Network Reconfiguration and Reactive Power Compensation Dispatch in Smart Distribution Systems

Abstract

A significant challenge is to design strategies to minimize electrical losses in smart distribution systems while observing voltage and feeder loading constraints. Unfortunately, few studies have solved the problem of simultaneously coordinating already installed capacitors banks with network reconfiguration problems. This book chapter presents two methodologies for solving the reconfiguration and reactive power compensation dispatch. Both methodologies are formulated as two-stage solve reconfiguration and reactive power compensation problems with the characteristic of having acceptable computational efficiency and loss reduction close to the optimal solution. In the first stage, network reconfiguration is carried out to discriminate radial configurations that do not satisfy voltage and overload restrictions. In the second stage, a reactive power compensation dispatch is applied to each feasible network configuration by connecting capacitor blocks successively until all available reactive capacity has been used or until a specified loss tolerance has been reached. Finally, switching each capacitor block is carried out using voltage linear sensitivities related to shunt reactive compensation to make a relatively low computational work during the process.

1. Introduction

During the last decades, the great growth in technologies of computers, microprocessors, telecommunications, incorporation of distributed generation (DG) and electronic equipment as ac/dc converters and flexible ac transmission systems (FACTS) at the distribution level has led to opportunities for an advanced supervising and control of electric energy delivery systems (distribution systems), particularly in distribution network automatization, but at the same time, it presents new challenges for accomplishing with the main objective of improving the automatization, control, and efficiency of real-life distribution systems, converging into the infrastructure named smart distribution center. Therefore, almost all the efforts have comprehensively investigated the optimal distribution reconfiguration problem [1].

Distribution systems mainly consist of transformers, several feeders composed of line sections, switches, diverse DG systems, and consumer loads.

In existing distribution systems, global loss minimization is a common objective function because of its high impact on their economic efficiency, and it may be basically carried out in two stages, which are as follows—(i) through feeder reconfiguration by opening or closing switching devices and (ii) reactive power compensation by capacitor bank commutations.

1.1 Distribution system reconfiguration

Distribution system reconfiguration is useful for either planning or real-time control to change the electrical conditions of primary supply feeders to reach, in some sense, an optimal operation point [2]. Reconfiguration is a process for modifying the topological structure of distribution feeders by changing the open/close status of sectional switches and feeder sections to find the minimum loss topology, maintaining voltages between their low and high limits simultaneously, and its radial structure, that is, there is only one path between two points in the same feeder [3]. Sectional switches and feeder sections will be named as switches throughout this chapter.

Many feeders are interconnected to the distribution system keeping a radial structure, which is maintained by properly controlling the status of the associated switches, which are classified as normally open or closed. These switches operate under a feeder fault condition to change their statuses to isolate the faulted feeder section and enable load transferences between adjacent feeders to redistribute current flows without any restriction violation [4]. A smart distribution center supports this process mainly with supervisory and control infrastructure, which helps raise the reliability and efficiency of electricity supplied to the final consumer [5].

Almost all feeders have a mix of industrial, commercial, and residential loads, which, through 24 h, show diverse load variations so that their peak load values occur at different moments. In this sense, reconfiguration permits to transfer of loads to a relatively lesser loaded feeder with the benefit of better voltage regulation and lower electrical losses; also, at the same, it increments security margins and quality of energy supplied [2]. Furthermore, under emergency conditions due to a short circuit in some distribution system points, an important goal is to minimize the close/open switching operations to reduce the load-restoration period. In both situations, the distribution system topology should remain radial [6].

From the above discussion, it is obvious that loss minimization implicitly includes operative cost reductions in the distribution system [7]. Hence, almost of reconfiguration methods have electric loss reduction as the main goal. Furthermore, distribution system reconfiguration is a combinatorial problem involving many open/close switch operations for real-life distribution systems [8]. In fact, a small 33-bus distribution system, and considering that each feeder section has a switch in its extremes, the open/close commutations are 435,897. Then, the development of reconfiguration methods must contain the next features [2]:

• Capacity for estimating loss changes resulting from the reconfiguration process, involving minimal computational work.

• There must be a useful criterion to avoid irrelevant switching actions to reduce the searching space and increment the problem solution efficiency.

Researchers have studied the distribution system reconfiguration problem during the last five decades, developing and using different solution methods. Some of the most reported are the next kind of approaches [9]:

• Heuristic methods

• Metaheuristic algorithms

• Conventional mathematical optimization algorithms

• Hybrid methods involving any of the three above

Heuristic methods (HM) are the most attractive because of their relative simplicity and suitability for operating in real-time environments; however, they do not always obtain the optimal global reconfiguration. This drawback is overridden by metaheuristic methods (MM), but they are more complicated in formulation with larger execution time requirements than HM. Therefore, many MM have been developed using ideas of nature behavior [9], which could be based on genetic algorithms [2], particle swarm optimization [3, 10, 11], tabu search [12, 13], simulated annealing [13, 14, 15], variable scaling hybrid differential algorithm [16], ant colony [17, 18], plant growth simulation [19, 20], bacterial foraging [21], gray wolf [22], salp swarm [23], symbiotic organism search, hybrid cuckoo search [24], harmony search [25], and binary gravitational search [26], among others. On the other hand, mathematical optimization algorithms solve the reconfiguration problem by using conventional optimization techniques, for example, OPF by Bender Decomposition [8], mixed-integer convex programming [27, 28], convex models [29], mixed-integer linear programming [30], and mixed-integer second-order cone programming [31].

Nowadays, with the proliferation of photovoltaic systems, many distribution systems could integrate distributed generation (DG), storage systems, and power electronics (STATCOM), so they have to be included in the reconfiguration formulation problem as in Refs. [11, 20, 24, 32, 33, 34, 35].

Furthermore, in some papers, multi-objective formulation problems are considered. In this sense, formulations include loss minimization and some other function, such as voltage profile enhancement [20, 36, 37, 38, 39, 40], load balancing [19, 38, 41, 42], branch current overloads [38], operation cost reduction [43], reliability [32, 44, 45], and outage costs [46].

1.2 Reactive power compensation dispatch

As pointed out before, reactive power compensation (RPC) dispatch is the second way of reducing distribution system electrical losses, so a common objective function is electrical loss minimization either at the planning or operation stage. Furthermore, this objective function is nonlinear and convex, permitting its reduction by sequential commutation of capacitor banks to find one point where its value is minimal until the next capacitor bank commutation causes an electrical loss increment again [47].

It is important to note that the joint application of reconfiguration and RPC strategies allows for obtaining lower losses than either separated so that methods that have been developed involve both strategies [17, 48].

From the planning perspective, loss minimization may be reached by solving an optimal reconfiguration and allocation capacitor problem [33, 34, 49, 50, 51, 52, 53], which is combinatorial and nonlinear, so its solution has been proposed by using heuristic methods, metaheuristic methods, and mathematical optimization methods [54].

On the other hand, in an operation environment, loss minimization may be achieved by solving a loss minimization problem by joint reconfiguration and RPC dispatch, which is carried out with capacitor banks already installed with the capacity to be managed from the distribution control center [55]. However, due to the emerging concept of distribution control centers, solutions methods involving reconfiguration and RPC dispatch in real-time are few [54] and are based on ordinal optimization theory [56], parallel metaheuristic [57], multiagent system [55], analytical partitioning method [58], modified binary gray wolf [35], and robust optimization model [59].

This chapter presents two methodologies for solving the reconfiguration and RPC dispatch. Both methodologies are formulated as two-stage reconfiguration and RPC dispatch problems. In the first stage, network reconfiguration is carried out to find a set of feasible radial configurations with the lowest losses, satisfying voltage, and overload restrictions. In the second stage, an RPC dispatch is applied to each feasible network configuration by connecting capacitor blocks successively until all available reactive capacity has been used or a specified loss tolerance has been reached. Finally, analysis of switching each capacitor block is carried out using voltage/shunt reactive compensation linear sensitivities to make a relatively low computational work during the process.

2. Network reconfiguration and reactive power compensation formulation problems

This section describes the methodology proposals for reconfiguration and RPC dispatch. Firstly, considerations for modeling and problem formulation are defined, bearing in mind the electrical loss minimization as the objective function, subject to various restrictions observed during the overall solution process.

To develop the problem formulation, it is necessary to make the next considerations, which are as follows—(i) a three-phase distribution system is operating under phase balance so that the network model is defined by the positive sequence circuit of its components; (ii) the distribution system may be supplied by one or more substations; (iii) RPC dispatch is realized by already installed capacitor banks and they can be commutated remotely from the distribution control center; (iv) capacitor banks only have either a commercial capacity of 300, 600, or 900 kVAr.

2.1 Problem formulation

The problem of electrical loss minimization in distribution systems considering reconfiguration and RPC dispatch can be formulated as follows:

$$\mathit{Min}{P}_{\mathit{\text{loss}}}=\sum _{k\in F_B}\frac{r_k}{r_k^2+x_k^2}(V_i^2+V_m^2-2V_i{V}_{m}cos\left({\theta }_i-{\theta }_m\right)$$E1

$$S.\mathit{\text{to}}{P}_{\mathit{Gi}}-P_{\mathit{Di}}-V_i\sum _{m\in i}{V}_m{Y}_{\mathit{im}}cos\left({\theta }_i-{\theta }_m-{\gamma }_{\mathit{im}}\right)=0,\forall i\in N$$E2

$$Q_{\mathit{Gi}}-Q_{\mathit{Di}}-V_i\sum _{m\in i}{V}_m{Y}_{\mathit{im}}sin\left({\theta }_i-{\theta }_m-{\gamma }_{\mathit{im}}\right)=0,\forall i\in N$$E3

$$V_i^{\mathit{Min}}\le V_i\le V_i^{\mathit{Max}},\forall i\in N$$E4

$$S_k\le S_k^{\mathit{Max}},\forall k\in F_B$$E5

$$S_{\mathit{\text{out}},t}\le S_{\mathit{\text{out}},t}^{\mathit{Max}},\forall t\in T_S$$E6

$$0\le b_c\le b_c^{\mathit{Max}},\forall c\in N_C$$E7

$$\left|F_B\right|=\left|N\right|-1$$E8

$$\left\{\left(i_1,m_1\right)\cup \left(i_2,m_2\right)\cdots \cup \left(i_{\mid F_B\mid },m_{\mid F_B\mid }\right)\right\}=N$$E9

where $b_c$ represents a capacitor bank susceptance, $F_B$ is the feeder section set, $r_k$ and $x_k$ are the series resistance and reactance of the k^th feeder section, respectively. $S_k$ is the apparent power flowing on the k^th feeder section and $S_k^{\mathit{Max}}$ is the rating of the k^th feeder section,$T_S$ is the set of supply transformers in the distribution system. $V_i$ and ${\theta }_i$ are the voltage magnitude and angle of complex voltage ${\overline{V}}_i$ at bus i, and $V_i^{\mathit{Max}}$, $V_i^{\mathit{Min}}$ are the maximum and minimum voltage magnitudes at bus i, respectively. $Y_{\mathit{im}}$ and ${\gamma }_{\mathit{im}}$ are the magnitude and angle of the complex nodal admittance associating busses i and m.

The objective function (1) accounts for the distribution system losses. The decision variables are the voltage magnitudes and angles of complex nodal voltages because their values define the losses at each feeder section. These decision variables are to be modified using reconfiguration and RPC dispatch.

Constraints (2) and (3) are the active (P) and reactive power (Q) balances at each bus $i\in N$ . Constraint (4) considers that all nodal voltage magnitudes should remain within limits. Constraint (5) takes care of apparent power flow that does not reach a value above its maximum limit through each feeder section. Because load transfers are possible when the distribution system is being reconfigured, constraint (6) is necessary for imposing a maximum limit, $S_{\mathit{\text{out}},t}^{\mathit{Max}}$, to the apparent power flow, $S_{\mathit{\text{out}},t}$, from the supply substation transformer t to distribution feeders. Constraint (7) refers to every capacitor bank c of the set $N_C$, denoted in terms of its susceptance; it may have a zero value when disconnected and connected by one or more steps (blocks) until it reaches its maximum value $b_c^{\max }$. Finally, constraints (8) and (9) refer to guarantee radiality and maximum spanning tree of the network, which are based on the concept of set cardinality, Constraint (8) indicates that cardinality of $F_B$, $\left|F_B\right|$, should be equal to the cardinality of N, $\left|N\right|$, minus 1, while Constraint (9) guarantees that the radial network is a spanning tree, due that the left-hand set should be equal to N.

This general formulation is cast as a nonlinear and combinatorial problem. It can be solved by heuristic, metaheuristic, mathematic optimization, and combinations of those mentioned above. In general, a power flow should be realized at each solution step. Then, it is important that solution methods can find the optimal configuration rapidly to reduce the number of power flow simulations and computing time. Also, computational efficiency can be improved if the power flow algorithm is efficient, like those developed for solving radial distribution power flows [54]. However, these algorithms are not useful for the proposal presented in this chapter due to sensitivity calculations.

2.2 Power flow and sensitivity calculations

Because sensitivity calculations are straightforward from the linearized power flow model solved in each iteration of the Newton–Raphson (NR) method, this is the algorithm used for the methodologies developed for solving the reconfiguration and RPC dispatch to minimize electrical distribution losses.

Equation sets (2) and (3) represent the power flow problem, which is solved iteratively applying the NR method by formulating and solving the next linear equation set expressed in the compact form [60]:

$${\left[\begin{array}{c}\begin{array}{l}\mathrm{\Delta }P\\ \end{array}\\ \mathrm{\Delta }Q\end{array}\right]}^{\left(l\right)}={\left[\begin{array}{cc}\frac{\partial P}{\partial \theta }& \frac{\partial P}{\partial V}V\\ \frac{\partial Q}{\partial \theta }& \frac{\partial Q}{\partial V}V\end{array}\right]}^{\left(l\right)}{\left[\begin{array}{c}\begin{array}{l}\mathrm{\Delta }\theta \\ \end{array}\\ \frac{\mathrm{\Delta }V}{V}\end{array}\right]}^{\left(l\right)}$$E10

Once solved (10), nodal voltage angles and magnitudes are updated as follows:

$${\theta }_i^{\left(l+1\right)}={\theta }_i^{\left(l\right)}+\mathrm{\Delta }{\theta }_i^{\left(l\right)},\forall i\in N$$E11

$$V_i^{\left(l+1\right)}=V_i^{\left(l\right)}+{\left[\mathrm{\Delta }{V}_i/V_i\right]}^{\left(l\right)}{V}_i^{\left(l\right)}\forall i\in N$$E12

where l is the iteration number, this process continues until a convergence tolerance is accomplished. In Eq. (10), the partial-derivative matrix is the Jacobian matrix and should be calculated at iteration l. Let ${\left[{\theta }^{\left(\mathit{\text{base}}\right)}{V}^{\left(\mathit{\text{base}}\right)}\right]}^T$, the power flow problem solution once that iterative process has been finished, where T indicates transposed. This vector solution represents the electrical system base case state, that is, the operating point defined by load, supply, electric network power flows, and losses. For sensitivity calculations, it is known as the base case. Suppose that this equilibrium point is perturbed with the commutation of a capacitor bank c installed at bus j, denoted by $\mathrm{\Delta }{b}_{\mathit{cj}}$. Therefore, changes in the state vector can be calculated by constructing the next linear equation set [61]:

$$\left[\begin{array}{cc}\frac{\partial P}{\partial \theta }& \frac{\partial P}{\partial V}\\ \frac{\partial Q}{\partial \theta }& \frac{\partial Q}{\partial V}\end{array}\right]\left[\begin{array}{c}\begin{array}{l}\frac{\mathrm{\Delta }\theta }{\mathrm{\Delta }{b}_c}\\ \end{array}\\ \frac{\mathrm{\Delta }V}{\mathrm{\Delta }{b}_c}\end{array}\right]=-\left[\begin{array}{c}0\\ ⋮\\ 0\\ -V_j^2\\ 0\\ ⋮\\ 0\end{array}\right]$$E13

The only nonzero entry in the right-hand vector indicates that capacitor bank c is connected at bus j. Vector ${\left[\mathrm{\Delta }\theta /\mathrm{\Delta }{b}_{\mathit{cj}}\mathrm{\Delta }V/\mathrm{\Delta }{b}_{\mathit{cj}}\right]}^T$ is known as the relative sensitivity vector between the state base case vector ${\left[{\theta }^{\left(\mathit{\text{base}}\right)}{V}^{\left(\mathit{\text{base}}\right)}\right]}^T$ and the perturbation scalar value $\mathrm{\Delta }{b}_{\mathit{cj}}$. The Jacobian matrix in Eq. (13) may be taken from Eq. (10) formed, ordered, and factorized in the last NR iteration, but noting that submatrices $\left[\left(\partial P/\partial V\right)V\right]$ and $\left[\left(\partial Q/\partial V\right)V\right]$ in Eq. (10) are divided by their corresponding V. After the sensitivity calculations, a new state vector can be obtained, by using the next equations, once that $\mathrm{\Delta }{b}_{\mathit{cj}}$ has been defined in terms of kVAr:

$${\theta }_i^{\left(\mathit{new}\right)}={\theta }_i^{\left(\mathit{\text{base}}\right)}+\left(\mathrm{\Delta }{\theta }_i/\mathrm{\Delta }{b}_{\mathit{cj}}\right)\mathrm{\Delta }{b}_{\mathit{cj}},\forall i\in N$$E14

$$V_i^{\left(\mathit{new}\right)}=V_i^{\left(\mathit{\text{base}}\right)}+\left(\mathrm{\Delta }{V}_i/\mathrm{\Delta }{b}_{\mathit{cj}}\right)\mathrm{\Delta }{b}_{\mathit{cj}},\forall i\in N$$E15

With these new values, power flows and distribution losses are recalculated to know if the capacitor block $\mathrm{\Delta }{b}_{\mathit{cj}}$ connection causes a decrease or increase in the objective function, that is, total distribution system losses. If the distribution system has installed a set of capacitor banks, denoted as {$b_{c1},\dots ,b_{{\mathit{cN}}_c}$}, Eq. (13) should be solved each time one capacitor bank or block is connected, which seems to be a high computational work; however, the Jacobian matrix in this equation remains constant and factored, so that, each solution of Eq. (13) requires only a forward and a backward substitution process, which represent much lower computational effort than the one related with a complete power flow calculation.

A flow chart of the power flow algorithm based on the NR method is shown in Figure 1, while sensitivity calculation is shown in Figure 2.

Figure 1.

Flow chart for solving power flow problems by Newton Raphson method.

Figure 2.

Flow chart for sensitivity evaluations of capacitor bank connections.

Notes about the Figure 1:

1. $\mathrm{\Delta }{P}_{\max }$ and $\mathrm{\Delta }{Q}_{\max }$ are defined by selecting the greatest values of $\mathrm{\Delta }{P}_i^{\left(l\right)}$ and $\mathrm{\Delta }{Q}_i^{\left(l\right)}$, $\forall i\in N$.

2. Expression $i\ne s$ signifies a node designated as slack, which contributes with all active and reactive powers plus electrical distribution losses for always keeping the balance power of restrictions (2) and (3). This node represents the supply point for all the system feeders in distribution systems. Furthermore, the slack node maintains constant values of $V_s$ and ${\theta }_s$ during the power flow solution process. $V_{\mathit{\text{spec}}}$ is a value around 1.0 per unit value.

3. lmax is the maximum number of iterations for the NR method, and currently, NR converges to the solution in a few iterations because of its quadratic convergence characteristic.

4. Convergence tolerances ${\mathit{tol}}_P$ and ${\mathit{tol}}_Q$ are defined as 0.001 or 0.0001.

5. Post-iterative calculations refer to the determination of power flows through each feeder section and electrical losses in the distribution system as follows:

   $$P_{\mathit{im}}=V_i\left[V_i{g}_{\mathit{im}}-V_m{y}_{\mathit{im}}cos\left({\theta }_i-{\theta }_m-{\phi }_{\mathit{im}}\right)\right]$$E16
   $$P_{\mathit{mi}}=V_m\left[V_m{g}_{\mathit{im}}-V_i{y}_{\mathit{im}}cos\left({\theta }_m-{\theta }_i-{\phi }_{\mathit{im}}\right)\right]$$E17

   where complex series admittance of feeder section connecting nodes i and m is calculated in rectangular and polar coordinates by Eqs. (18) and (19), respectively.

   $${\overline{y}}_{\mathit{im}}=g_{\mathit{im}}+{\mathit{jb}}_{\mathit{im}}=\frac{r_{\mathit{im}}}{r_{\mathit{im}}^2+x_{\mathit{im}}^2}+j\frac{x_{\mathit{im}}}{r_{\mathit{im}}^2+x_{\mathit{im}}^2}$$E18
   $${\overline{y}}_{\mathit{im}}=y_{\mathit{im}}\angle {\phi }_{\mathit{im}};y_{\mathit{im}}\sqrt{g_{\mathit{im}}^2+b_{\mathit{im}}^2};{\phi }_{\mathit{im}}={tan}^{-1}\left(b_{\mathit{im}}/g_{\mathit{im}}\right)$$E19

   Active power losses through the feeder section connecting nodes i and m are calculated by Eqs. (16) and (17), which, after some algebraic operations, are expressed as:

   $$P_{\mathit{\text{loss}},i-m}=g_{\mathit{im}}\left[V_i^2+V_m^2-2V_i{V}_{m}cos\left({\theta }_i-{\theta }_m\right)\right]$$E20

   Eq. (20) is the same as Eq. (1) but $g_{\mathit{im}}$ written in terms of $r_{\mathit{im}}$ and $x_{\mathit{im}}$.

6. In addition, among other post-iterative calculations, sensitivity evaluations when connected to capacitor banks may be included in the next explanation.

2.2.1 Sensitivity Evaluation to Find the Maximum Loss Reduction

By using the power flow solution obtained by the NR algorithm, sensitivities for capacitor bank connections may be calculated straightforwardly. RPC dispatch assumes that the already installed capacitor banks have initially disconnected one, two, three, or more blocks. A power flow is performed to determine the initial distribution losses. After, capacitor blocks will be connected successively. The process calculates sensitivities for each block connection evaluating the new state distribution system, power flows, and distribution losses. Once the first block pertaining to every one of the capacitors installed were connected, their corresponding distribution losses were ranked from the lowest to the highest. The capacitor block associated with the first position is added to a connected set of capacitor blocks. Then, a new power flow is carried out to refresh the Jacobian matrix to continue the sensitivities calculation to evaluate the connection of the next capacitor block remaining as disconnected in each node where capacitors exist in the distribution system. This procedure is carried out until no more capacitor blocks are disconnected or optimal distribution losses are found. The flow chart of Figure 2 resumes this procedure, where $n_c$ = $\left|N_C\right|$, that is, the number of capacitor banks.

To observe the behavior of distribution losses with capacitor bank connections, Figure 3 shows a distribution feeder whose section parameters are all equal on a per unit basis: r = 0.1 and x = 0.07 over a base of 10 MVA and 13.8 kV. Node 0 is the supply point with no load, while all the other nodes have a uniform load of 120 kW and 60 kVAr.

Figure 3.

Distribution system with equal section parameters and uniform load.

The analysis was realized by simulating the connection of a capacitor bank in nodes 1 through 12 to find the optimum capacitor bank and its location. For the sake of clarity, Figure 4 shows the results only for nodes 7, 8, and 9, which showed lower losses. Note that distribution losses were plotted from 300 kVAr to 750 kVAr, with a linear distribution loss reduction up to 480 kVAr capacitor bank connection in the three nodes, where node 9 presents the lowest distribution losses. After this point, distribution losses become more nonlinear until reaching a minimum value of around 116.5 kW with 620 kVAr in node 9, 116.0 kW with 660 kVAr in node 8, and 116.4 kW with 680 kVAr in node 7. From this analysis, important concluding remarks are as follows—(i) the node with the initial lowest distribution losses could not be the same when the minimum is found after capacitor connections are performed; (ii) minimum loss values reached with capacitor connections are very close to each other, and the same occurs with their kVAr capacities, which are very similar; (iii) the block capacitor connection effect is steadily decreasing over distribution losses until it arrives at a practically zero value, and after this point, the effect tends to be negative; (iv) lately, it can be considered that the optimal distribution losses can be found with an error below of 1 kW, so that, this tolerance is used as stopping criterium for both methodologies explained next.

Figure 4.

Distribution electrical losses versus capacitor bank connection.

3. Methodologies

Formulations proposed in Refs. [5, 47] solve the network reconfiguration problem in distribution systems following a similar strategy. First, the initial radial distribution system is converted to a meshed network by closing all the system switches; then one switch is opened according to the lowest apparent power flow criterion. Once this action is finished, the next switch with the lowest apparent flow is opened, and so on. This process continues until the resulting configuration is completely radial, and its graph is a spanning tree. This algorithm can find the optimal or suboptimal reconfiguration, that is., with the lowest or almost lowest distribution of electrical losses. An alternative strategy consists in looking for the two or more reconfigurations whose switches cause the lower distribution losses instead of searching for only one reconfiguration at each step. This will lead to a feasible reconfiguration set. Furthermore, as explained before, the resultant distribution losses may be further reduced with the application of RPC dispatch to those reconfigurations obtained before, which have minor losses and are very close to each other, as in the example described above. Thus, once the reconfiguration process has finished (Stage 1), the result is a feasible reconfiguration set containing a reduced reconfiguration alternative number, which passes through the RPC dispatch procedure for obtaining the optimal loss reconfiguration (Stage 2).

3.1 Feasible reconfiguration search algorithm

The basic algorithm developed in Ref. [5] looks for only one reconfiguration, which, almost in all cases, yields the one with the lowest distribution losses. In this chapter, this algorithm is modified for searching various reconfigurations based on the selection of the two switches having the least apparent power flow through them.

With a view to clarifying explanation, consider that every distribution system, for simplicity, is operated as a radial network and supplies all the electricity consumers.

In general, distribution systems can be represented by graphs. For example, the distribution system has a radial configuration if there is only one path between the supply and load points. Furthermore, for making possible the optimal operation with the least electrical losses, distribution systems have normally open switches, which can be closed when some operational condition change causes electrical losses to increase significantly, and the same number of normally closed switches should be opened for maintaining distribution system radiality providing that there is not any node isolated from the rest.

Let us assume that switches can be opened or closed from the distribution control center, so the distribution electrical network topology may be updated depending on the prevailing operational conditions in the distribution system.

In terms of graph theory, a graph is formed by branches and links, which connect vertices (nodes), so that if the latter are separated from the graph, the result is a graph with only one trajectory between any two nodes in the graph and the same occurs with a radial distribution system.

An assumption can be made—normally closed switches can be considered branches, while normally open switches are regarded as links. Defining the branch number as B, link number as L, and the extreme points of every switch as nodes, whose number is M, therefore, the tree graph has M − 1 = B branches [62]. Also, it is called a connected tree graph or spanning tree if this graph includes all radial distribution system nodes. Networks whose graphs are not tree graphs but contain all their nodes, that is, they do not have isolated nodes. Therefore, they are just connected graphs, as in the case where one or more of the normally open switches are closed, creating a meshed distribution system.

With the graph information and using a breadth-first search algorithm (BFA), the search process for finding the feasible reconfiguration set (FRS) is described as follows.

The BFA organizes its search in levels by opening some of the normally closed switches from the base case. Figure 5 illustrates this procedure.

Figure 5.

Flow chart for finding the feasible reconfiguration set.

The procedure begins at level 0 with no open switches, denoted as the set OS = {}. Therefore, considering the two switches, S₁ and S₂, with the lowest apparent power flow, the first level is formed with the sets: OS = {S₁, S₂}, ${\mathit{FR}}_1^{\left(1\right)}$ ={S₁}, ${\mathit{FR}}_2^{\left(1\right)}$ = {S₂}, which are used for the level 2, giving the next sets: OS = {S₁, S₂, S₃, S₄}, ${\mathit{FR}}_1^{\left(2\right)}$ = {S₁, S₃}, ${\mathit{FR}}_2^{\left(2\right)}$ = {S₁, S₄}, ${\mathit{FR}}_3^{\left(2\right)}$ = {S₂, S₅}, ${\mathit{FR}}_4^{\left(2\right)}$ = {S₂, S₆}, and so on. The number of levels is L, so that, at the last level, OS = {S₁, …, S_R}, where R is the number of open switches at this level.

On the other hand, the construction of feasible reconfigurations, which have only open switches, is carried out with the progress informing ${\mathit{FR}}_1^{\left(1\right)}$, ${\mathit{FR}}_2^{\left(1\right)}$, ${\mathit{FR}}_1^{\left(2\right)}$, ${\mathit{FR}}_2^{\left(2\right)}$, ${\mathit{FR}}_3^{\left(2\right)}$, …, etc. The number of reconfigurations created by brute force is 2^L, but any violation in either constraint (4)–(6), (8), (9), or by the existence of similar reconfigurations during the searching process, may reduce this number substantially at the end of stage 1.

3.2 Methodology 1

The proposal of Methodology 1 is built upon the method described in Refs. [47, 63] to obtain the compensation scheme. This proposal aims to obtain an accurate solution similar to other methods but tries to maintain a reasonable computational efficiency to be used as a tool for the operation of electrical distribution systems. This proposal looks for the optimal reconfiguration by performing two stages, where the first is related to the search of the FRS using the process illustrated in Figure 5. In contrast, the second one realizes the RPC dispatch considering the FRS. The steps of stages 1 and 2 in Methodology 1 are described below.

Stage 1. Determination of a feasible configuration set.

1. All the system switches are closed, forming a meshed system. Also, the FR and OS sets are initialized as empty sets.

2. A power flow problem is solved, and the two switches with the lowest apparent power flow through them are chosen to form the first level branches, named S₁ and S₂.

3. The selected switches S₁ and S₂ (which do not generate a disconnected graph) are opened separately, and a power flow is executed for each of these resulting network reconfigurations. Suppose the resultant reconfiguration violates some of the constraints (4), (5), and (6). In that case, it is discarded for continuing with the process, and the corresponding switch is closed again for subsequent calculations. Otherwise, go directly to the next step.

4. Open the actual switch at the first level, run the power flow algorithm, and select the next two switches with the lower apparent power flow. Assume that these new switches in the second level are defined as S₃ and S₄, which now will be investigated, and the results will create two branches of the decision-making tree at the third level and so on. This procedure continues until the configuration accomplishes constraints (8) and (9), that is, it is radial and a spanning tree, which will be saved in the feasible reconfiguration set (FRS). Then go to the next step.

5. If the power flow for each of the two switches selected in step 2 was already executed, go to the next step; otherwise, go to step 3.

6. Once the feasible reconfiguration process is terminated, all the feasible reconfigurations are ranked from the lowest to the highest distribution losses. Therefore, after this process, only the first reconfigurations (10 or less, depending on the distribution system size) are considered for performing Stage 2 and Stage 1 finishes.

Stage 2. Reactive power compensation dispatch.

As pointed out before, a node where there is capacitor block(s) installed is consider a feasible system node; also, only capacitor blocks of 300, 600, or 900 kVAr are considered for the RPC dispatch, which is carried out following the next steps:

1. A reconfiguration is selected from the feasible reconfiguration set obtained at the first stage.

2. Independently of the actual state, all the already-installed capacitors in the distribution system are disconnected, and a power flow is executed to compute system losses.

3. A capacitor block is connected to its feasible system node and analyzed by performing a power flow.

4. If there are more feasible nodes to investigate, step 3 is repeated. Otherwise, proceed to the next step.

5. Distribution losses obtained by connecting a capacitor block in each feasible node are compared. The capacitor block that obtains a greater loss reduction is permanently connected.

6. A power flow is executed with the capacitor block connected. If distribution losses were reduced above 1 kW, go to step 3; otherwise, the process is finished.

3.3 Methodology 2

Methodology 1 may be improved, without affecting its accuracy, by considering for the first and subsequent intermediate levels of the decision-making tree, a limit of no more than 10 possible reconfigurations for reducing computational work in the last levels, because, at that levels, they have less influence over distribution loss reductions when selecting different combinations. In addition, the configurations resulting from this process must have lower losses and be close to each other to be taken into account as feasible reconfigurations at Stage 2.

With the above considerations, by making some changes to Methodology 1, Methodology 2 resulted, whose description is as follows.

The experience gained working with Methodology 1 is that, at the first level, considering only two switches with the lower apparent power flow may lead to suboptimal results when the reconfiguration process finishes. Therefore, in level 1, Methodology 2 includes five switches with the lowest apparent power flow. From this point, Methodology 2 continues normally since computational efficiency degrades if this criterion prevails in the next levels.

On the other hand, to improve the computational efficiency of the first stage, the search space at each level is limited so that the decision-making tree does not grow excessively (which happens if there are many link switches).

Furthermore, an additional computational efficiency improvement is disregarding, from the second level and the next ones, those reconfigurations that do not accomplish a tolerance margin of 3%, based on the difference in losses between the reconfiguration with the lowest losses and all other reconfigurations obtained in the correspondent level.

Finally, in Stage 2, of Methodology 2, investigating the connection of capacitor blocks by sensitivity calculations instead of using the complete power flow algorithm may improve the computational efficiency without losing accuracy.

Performing the previous modifications to Methodology 1, the steps of each stage of methodology 2 are defined as follows.

Stage 1. Determination of a set of feasible reconfigurations.

1. All the system switches are closed, forming a meshed system. Also, the FR and OS sets are initialized as empty sets.

2. A power flow problem is solved, and the five switches (which do not generate islands when opened) with the lowest apparent power flow through them are chosen.

3. The selected switches are opened separately, and a power flow is executed for each of these resulting network reconfigurations. If the resultant reconfiguration violates any of the constraints (4), (5), or (6), it is discarded, and the corresponding feeder section is closed again to go forward with the process of finding the feasible reconfiguration set. Also, reconfigurations that result in one previously defined as feasible are removed, and their corresponding switch is closed again. If these constraints are not violated, go to the next step.

4. The losses obtained for each different reconfiguration are compared. At the first level of the decision-making tree, the three reconfiguration options with the greatest losses are eliminated so that only two remain.

5. If the actual reconfiguration accomplishes constraints (8) and (9), that is, is radial and a spanning tree, save it in the FRS. Also, if all the reconfiguration alternatives are exhausted, go to step 6; otherwise, go to step 2.

6. All the feasible reconfigurations are ranked considering their distribution losses from the lowest to the highest. The reconfigurations that present a distribution loss difference greater than a tolerance margin of 3% concerning the reconfiguration with the lowest losses are eliminated. The remaining reconfigurations form the final FRS, and they will be passed through the RPC dispatch at Stage 2.

Stage 2. Reactive power compensation dispatch.

Under the same considerations of Stage 2 of Methodology 1, the RPC dispatch is carried out following the next steps:

1. A reconfiguration is selected from the feasible reconfiguration set obtained at the first stage.

2. Independently of the actual state, all the already-installed capacitors in the distribution system are disconnected, and a power flow is executed to compute system losses.

3. A capacitor block is connected to its feasible system node and analyzed by the sensitivity algorithm to estimate the new losses.

4. If there are more feasible nodes to investigate, step 3 is repeated; else, proceed to the next step.

5. Distribution losses obtained by connecting a capacitor block in each feasible node are compared. The capacitor block that obtains a greater loss reduction is declared as permanently connected.

6. A power flow is executed with the capacitor block connected. If distribution losses were reduced above 1 kW, the Jacobian matrix is updated for sensitivity calculations and goes to step 3; otherwise, the process is declared finished.

4. Numerical examples

This section reports numerical examples from two case studies. The proposed methodologies are first applied to the IEEE 16-bus system for illustrative purposes. This benchmark is useful to analyze comprehensively the effects of the novel modeling aspects featured by the proposed methods. Subsequently, the performance of both methodologies on the Taiwan Power Company distribution system is investigated.

4.1 IEEE 16-bus distribution system

In addition to being widely used in the literature, the IEEE 16-node distribution system has a reactive compensation scheme, so it is ideal for observing the behavior of algorithms that perform reconfiguration and RPC dispatch for distribution systems. This system is a three-phase distribution system with three feeders, 16 nodes, 13 load points, 13 normally closed switches, and 3 normally opened switches, and the operation nominal voltage is 12.66 kV. Figure 6 shows the corresponding single line diagram, where switches S14, S15, and S16 define the initial operative condition of this system. Suppose that the load pattern changes so that nodal voltage and angle profiles are modified, causing distribution losses that may not be optimal with the actual configuration and RPC dispatch.

Figure 6.

IEEE 16-node distribution system.

A set of feasible reconfigurations are obtained at Stage 1. Then, the solution with the lowest losses and the resultant RPC dispatch with capacitor banks is obtained at Stage 2.

First, the three normally open switches are closed; thus, a meshed network is formed. Then, there are three meshes in the system, so the same number of switches must be opened for the system to be radial again. Therefore, the decision-making tree consists of three levels. For level 1, the two switches, S7 and S16, are selected because they have the lowest apparent power flow through them. After, a power flow is carried out for switches S7, and S16 opened separately, and the next two switches with the lowest apparent power flow are selected.

Figure 7 illustrates the feasible reconfiguration selection process. Note that losses were increasing from the top level to the bottom level. This is due to meshed networks presenting lower losses than radial networks. However, between the five feasible reconfigurations in level 3, ${\mathit{FR}}_4^{\left(3\right)}$ = {S7, S16, S8}, presents the lowest distribution losses, 466.1 kW, which is to be the first candidate to obtain the optimal value at the end of Stage 2 because of the distribution losses given by the other feasible reconfigurations.

Figure 7.

Generation of feasible reconfigurations by levels for the system of Figure 4.

On the other hand, three repeated options are eliminated (alternatives with dotted lines), one of them at level 2. Therefore, there is no reason why it must be investigated at lower levels. At the process ending, only five feasible reconfigurations passed to Stage 2.

The resulting reconfiguration options make up a set of feasible configurations, as shown in Table 1.

Feasible reconfiguration	Open switches	Losses (kW)
FR1	S7, S16, S8	468.33
FR2	S7, S16, S1	485.77
FR3	S7, S4, S8	481.82
FR4	S7, S4, S14	499.78
FR5	S16, S8, S1	496.00

Table 1.

Feasible reconfigurations obtained (first stage) using methodology 1.

Other methodologies applied to this example selected a reconfiguration with the lowest losses open switches S7, S16, and S8 (FR1), which presents the lowest loss value of 468.33 kW. Therefore, Methodology 1 includes this optimal solution among feasible reconfigurations. Then, stage 2 is applied to know the final losses with RPC dispatch to the five feasible reconfigurations.

For this system, to apply Methodology 2, the five switches with the lowest apparent power flow are selected to be the decision-making tree first level, S7, S16, S4, S8, and S1. The distribution losses obtained with switches S7 and S16 open are less than those obtained when opening the other ones. Therefore, these two switches are selected to begin the decision-making tree and continue the reconfiguration process. The set of feasible reconfigurations is shown in Table 2.

Feasible Reconfiguration	Open switches	Losses (kW)
FR1	S7, S16, S8	468.33
FR2	S7, S4, S8	481.82

Table 2.

Feasible configurations found using methodology 2.

Unlike the feasible reconfigurations obtained by Methodology 1, only two configurations resulted from applying Methodology 2. This is because, when considering a tolerance margin of 3%, as indicated in the last step of Stage 1, configurations whose difference in distribution losses concerning the lowest losses obtained is more than 3%, that is, 14.049 kW, are eliminated from the feasible reconfiguration set. The only feasible reconfiguration that complies with this tolerance is the one corresponding to the opening of switches S7, S4, and S8, which corresponds to the FR3 when applying Methodology 1.

The total reactive capacity compensation defined for the original system is 11.4 MVAr. Therefore, 11.4 MVAr will also be used as a limit for the RPC dispatch or until the loss reduction is less than 1 kW. Each capacitor block is 300 kVAr. Hence, 38 blocks can be used to cover the total reactive capacity. The results are shown in Table 3.

	Methodology 1					Methodology 2
Feasible reconfiguration	FR1	FR2	FR3	FR4	FR5	FR1	FR2
Losses without reactive compensation (kW)	606.6	628.0	627.2	650.5	634.3	606.6	627.2
Losses with reactive compensation (kW)	461.5	480.2	471.9	492.3	479.7	461.5	471.9
Loss reduction (kW)	145.1	147.8	155.3	158.2	154.6	145.1	155.3

Table 3.

Results of stage 2 applying both methodologies.

As can be noted, with reconfiguration FR1, the minimum losses are obtained before and after performing the RPC dispatch. In addition, it is observed that the higher the losses before starting Stage 2 tend to be reduced the more when applying for reactive compensation. However, this fails to change the result of the combination of reconfiguration and RPC dispatch with minimal losses.

The initial loss difference without connected capacitors between FR2 and FR5 configurations is 6.3 kW or 1.003% over FR2. Therefore, if these two reconfigurations changed positions and FR5 obtained lower losses than FR2 with the capacitors connected, it means that a difference of approximately 1% between the reconfiguration of lower losses and the other feasible reconfigurations is not enough to guarantee that the positions according to lower losses remain the same at the end of Stage 2. In addition, the difference between the losses of the FR2 and FR3 configurations before applying the RPC dispatch is only 0.8 kW, and after applying it, the difference results in 8.3 kW.

The solution to the problem of reconfiguration and RPC dispatch for this case is to use FR1 with the RPC dispatch scheme shown in Table 4.

Node	Methodology 1	Methodology 2
5	0.6	0.9
6	0.3	0.3
7	1.2	1.2
8	0.6	0.6
9	3.3	3.3
10	0.6	0.9
11	0.3	0.3
12	2.4	2.4
14	0.6
15	0.6	0.3
16	0.9	1.2
Losses (kW)	461.44	461.45

Table 4.

Resulting RPC dispatch (MVAr) scheme: IEEE 16-bus system.

From Table 4, it can be observed that the difference between methodologies is minimal both in the RPC dispatch scheme and in the final losses. This means that it is adequate to replace the power flow simulations with sensitivity calculations when evaluating losses with the connection of each block of the capacitor complete set.

Methodology 1 determines a different location of capacitor banks than Methodology 2, but the same amount of reactive compensation (11.4 MVAr) is used.

Figure 8 depicts the voltage magnitudes resulting from the base case and Methodologies 1 and 2. Note that, in general, the voltage profile is elevated with respect base case. This is because values of 1 p.u. for nodes 1, 2, and 3 are supply points (see Figure 6), and their voltages always are constant. On the other hand, nodes 8, 10, and 11 present the lowest values around 0.97 p.u. and below, whereas the highest voltage magnitude values are presented in nodes 12, 13, 14, 15, and 16. This is because these nodes initially have capacitors connected with higher capacities than those obtained with the RPC dispatch. This situation illustrates that it is better for distribution loss reduction to have deployed capacitor banks in various nodes instead of concentrating them in a few nodes. Finally, it is worth observing the voltage magnitude scale in the vertical axis, and it can be seen that the voltage magnitude differences shown are relatively small.

Figure 8.

IEEE 16-bus distribution system voltage magnitudes.

4.2 Taiwan power company distribution system

The Taiwan Power Company (TPC) distribution system is a three-phase system, 11.4 kV, 11 feeders, 83 normally closed switches, and 13 normally open tie switches. Figure 9 depicts the network diagram; dotted lines represent normally open tie switches.

Figure 9.

TPC distribution system.

As there are 13 tie-lines, 13 levels for the decision-making tree determine the feasible configurations. With Methodology 1, only the investigation in the last level resulted in 682 feasible reconfigurations. Performing this number of power flow simulations only at the last level’s decision-making tree is not convenient for finding FRS. Moreover, for distribution systems with many switches, the efficiency of the proposed methodologies could be similar or even worse than that of almost metaheuristic methods normally used for planning. This is why only 10 configurations with the lower distribution losses are selected, as pointed out before for Stage 1 of both methodologies. Keeping in mind this feature, the result for the current distribution system is shown in Table 5, after Stage 1 was realized.

FR	Methodology 1		Methodology 2
	Open switches	Losses	Open switches	Losses
Base	89, 90, 92, 93, 94, 95, 96, 84, 85, 87, 88, 98, 91	343.5 + 254.0 = 599.5	89, 90, 92, 93, 94, 95, 96, 84, 85, 87, 88, 98, 91	343.5 + 254.0 = 599.5
FR1	42, 89, 55, 33, 13, 82, 29, 7, 98, 64, 92, 71, 90	356.2 + 235.3 = 591.5	42, 89, 55, 33, 13, 83, 93, 7, 98, 64, 71, 92	348.9 + 239.3 = 588.2
FR2	42, 89, 55, 33, 13, 83, 29, 7, 98, 64, 92, 71, 90	350.9 + 239.3 = 590.2	42, 89, 55, 33, 13, 83, 29, 7, 98, 64, 92, 71, 90	350.9 + 239.3 = 590.2
FR3	42, 89, 55, 34, 13, 82, 29, 7, 98, 64, 92, 71, 90	348.7 + 239.3 = 588.0	42, 89, 55, 34, 13, 83, 93, 7, 98, 64, 92, 71, 90	347.8 + 239.3 = 587.3
FR4	42, 89, 55, 34, 13, 82, 93, 7, 98, 64, 92, 71, 90	354.1 + 235.3 = 589.4	95, 89, 55, 33, 13, 83, 29, 7, 98, 64, 92, 71, 90	350.8 + 239.3 = 590.1
FR5	42, 89, 55, 34, 13, 83, 29, 7, 98, 64, 92, 71, 90	348.7 + 239.5 = 588.0	42, 89, 55, 34, 13, 83, 29, 7, 98, 64, 32, 71, 90	354.5 + 239.3 = 593.8
FR6	42, 89, 55, 34, 13, 83, 93, 7, 98, 64, 92, 71, 90	347.8 + 239.5 = 587.3	42, 89, 55, 34, 13, 83, 29, 7, 98, 64, 92, 71, 90	348.7 + 239.3 = 588.0
FR7	95, 89, 55, 34, 13, 82, 29, 7, 98, 64, 85, 71, 90	354.4 + 235.3 = 589.7	42, 89, 55, 34, 13, 83, 29, 7, 98, 64, 71, 32, 90	355.2 + 239.3 = 594.5
FR8	95, 89, 55, 34, 13, 82, 93, 7, 98, 64, 92, 71, 90	353.3 + 235.3 = 588.6	42, 89, 55, 33, 13, 83, 93, 7, 98, 64, 92, 71, 90	349.1 + 239.3 = 588.4
FR9	95, 89, 55, 34, 13, 83, 29, 7, 98, 64, 92, 71, 90	349.0 + 239.3 = 588.3	42, 89, 55, 33, 13, 83, 93, 7, 98, 64, 71, 32, 90	353.2 + 239.3 = 592.5
FR10	95, 89, 55,34, 13, 83, 93, 7, 98, 64, 92, 71, 90	347.9 + 239.3 = 587.2	95, 89, 55, 34, 13, 83, 29, 7, 98, 64, 92, 71, 90	349.0 + 239.3 = 588.3

Table 5.

Ten feasible configurations with the lowest distribution losses (kW).

The FRS obtained by Methodology 1 is almost completely different from that resulting from Methodology 2; only three FRs are equal: FR2-FR2, FR5-FR6, and FR9-FR10 defined by methodologies 1 and 2, respectively. Also, the configuration with the minor losses of Methodology 1 reports lower losses than that of Methodology 2. This is because, during the process, the combination of open switches that leads to the configuration of lower losses for Methodology 1 is eliminated when using Methodology 2, since at one level, the limit of 10 possible combinations is exceeded, and that option has higher losses than the 10 with which the process is continued. Despite this, configurations with similar losses were determined since the lowest loss FR obtained by Methodology 1 is 587.2 kW. The one obtained by Methodology 2 is 588.0 kW (the difference is only 0.8 kW or 0.18%). Finally, note that all 10 FRs cause a decrease in electrical losses compared with FR0, denoted as Base.

On the other hand, the combinations formed in the decision-making tree (from the first level to the last) using Methodology 1 are 1135. In the case of Methodology 2, there are only 160 combinations. Because a power flow simulation must be performed for each combination, the difference in computational work between both methodologies applied to the TPC distribution system is too great (975 power flows). However, in general, it may be compensated because there is a small difference between the accuracy of both methods.

The effect of connecting capacitors to each feasible configuration in terms of total distribution losses is reported in Table 6. It should be noted that methodologies 1 and 2 reached practically the minimum electrical loss level with a difference of only 0.1 kW, but, as pointed out before, Methodology 2 is much more efficient than Methodology 1.

FR	Methodology 1				Methodology 2
	Before RPC dispatch		After RPC dispatch		Before RPC dispatch		After RPC dispatch
	Losses	Rank	Losses	Rank	Losses	Rank	Losses	Rank
FR1	591.5	10	439.7	10	588.2	3	437.5	3
FR2	590.2	9	438.9	8	590.2	7	438.9	8
FR3	588.0	4	437.9	5	587.1	1	437.3	1
FR4	589.4	7	438.9	9	590.1	6	438.8	7
FR5	588.0	3	437.9	4	593.8	9	439.4	9
FR6	587.3	2	437.3	2	588.0	2	437.9	5
FR7	589.7	8	438.8	7	594.5	10	439.5	10
FR8	588.6	6	438.2	6	588.4	5	437.5	2
FR9	588.3	5	437.8	3	592.5	8	438.5	6
FR10	587.2	1	437.2	1	588.3	4	437.8	4

Table 6.

Distribution losses (kW) and rank with the RPC dispatch for the 10 FRs.

The RPC dispatch scheme with Methodology 1 obtained the lowest distribution losses using FR4, whereas Methodology 2 applied RPC dispatch to FR3. Both solutions are reported in Table 7. Note that total electrical distribution losses decreased from 587.2 kW without RPC dispatch to 437.2 kW with RPC dispatch applying Methodology 1, that is, a reduction of 150 kW or 25.54%.

Node	Reactive Power Compensation (MVAr)
	Methodology 1	Methodology 2
6	1.2	1.2
21	1.8	1.8
28	1.8	1.8
37	0.9	0.9
51	2.1	2.1
63	0.3	0.3
71	2.1	2.1
75	1.5	1.5
81	1.2	1.2
Total compensation (MVAr)	12.9	12.9
Losses (kW)	437.2	437.3

Table 7.

Resulting RPC dispatch (MVAr) scheme: TPC distribution system.

Figure 10 depicts the voltage profile for the initial configuration and the proposed methodologies. It can be noticed that the results obtained by the proposed methodologies are quite similar. Both methodologies find a better voltage profile since the minimum voltage is 0.9642, p.u. whereas in the base case, it reaches 0.9466 p.u.

Figure 10.

TPC distribution system voltage magnitudes.

5. Conclusions

Applying reconfiguration and RPC dispatch with capacitor banks simultaneously, even when different formulations are used to solve both problems, allows for a greater reduction in electrical distribution losses. Furthermore, if reconfiguration and RPC dispatch are executed sequentially, the optimal solution may be obtained.

The operation of electrical distribution systems with already installed reactive compensation requires an RPC dispatch to reduce electric losses or improve voltage profiles. Therefore, only nodes with already connected capacitors should be considered for the study. In this regard, the efficiency of the proposed methodologies is increased due to the fact that the search space is reduced and with it the number of necessary operations.

Sensitivity analysis instead of power flow increases the computational efficiency to calculate changes in capacitive compensation.

Both methodologies are a competitive option for distribution loss optimization because of their relatively low computational work. Also, Methodology 2 has higher computational efficiency and accuracy comparable with Methodology 1.

From the results of Stage 1 with the two distribution systems analyzed, it is convenient to limit the FRS to 10 or fewer and consider only those that meet a 3% tolerance with respect to that of lower losses, since only those that report the lowest losses are most likely to be part of the solution at the end of the capacitive compensation process.

When RPC dispatch is performed, a minimum tolerance margin is defined to reduce losses for planning or operational decisions on the distribution system to increase computational efficiency. This loss reduction margin (which also improves the voltage profile of the system in general) must justify the investment made in the capacitor banks and in their connection to the busses that require it.

Acknowledgments

This work was supported by the Tecnologico Nacional de México under Project 13322.21-P.