grouped together. In [13], a modified version of combined Chaotic PSO and TLBO methods is used to solve the distribution network reconfiguration problem with loss objective function. In [14] the service restoration and other related issues are formulated in an optimization problem while a heuristic method is proposed to solve it considering the practical rules. In [15] a method for distribution network restoration is presented based on connecting to the adjacent feeder by normally open switches. The authors use group restoration, zone restoration and load transfer to increase the service restoration capacity. In [16], a multi-criteria decision making service restoration is presented to cope with the conflicting objective functions in two steps i.e. candidate set generation and fuzzy decision making.

The abovementioned techniques have provided pertinent simulation results for the sectionalising switch placement problem. However, the impact of DGs installation hasn’t been investigated in these research studies which can significantly affect the distribution networks especially voltage deviation of network buses and system loss. Their influences can be positive or negative based on their locations and network operation modes. In other hands, they can lead to a bidirectional power flow in distribution networks, so that their penetration can impact on the optimal switch placement. It means that simultaneous optimizing of DGs and sectionalizing switches placement can help to minimize cost, loss and voltage deviation which is conducted in this paper. A multi-objective sectionalizing switch and DG placement taking the critical system condition into account is proposed in [17]. A mixed integer non-linear programing approach is developed in [18] in order to localize the sectionalizing switches in island DGs operation. The long-term and short-term interruptions costs and switch cost are considered from the reliability perspective.

There are several drawbacks related to previous researches: 1) since the majority of the proposed approaches are based on heuristic algorithms or employ heuristic rules, they suffer from guaranteeing the global or near-global optimality of the obtained solutions; 2) since three optimization problem i.e. optimal switch placement, DG placement and centralized Vot/Var control problems can affect each other, so investigating these problems independently may be far from the optimal conditions. In other word, the previous researches do not consider a comprehensive model including the voltage deviation, the loss and the reliability indices as the objective functions and do not control the outage of DGs and capacitors after restoration process while these objectives have considerable effects on power quality and reliability of distribution networks.

To cover these drawbacks this paper presents a multiobjective optimal switch and DG placement problems considering the centralized Volt/Var control as an optimization problem. In this regard, a cost function is introduced to formulate the customer interruption costs in which the load-point reliability indices should be calculated using the switching time as well as fault clearance time or distribution element repair time. In the second stage, the DG placement problem is solved to optimize other objective functions comprise of power loss and voltage deviation as the volt/var control process. Moreover, the DGs and capacitors switch off during the restoration procedure by changing the status of tie-switches in order to balance active and reactive power between generator and consumer sides and in the other hand to avoid the system loss much more than pre-restoration time. Furthermore, the number of outage DGs and capacitors are considered as the objective functions to reduce this phenomenon. To handle all objective functions and network constraints, a new meta-heuristic multi-objective optimization algorithm, named enhanced sine cosine algorithm (ESCA), is proposed in this study and tested on the single objective version of the proposed problem to evaluate its performance and provides the extreme points of the trade-off surface, and is then applied to the proposed multi-objective framework. The solution of the analysis consists of the optimal number and location of DGs and sectionalizing switches. The effectiveness of the proposed framework is illustrated by applying it to the 33-bus IEEE power distribution test system.

The rest of this paper is outlined as follows. The proposed formulation for the switch placement problem and DG sitting under volt/var problem is presented in Section 2. The explanation of proposed ESCA algorithm is provided in Section 3. The proposed solution methodology is described in Section 4. Simulation results are given and thoroughly discussed in Section 5. Finally, the paper concludes in Section 6.

2 Problem formulation

2.1 Decision variables

$\begin{matrix} X = [\begin{matrix} Loc . Sw & Loc . DG \end{matrix}]_{1 \times N var} \\ Loc . Sw = [\begin{matrix} Loc . {Sw}_{1} & \dots & Loc . {Sw}_{ls} & \dots & Loc . {Sw}_{2 N_{br}} \end{matrix}]; \\ Loc . {Sw}_{ls} \in {0, 1} \\ Loc . DG = [\begin{matrix} Loc . {DG}_{1} & \dots & Loc . {DG}_{ld} & \dots & Loc . {DG}_{N_{DG}} \end{matrix}] \\ N var = 2 N_{br} + N_{DG} \end{matrix}$ (1) In above equation Xis the vector of decision variables, Loc.Sw and Loc.DG are the vectors contain the location of sectionalizing switches and DG units, respectively. Loc . Sw_ls and Loc . DG_ld define the location of the ls^th sectionalising switch and ld^th DG units, respectively.

2.2 Objective functions

The multiple objective functions during the normal and contingency conditions which should be minimized are formulated as Equations (2)– (8).

Minimizing total power loss during normal and post restored condition

$F_{1} (X) = \sum_{i = 1}^{N_{br}^{norm}} R_{i} \times {| I_{i}^{norm} |}^{2}$ (2) $F_{2} (X) = \sum_{j = 1}^{N_{br}^{norm}} \sum_{i = 1}^{N_{br}^{por}} R_{i} \times {| I_{i, j}^{por} |}^{2}$ (3)

When a contingency condition is occurred, a blackout imposes on some network braches. In this condition, the sectionalizing switches adjacent to faulted branch is opened which is called pre-restored (PRR) condition in this paper. Then energy of the un-faulted zone is restored by closing the normally open switches (tie switches) while the faulted zones are isolated. The second objective function (F₂ (X)) is calculated in this condition which is called post restored (POR) condition in this paper. In Equations (2)-(3) R_i is the resistance of i^th network branch. $N_{br}^{norm (por)}$ is the number of network branch in normal condition (post restoration condition). $I_{i}^{norm .}$ is the current of ith branch in normal condition and $I_{i, j}^{por}$ is the current of branch i when a fault is occurred in branch j in post restoration condition [19].

Minimizing the voltage deviation during normal and post fault condition

$F_{3} (X) = \sum_{i = 1}^{N_{bus}^{norm}} | \frac{V_{i}^{N} - V_{i}}{V_{i}^{N}} |$ (4) $F_{4} (X) = \sum_{j = 1}^{N_{br}^{norm}} \sum_{i = 1}^{N_{bus}^{por}} | \frac{V_{i}^{N} - V_{i, j}}{V_{i}^{N}} |$ (5)

The 4th objective function is calculated in similar condition as the 2nd objective. $V_{I}^{N}$ , V_I and V_i,j are the nominal voltage of bus i, voltage of bus i in normal condition and voltage of bus i when a fault is occurred in branch j.

Minimizing the total number of switching DG units and capacitors off

$\begin{matrix} F_{5} (X) = \sum_{j = 1}^{N_{br}^{norm}} \sum_{i_{DG} = 1}^{N_{DG}} (U_{i_{DG}}^{por} + U_{i_{DG}}^{prr}) \\ + \sum_{j = 1}^{N_{br}^{norm}} \sum_{i_{Cap} = 1}^{N_{Cap}} (U_{i_{Cap}}^{por} + U_{i_{Cap}}^{prr}) \\ U_{i_{DG}}^{por (prr)} = {\begin{matrix} 1 & \begin{matrix} if i^{th} DG is switched off \\ during por (prr) \end{matrix} \\ 0 & else \end{matrix} \\ U_{i_{cap}}^{por (prr)} = {\begin{matrix} 1 & \begin{matrix} if i^{th} capacitor is switched off \\ during por (prr) \end{matrix} \\ 0 & else \end{matrix} \end{matrix}$ (6)

This objective is named SDGCO for simplicity hereinafter. N_DG and N_cap are the number installed DG units and capacitors in network, respectively.

Minimization the total switch cost

$\begin{matrix} F_{6} (X) & = & \sum_{i = 1}^{2 N_{br}^{norm}} {(IC}_{i} + {CI}_{i} + \sum_{t = 1}^{T} {MC}_{i} \times (1 + DR)^{- t}) \\ \times Loc . {Sw}_{i} \end{matrix}$ (7)

IC_i, CI_i and MC_i are the installation, the capital investment and the maintenance costs of the ith sectionalizing switch. DR is the Annual discount rate.

Minimize the reliability cost

$\begin{matrix} F_{7} (X) = \sum_{t = 1}^{T} \sum_{f = 1}^{N_{f}} \sum_{i = 1}^{N_{i}} \sum_{j = 1}^{N_{j}} \sum_{k = 1}^{N_{k}} {ECOST}_{i, j, k, f, t} \\ {ECOST}_{i, j, k, f, t} \\ = L_{j, k, f} (1 + q)^{t - 1} C_{i, j, k, f} (r_{i}) (1 + DR)^{- t} λ_{i, f, t} \end{matrix}$ (8)

The reliability cost refers to customer’s damaged cost during the PRR and the POR conditions. N_f, N_i, N_j, N_k are the number of feeders, possible fault locations, load points and customers types, respectively. L_j,k,f is the Average customers of k-type load at load point j and feeder f. q is the annual increasing rate of load, C_i,j,k,f (r_i) is the customer’s damage cost function due to outage time r_i ; λ_i,f,t is the fault rate.

The network losses and the voltage deviation are calculated as the DG placement objective functions during the normal operation condition (F₁ (X) and F₃ (X)). Furthermore, F₂ (X) and F₄ (X) which calculates the network losses and the voltage deviation of network buses during the POR conditions will be affected by the location of DG units and sectionalizing switches. The F₅ (X) tries to join the optimal DG and the switch placement problems. In other hand, the DG units and the capacitors should be switched off during the PRR and POR conditions if one or more following states be occurred: a) they are isolated from the main bus, b) network losses be greater than the loss of normal operation condition, and c) the generated active and reactive power in network are greater than the network demands. So the location of DG units and the sectionalizing switches affect the F₅ (X), simultaneously. If the above mentioned conditions be occurred so that the DG units force to switched off; they are selected based on decision variable; i.e. the DG unit in Loc . DG₁ is selected in the first step, then Loc . DG₂ and so on. This idea permits optimization algorithm to change the order of DG units without dedicating any additional decision variables. F₆ (X) defines the installation and capital investment cost of sectionalizing switches besides the maintenance cost of them during the switch life time. All in all, objectives 2, 4 and 5 encouraged us to formulate the optimal switch placement besides the optimal DG placement considering the technical and economical objective functions.

2.3 Problem constraints

The proposed optimization problem deals with some technical and operational constraints which are formulated based on equations (9)– (18):

Customer damage cost based on switching and repair time

$\begin{matrix} C_{i, j, k, f} (r_{i}) - C_{i, j, k, f} (r_{i}^{'}) ⩾ 0 \\ C_{i, j, k, f} (r_{i}) - C_{i, j, k, f} (r_{i}^{″}) (1 - x_{i}) ⩾ 0 \end{matrix}$ (9)

Constraint (9) confirms that the number and the location of switches can affect customer damage cost due to fault events. In this regard, the customer damage cost for load point j imposed by fault at location i is at least equal to interruption cost corresponding to switching action time ( $r_{i}^{'}$ ) for isolating the faulty area (first line of (9)). If closing the normally open tie switches cannot restore the load point j due to fault occurrence at location i, then this load point experience an interruption with repair time duration while its customer damage cost function satisfies the second line of (9).

Limit on the number of switching operations

\sum_{i = 1}^{N_{s}} | S_{i} - S_{i, old} | ⩽ N_{switch}

(10)

S_i and S_i,old are the new and base status of switch i. Power flow equations

P_{i} = \sum_{\begin{matrix} j = 1 \\ i \neq j \end{matrix}}^{N_{bus}} V_{i} V_{j} Y_{ij} cos (θ_{ij} - δ_{i} + δ_{j}) i = 1, 2, . . ., N_{bus}

(11)

Q_{i} = \sum_{\begin{matrix} j = 1 \\ i \neq j \end{matrix}}^{N_{bus}} V_{i} V_{j} Y_{ij} sin (θ_{ij} - δ_{i} + δ_{j}) i = 1, 2, . . ., N_{bus}

(12)

Limits of apparent power flow

| S_{ij}^{Line} | < S_{ij, max}^{Line} i \neq j i, j \in {1, 2, . . ., N_{br}}

(13)

Limits of bus voltages

V_{i, min} ⩽ V_{i} ⩽ V_{i, max} \forall i \in {1, 2, . . ., N_{bus}}

(14)

Limits of transformer currents

| I_{i, trn} | ⩽ I_{i, trn}^{max} \forall i \in {1, 2, . . ., N_{trn}}

(15)

Limits of feeder currents

| I_{f} | ⩽ I_{f}^{max} \forall f \in {1, 2, . . ., N_{f}}

(16)

Constraint for distribution network radiality

$N_{br} = N_{bus} - N_{sub}$ (17)

N_sub is the number of network substations.

Closing the normally open tie switches after isolating the faulty area change the network topology. This change should keep the network radiality.

Constraint for switch in branch of each loop

$\begin{matrix} \sum_{i = 1}^{N_{br, k}} {NSW}_{i} ⩾ min {2, 1 + {NSW}_{c, i}} k = 1, 2, . . ., N_{loop} \\ {NSW}_{i} = {\begin{matrix} 1 \begin{matrix} if branch i in loop k \\ has two switches \end{matrix} \\ 0 else \end{matrix} \\ {NSW}_{c, i} = {\begin{matrix} 1 \begin{matrix} if branch i in loop k \\ is common between two loops \end{matrix} \\ 0 else \end{matrix} \end{matrix}$ (18)

The achieved network resulted from optimal switch placement problem will be considered as the input network to feeder reconfiguration problem in other projects. So, in each loop of the network, at least on branch should have two switches at both sides to ensure that the network has an acceptable condition for other projects. In other hand this constraint make all tie switches useable during feeder reconfiguration analysis. N_br,k is the number of branches in loop k and N_loop is the number of loop in network.

3 Strategy for Volt/Var control during POR and PRR

Volt/Var control is one of the main subjects related to distribution network in normal and contingency condition. In normal operation condition, the transformer’s tap position and the output power of capacitors and DG units are used to control the voltage of network buses and the reactive power at substation. However, during the contingency condition the voltage of network buses and the substation active and reactive power should be managed in a different manner. In this regard, a new strategy is proposed in this paper to handle the above mentioned criteria. Based on this method, in PRR condition the DG units are switched off, while the active power flows from the distribution network to the substation. This process is renewed for calculating the exchange reactive power with the main grid to define the on/off situation of capacitors. The active and reactive powers of the substation are analyzed in POR to determine the on/off situation of DG units and capacitors. The number of switched off DG units and capacitors in PRR and POR is used to calculate the Equation (6). Furthermore, since the network topology is changed in POR, so the power flow and the network losses are undergone a positive or negative change. Here, the DG units are investigated to determine if switching any of them off can decrease the network losses or not. The number of DG units which are switched off in this process is also used to calculate Equation (6). The procedure of calculating the objective functions besides the strategy of implementing the Volt/var control concept are shown in Fig. 1.

Fig.1

Process of calculating the objective functions besides the proposed Volt/Var control strategy.

4 Enhanced Sine Cosine Algorithm (ESCA)

4.1 Original SCA

The SCA is a new population based meta-heuristic algorithm derived by Mirjalili in [20]. Similar to each optimizations algorithm, the exploitation and exploration is considered in SCA as well. The exploration is deployed to combine the random solutions in the set of solutions sharply with a high degree of randomness to discover the talented areas of the problem search space. The exploitation is implemented in a way that there are steady variations in the random solutions. In this context, the accuracy of the SCA can be pointed out in the following form: i) The original SCA provides multiple initial random agents and needs them to oscillate outwards or toward the global or near-global solution using sin and cosine functions; ii) The exploitation is deployed once these two functions return a value between 1 and – 1; iii) Exploration is conducted while these functions provide a value more than 1 or less than – 1; iv) In order to balance the exploration and exploitation, a specified random and adaptive variables are appended into the SCA; and v) The best approximation of the global or near-global optimum is stored in a variable as the terminus point and it never lost during the optimization procedure.

Phase 1: Generate random population:

Similar to any other population-based optimization methods, the SCA starts the search process with a random solution.

Phase 2: Evaluate and update population:

All the members of the randomly generated population are evaluated frequently by a definite fitness function. In order to describe the original SCA, consider a problem search space with NP individuals (X_m m = 1, …, NP) with discrepant value or fitness function F (X_m). To this end and to involve the decision maker’s (DM’s) preferences, the membership value of each candidate is calculated using (19). The best solution which is the nearest one to the DM’s requirements is the optimal solution to the max-min equation provided in (20). $μ_{F_{q} (X)} = {\begin{matrix} 1, F_{q} (X) ⩽ F_{q}^{min} \\ \frac{F_{q}^{max} - F_{q} (X)}{F_{q}^{max} - F_{q}^{min}}; F_{q}^{min} ⩽ F_{q} (X) ⩽ F_{q}^{max} \\ 0 F_{q} (X) ⩾ F_{q}^{max} \end{matrix}$ (19) $max_{X \in S} {min_{q = 1, . . ., 7} {μ_{F_{q} (X)}}}$ (20) with S as the set of candidates in the search space. Further, each solution m at iteration k is related to a fitness value which is determined as follows: $min_{q = 1, . . ., 7} {μ_{F_{q} (X)}}$ (21)

All the members of the randomly generated population are enhanced by a set of rules for exploitation and exploration stages. These are updated according to a switching parameter as follows: $X_{m}^{k + 1} = {\begin{matrix} X_{m}^{k} + {rand}_{1} sin ({rand}_{2}) | {rand}_{3} P_{m}^{k} - X_{m}^{k} | {rand}_{4} < 0.5 \\ X_{m}^{k} + {rand}_{1} cos ({rand}_{2}) | {rand}_{3} P_{m}^{k} - X_{m}^{k} | {rand}_{4} ⩾ 0.5 \end{matrix}$ (22)

With $X_{m}^{k}$ as the position of mth candidate in kth iteration, rand₁ to rand₄ as random numbers (rand_i ∈ [0, 1]) and as the destination point in population of kth iteration. It is to be noticed that rand₁ is proposed to lead the area of next position that might be between the candidate and the corresponding destination or outside it. To balance exploration and exploitation, this parameter is tuned dynamically as follows: ${rand}_{1} = a - k \frac{a}{k_{max}}$ (23) with a and k_max as a constant factor and maximum number of iteration, respectively. rand₂ is designed in order to choose how far the candidate movement must be outward or toward the destination. rand₃ is considered to be a random weighting parameter. rand₄ is a switching coefficient which is implemented to do the sine and cosine switching action in (22).

4.2 Proposed modifications on ESCA

As compared to the other meta-heuristic algorithms, SCA has the key recompenses that can be deployed in solving complicated optimization problems such as the one proposed in this paper. Some of these proses are simple theory, lower executional difficulty, easy application, greater reliability, minimal storage requirement and no need to tune many parameters. Despite these characteristics, the unacceptable interactions in the above mentioned phases may lead to trap agents in local optima. Consequently, a new self-adaptive modified technique is derived to overcome this shortage. It is to be noticed that the fundamental impression behind this algorithm is to instantaneously deploy multiple efficient strategies from the pool according to their previous operations in generating potential solutions and applied to make the mutation process. Thus, one mutation method is chosen from the strategy pool according to its score for each candidate solution extracted from the second phase of SCA. In this study, four mutation approaches are deployed in order to optimize the proposed problem. They can be explained hereinafter:

Mutation Strategy 1:

$\begin{matrix} X_{m, mut 1}^{k} = X_{best}^{k} + {rand}_{5} (X_{m, r_{1}}^{k} - X_{m, r_{2}}^{k}); \\ m = 1, . . ., N_{1}^{k} \end{matrix}$ (24) $\begin{matrix} Mutation Strategy 2 : X_{m, mut 2}^{k} = X_{m, r_{1}}^{k} + \\ {rand}_{6} (X_{m, r_{2}}^{k} - X_{m, r_{3}}^{k}) + {rand}_{7} (X_{best}^{k} - X_{m, r_{4}}^{k}); \\ m = 1, . . ., N_{2}^{k} \end{matrix}$ (25) Mutation Strategy 3: $\begin{matrix} X_{m, mut 3}^{k} = X_{m, r_{1}}^{k} + {rand}_{8} (X_{m, r_{2}}^{k} - X_{m, r_{3}}^{k}); \\ m = 1, . . ., N_{3}^{k} \end{matrix}$ (26) $\begin{matrix} Mutation Strategy 4 : \\ X_{m, mut 4}^{k} = X_{m, r_{1}}^{k} + {rand}_{1} (X_{best}^{k} - X_{worst}^{k}); \\ m = 1, . . ., N_{4}^{k} \end{matrix}$ (27)

With $N_{1}^{k}$ , $N_{2}^{k}$ , $N_{3}^{k}$ and $N_{4}^{k}$ as the number of individuals that select the method 1, 2, 3, and 4, respectively. Four agents r₁, r₂, r₃ and r₄ are randomly chosen from the current population in order to consistently cover the search space. Additionally, $X_{best}^{k}$ and $X_{worst}^{k}$ are the best and worst vectors among population, respectively.

The criteria of choosing these four mutation methods are that they have different features that cover varied circumstances. All the candidates from the population have a casual to be altered according to the mutations’ scores.

In the proposed ESCA, the mth candidate solution has a probability vector Prob_m = [Prob_m1, …, Prob_mj, …, Prob_m4], where Prob_mj signifies the probability that the jth mutation method might be adopted by the mth solution. There is another variable as a flag vector Flag_m = [Flag_m1, …, Flag_mj, … , Flag_m4], where Flag_mj equals to one if the jth method is chosen. Otherwise, it is 0. Initially, they are considered to be equal to Prob_mj = 0.25, and Flag_mj = 0. In each iterate of optimization algorithm, Tprob_mj, the probability accumulator of the mth candidate with the jth technique, is calculated as follows: ${Tprob}_{mj} = {\begin{matrix} {Tprob}_{mj} + {IFlag}_{m} W_{m} if {Flag}_{mj} = 1 \\ {Tprob}_{mj} Otherwise \end{matrix}$ (28) ${Tprob}_{mj} = \frac{{Tprob}_{mj}}{\sum_{j = 1}^{4} {Tprob}_{mj}}$ (29) with IFlag_m as the improvement flag and IFlag_m = 1 if the fitness function of the mutated candidate is better than that of the original one. Otherwise, IFlag_m = 0.1; IFlag_m indicates the weight that is created according to the sorted fitness function, that is, NP - Seq_m + 1/Σ (NP - Seq_m + 1) for the Seq_mth best candidate, where Seq_m is the sequence number of the mth candidate according to the sorting fitness functions of all solutions. Moreover, Tprob_mj is equal to Prob_mj at the beginning. Eventually, in order to update the probability of the jth mutation method of the mth agent following a fixed learning interval LI, the following equation can be implemented: ${Prob}_{mj} = 1 / LI \times \sum_{m = 1}^{LP} {Tprob}_{mj}$ (30) After which, Prob_mj is normalized by ${Prob}_{mj} = {Prob}_{mj} / \sum_{j = 1}^{4} {Prob}_{mj}$ (31)

5 Results and discussion

The developed procedure for technical and economic studies on distribution system is conducted to IEEE 33-bus distribution test feeder and different scenarios are designed to evaluate the network and objective functions characteristics. In this regard, each scenario is presented in distinct section and its results are compared to other scenarios.

5.1 Description of test system

The single line diagram of the test system is presented in Fig. 2. This system feeds the 3720 kW and 2300 Kvar load demand from the substation transformer. Each line of the network is candidate for two switches installation while the 7 DG units are optimally located on network buses. When a permanent failure is occurred on each network component the Circuit breaker at the substation is inevitably switched off to de-energized the fault location. This breaker will kept open while the fault is prevail. So, the closest switches to the fault location are opened to separate the fault location out from the other network sections. In the next step, the tie-lines or tie-switches are closed as the substitution path of power flow to energize the un-fault sections of network. The failure rate of network branches is changed linearly from 0.1 (fr/year) to 0.4 (fr/year) corresponding to it’s impedance. So, the branches with the minimum and the maximum values of impedance have the 0.1 and the 0.4 for their failure rates, respectively [21]. The switching time action and the repairing time are assumed to be 30-min and 360-min respectively.

Fig.2

33-bus IEEE distribution test feeder.

5.2 Discussion of different scenarios

In this section the numerical results of each scenario and the explanations about each of them are presented.

Scenarios S1 and S2 are the single objective cases.

Single objective switch placement considering cost (S1)

In this scenario the optimization problem is solved to find the best combination of switches location in distribution network from the total switches and reliability cost objectives. Since F₆ (X) and F₇ (X) try to minimize the cost objective functions which should be minimized by the system operator, so these two objectives can be aggregated as a single cost objective. Indeed, this objective try to compromise between the total investment, installation and maintenance cost of switches and the damage cost which should be paid as the penalty cost to customers who experience the outage duration corresponding to switching or repair time action. The cost value for this scenario is 391648$ for 25 switches. If all branches have two switches at both sides the cost objective is 697823$ for 64 switches.

Single objective DG placement considering Loss / Voltage deviation objective (S2)

In this scenario optimal DG placement problem is solved to find the best location of these units from the loss/voltage deviation minimizing point of view. The best value for loss is 11.06 kW when the DG units are located at buses 8, 11, 15, 24, 25, 29, and 32. Also, the best value for voltage deviation objective is 0.0704 p.u when the DG units are located at buses 5, 6, 9, 16, 27, 31 and 33. When the loss objective is minimized, the voltage deviation objective is increased to 0.142 p.u and the loss is changed to 21.76 kW when the voltage deviation objective is minimized.

Scenarios S3 and S4 are the multiobjective cases which the switch placement problem is jointed to other proposed problems.

Multi objective total switch cost, SDGCO and loss objectives (S3)

In this scenario the optimization problem is solve to coordinate the best location of switches and DG units to minimize the switch and reliability cost besides minimizing the number of DG/capacitors switching off during the contingency conditions while the network losses during the normal condition is as the third objective function. It should be noted that the switch and the SDGCO cannot construct a multi objective problem. Table 1 shows the results of this scenario.

Table 1
Results of Multiobjective total switch cost, SDGCO and loss objectives (S3)

Loss (kW) Vol. Dev. (p.u) Loss Tr.* Vol. Tr.* Cap/DG Out. Sw. Cost ($)

S3 42.37 0.62 1181 17.11 29 595931

Base case 11.06 0.0704 366 5.47 54 391648

	Loss (kW)	Vol. Dev. (p.u)	Loss Tr.*	Vol. Tr.*	Cap/DG Out.	Sw. Cost ($)
S3	42.37	0.62	1181	17.11	29	595931
Base case	11.06	0.0704	366	5.47	54	391648

*Tr.:the target value in transient condition. The values of network losses and voltage deviation are added by a penalty value.

Indeed if the loss and/or voltage deviation objective don’t consider as the objective function besides the SDGCO, the best location for switches is as the previous scenario and the location of DG units should coordinate to switches.

About the ‘Base case’ some points should be regarded. The 2nd and 3rd columns relate to best values of loss and voltage deviation objectives in optimal DG placement problem. If the switches and DGs location be selected form S1 and S2, respectively, the values of 4th, 5th and 6th columns are achieved.

As can be seen, decreasing the value F₅ (X) deteriorates the situation of loss, voltage deviation and switch cost objectives. On the other hand, if the optimal switch and DG placement problems be solved separately, the more DG units and capacitors are switched off during the contingency condition compared to solving the problem as S3. Figure 3 shows the voltage profile of network buses corresponding to best solution of voltage deviation objective (S2) compared with the worst condition during the contingency condition matching with S3 (when line between buses 3 and 4 is out of service). Also, the network losses during contingency condition corresponding to each fault occurrence on network branches comparing the minimum network losses are depicted in Fig. 4. The ‘Base Case Loss’ in this figure is achieved when the switches and the DGs location are selected from S1 and S2, respectively. ‘S2 Loss’ is the minimum network losses achieved by S2.

Fig.3

Best voltage profile of S2 compared to worst voltage profile of S3.

Fig.4

Network losses of two scenarios and the base loss of network.

Increasing the network losses during the contingency condition and the number of switching DG units and capacitors off are two new major concepts which can join the optimal switches and DG placement problems. So, some other practical scenarios are defined to help the distribution system planners.

Multiobjective switch placement and Volt/Var control (S4)

As can be seen in scenario S3, the loss objective during the contingency condition and the voltage deviation objective in normal condition are increased. So, the network condition from VVC problem point of view is deteriorated. In this scenario, the optimal switches and DG units placement problem is done while the loss and the voltage deviation objectives in normal and contingency conditions and the total switches cost are considered as objective functions. In other word, the SDGCO objective is neglected in this scenario. This scenario can be interpreted as a normal/contingency Volt/Var control problem. Table 2 shows the results of S4.

Table 2

Results of multiobjective switch placement and Volt/Var control (S4)

	Loss (kW)	Vol. Dev. (p.u)	Loss Tr.	Vol. Tr.	Cap/DG Out.	Sw. Cost ($)
S4	14.17	0.18	383	4.94	73	593513
Base case	11.06	0.142	366	5.47	54	391648

As can be seen from Table 2 the loss objective has the acceptable condition compared to best value of network losses (Base case). If the system operator wants to decrease the network losses to 11.06 kW, the voltage deviation objective is 0.142 p.u; in this condition the loss and the voltage deviation values in contingency condition are 366 and 5.47, respectively. As can concluded, this scenario just can decrease the contingency voltage deviation value compared to condition which the DG placement problem with loss objective and switch placement problem with total cost objective are solved independently. The reason lies within the fact that the network’s tie-lines are not placed considering the voltage profile of the network during contingencies, so, decreasing this value can deteriorate the value of other objectives.

Scenarios S5– S7 are formulated as the multiobjective cases while the location of switches is based on S1.

Multiobjective loss and SDGCO objectives (S5)

In this scenario the network losses in normal and contingency conditions beside the SDGCO are as the objective functions while the switches location is as the S1. Table 3 shows the results of S5.The results of S5 confirm the severe conflict between the network losses in normal and contingency conditions with the SDGCO. Figure 5 shows the network losses during each line contingency compared to the best network losses. The best and the worst voltage profiles during the contingency conditions compared to best voltage profile of S2 are depicted in Fig. 6a and b.

Fig.5

Network losses during each line contingency (S5) compared to the best network losses.

Fig.6

a) best voltage profile of S5 compared to best voltage profile of S2 b) worst voltage profile of S5 compared to best voltage profile of S2.

Table 3

Results of multiobjective loss and SDGCO objectives (S5)

	Loss (kW)	Vol. Dev. (p.u)	Loss Tr.	Vol. Tr.	Cap/DG Out.	Sw. Cost ($)
S5	19.63	0.325	616	8.88	39	391648
Base case	11.06	0.0704	366	5.47	29	391648

Multi objective voltage deviation and SDGCO objectives (S6)

In this scenario the voltage deviation during normal and contingency conditions besides the SDGCO are considered as the objective functions. Table 4 shows the results of this scenario.

Table 4

Results of multi objective voltage deviation and SDGCO objectives (S6)

	Loss (kW)	Vol. Dev. (p.u)	Loss Tr.	Vol. Tr.	Cap/DG Out.	Sw. Cost ($)
S6	21.22	0.238	700	7.99	52	391648
Base case	11.06	0.0704	366	5.47	29	391648

Comparing the results for SDGCO objective in Tables 3 and 4, confirm the more conflicting manner between SDGCO and voltage deviation objective compared to loss objective. The best and the worst voltage profile during the contingency condition besides the voltage profile in normal condition corresponding to this scenario are as Fig. 7. In this figure the voltage of out of service buses during the contingency condition are set to zero. As can be seen the voltage of some buses during the contingency condition differ by 0.02 p.u from their normal values.

Fig.7

Voltage profiles of S6 beside the voltage profile during normal condition.

Multiobjective loss, voltage deviation and SDGCO objectives (S7)

This scenario combines the two previous scenarios in a multiobjective problem with three objective functions. Since the SDGCO has sever effect on achieved results, so, firstly the optimization process tries to keep this objective in an acceptable condition and secondly two other objectives are optimized. Indeed, a small change in loss and voltage deviation objectives can cause a big change in SDGCO. All objectives including the loss, the voltage deviation and the SDGCO have the conflicting manner. The results of this scenario which is depicted in Table 5 confirm this statement.

Table 5

Results of multiobjective loss, voltage deviation and SDGCO objectives (S7)

	Loss (kW)	Vol. Dev. (p.u)	Loss Tr.	Vol. Tr.	Cap/DG Out.	Sw. Cost ($)
S7	23.07	0.444	666	11.56	36	391648
Base case	11.06	0.0704	366	5.47	29	391648

The best and the worst conditions for voltage profile during the contingency condition besides the best voltage profile in normal condition corresponding to S7 are depicted in Fig. 8.

Fig.8

Voltage profile of S7.

As can be seen from Fig. 8 there is a considerable gap between the best voltage profile and the voltage profile during the contingency condition. It should be noted that customers experience this voltage profile during the system repair time.

Scenario S8 aggregates all proposed problem in a multiobjective formulation.

Multi objective switch placement, SDGCO and Volt/Var control (S8)

This scenario tries to compromise all objectives in a multi objective framework. As can be concluded from previous scenarios, the importance factor of SDGCO objective should be decreased to control the multi objective problem. Table 6 shows the results of this scenario while the importance factor of SDGCO is less than other objectives. The experiences in previous scenarios help the system operator to manage the multi objective formulation for achieving the better solutions. Since, the SDGCO objective has a sever conflict with other objectives, the importance factor of this objective selects smaller than other objectives. Also, the importance factors of the loss and the voltage deviation objectives during the contingency conditions are slightly smaller than the corresponding values in normal conditions. In other word, the sorted objective functions based on their importance factor are: 1) total cost of switches, loss and voltage deviation objectives during the normal condition, 2) loss and voltage deviation objectives during the contingency conditions, and 3) SDGCO objective.

Table 6

Results of multi objective switch placement, SDGCO and Volt/Var control (S8)

	Loss (kW)	Vol. Dev. (p.u)	Loss Tr.	Vol. Tr.	Cap/DG Out.	Sw. Cost ($)
S8	14.66	0.085	411	3.78	55	524522
Base case	11.06	0.0704	366	5.47	29	391648

As can be seen from Table 6 the loss and the voltage deviation objectives in normal condition have acceptable levels compared to their best values (depicted as ‘Base case’). Decreasing the voltage deviation objective during the contingency condition increases the value of loss objective in this condition. The results of this scenario confirm that achieving acceptable results for implementing in network needs a suitable view about the objective function’s manners which are discussed in scenarios S1 to S8.

6 Conclusion

In this paper the optimal switch and DG placement problems besides the Volt/Var control are formulated as a multiobjective optimization framework. The objective functions includes the network losses and the voltage deviation of network buses during the normal and contingency conditions, the number of switching DG units and capacitors off during the contingency condition, and the total cost related to switch devices and the reliability index.

These objectives are minimized in a multiobjective formulation using a modified version of Sine Cosine Algorithm while some practical constraints are satisfied. The implementation of centralized Volt/Var control is done using a suitable strategy during pre-restored (PRR) and post restored (POR) conditions. The results section is organized through different scenarios to clarify different aspects of proposed optimization problem. The achieved results in each scenario are compared to the base case while the performance of each objective function is discussed completely. Also, results confirm that the considered problems can affect each other so considering them in a unique optimization problem has more adoption with practical conditions.

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Multi-Objective Coordination of Local and Centralized Volt/Var Control with Optimal Switch and Distributed Generations Placement

Abstract

Keywords

1. Introduction

2 Problem formulation

2.1 Decision variables

4.1 Original SCA

5.1 Description of test system

Table 1 Results of Multiobjective total switch cost, SDGCO and loss objectives (S3) Loss (kW) Vol. Dev. (p.u) Loss Tr.* Vol. Tr.* Cap/DG Out. Sw. Cost ($) S3 42.37 0.62 1181 17.11 29 595931 Base case 11.06 0.0704 366 5.47 54 391648

References

Table 1
Results of Multiobjective total switch cost, SDGCO and loss objectives (S3)

Loss (kW) Vol. Dev. (p.u) Loss Tr.* Vol. Tr.* Cap/DG Out. Sw. Cost ($)

S3 42.37 0.62 1181 17.11 29 595931

Base case 11.06 0.0704 366 5.47 54 391648