Minimization of Transmission Line Losses Through System Topology Reconfiguration

Abstract

This research proposes a methodology for minimizing losses in transmission lines (TLs), considering the reconfiguration of the architecture of the electrical power system (EPS). The implementation of this methodology redirects the power flow with optimal switching through its TL to guarantee the stability of the voltage, angle, frequency, and power balance in order to minimize losses that affect the reliability and quality of the system. Optimal transmission switching (OTS) allows various types of analysis to be carried out; the loadability of the lines, response times, and operating costs, among other aspects, can be improved. This article proposes minimizing the losses in the transmission lines with OTS by using AC power flows as a mixed-integer nonlinear problem (MINLP). Several test scenarios evaluate the method’s effectiveness, determining the optimal topology for corrective control that optimizes power flows in different situations. It is proven that this approach reduces losses compared to a base case by 99%, even in the face of N − 1 or random contingencies, without losing the load and while maintaining the same active power dispatch and, finally, verifying a strategic increase in the dispatch of reactive power to maintain operating parameters within stable limits.

1. Introduction

Electric power systems (EPSs) are experiencing continuous progress in their components, including generation, transmission, distribution, and demand. This advance aims to reduce operating costs and losses, guaranteeing the reliability and quality of the service. The transmission system serves the function of being the link between power generation and load. As the number of users increases, transmission networks become more complex, giving rise to terms such as overloaded or underloaded lines, discrepancies in voltage profiles, and increased active and reactive power flows. This increase leads to the generation of power losses and voltage drops [1].

The losses in the EPS are defined as the difference between the power generated and the power consumed by the load. Losses occur due to heat dissipation in the lines or in the cores of the transformers, or due to different physical phenomena in the system’s elements while they are energized. We can consider the losses as energy losses at an instant and intervals, called energy losses. Losses can be classified as technical or non-technical; technical losses are produced via energy transport or transformation caused by the corona effect, joule effect, eddy currents, and hysteresis. Non-technical losses are reflected in the systems’ effectiveness and precision in measurement equipment, according to the electrical company. The determination of losses in transmission and sub-transmission systems is carried out through power flow analysis, in which the results are obtained from the voltages and powers in the nodes of the system. To achieve this, it is essential to have detailed knowledge of all line parameters. In the case of transformers, it is necessary to have complete information on iron and copper losses. The energy losses in the lines and the transformers can be obtained using the load duration curve or the loss factor. The analysis of losses in the EPS needs to mitigate failures, and the best option is the optimal use of generation and transmission, dispensing reagents, and minimizing elements; through optimal power flows, adequate use can be achieved and made ideal in the system [2,3,4].

Several researchers suggest the integration of compensators in the nodes of the transmission system to reduce reactive power. These studies also address the optimal placement of an SVC (static synchronous compensator) to improve voltage stability and reduce drifts. These optimizations are achieved through techniques such as HPSO (hybrid particle swarm optimization), genetic algorithms (GAs), bee colonies, and multi-criteria decision-making methods (MCDMs), among others. Applying these methodologies not only contributes to the stability of the electrical system but also reduces generation costs and losses in the transmission system [5,6].

Implementing strategies in the topology of the transmission system can directly impact various aspects, such as voltage adjustments and the control of line overloads. These adjustments and controls are crucial to optimizing the system’s operation, and they directly apply to economic dispatch (DE). By efficiently managing the system’s topology, we seek to minimize the associated operating costs, thus contributing to a more efficient and economical operation in transmitting electrical energy [4].

The current studies on optimal transmission line switching (OTS), also known as optimal transmission switching, have significantly impacted the annual production of developed countries worldwide. This trend is gaining momentum in Latin America, where new research collaborations are being established with countries that have seen notable progress in the field of electrical transmission [7,8]. Optimal transmission line switching (OTS) has made it possible to address direct solutions to the cost problem in DE [4]. Therefore, the studies focused on different adaptation measures to mitigate the problems linked to the OTS, such as overloadability in lines or generation, the optimization of operating costs, the best switching option, response times, large-scale medium and high tests in the SEP, the identification of poorly conditioned systems, island formation, congestion relief, and cascade failure blocking, among others [8,9,10].

The OTS shows a security solution that addresses the needs that arise from the increase in demand, so guaranteeing electrical energy has led to increased restrictions being proposed to cover new objectives based on the reliability and quality of the system [11,12]. Therefore, reconfiguring the system through its topology when there are different unforeseen contingencies reduces the losses reflected in the loads [13,14].

The mathematical models investigated support different analysis methods such as mixed-integer linear optimization for DC flows, for which reactive power or the different voltage effects are not considered, so their analysis time is reduced [15,16], while other methods with greater precision perform optimal AC flows with mixed-integer nonlinear optimization, they increase their restrictions and their analysis time by taking into account all the parameters of the EPS [17,18]. On the other hand, there are methods in which hybrid models are integrated to reduce complexity and analysis response times for real systems at different scales and to integrate non-conventional renewable energies (NCREs) [19,20].

The above models make it possible to visualize the advantages and contributions of each one so that considering losses in the system due to different contingencies entails analyzing all the topological variants that may exist, adapting mechanisms to deal with these uncertain scenarios and thus finding the best solution to these problems [21,22].

Losses in transmission systems are varied, such as those found due to overload in the lines’ voltage deviations, among others [23], so evaluating the losses in electrical power systems through the corrective control of the system topology for the minimization of losses in transmission lines is the proposal of the article. This study presents a proposed approach using a mixed-integer nonlinear programming (MINLP) method based on alternating current optimal power flow (OPF-AC). The evaluation will focus on losses in electrical power systems through the corrective control of the system topology to identify an optimal solution to minimize losses in transmission lines.

Motivation and Main Contribution

The main motivation for the development of the present research is based on the fact that different methodologies for OTS have been presented in the scientific literature, but each of them has the same common basis, which is the minimization of power-system operating costs. However, technical conditions have not been identified, as is the case of the present proposal, where the minimization of power system losses is analyzed with a consideration of OTS.

In addition to this, to test the robustness of the proposed methodology, N − 1 contingencies were randomly generated, and the methodology of the present investigation was tested on this new operational state, managing to find that the system can be optimized in the face of this failure state.

The following can be considered as the main contributions:

• An OTS methodology considering loss minimization as the objective function.
• Evaluating all electrical variables since the proposed OTS is based on the optimal AC power flow equations, including reactive power.
• Verification of the behavior of the power system in N − 1 contingency scenarios and, on this, the application of the proposed OTS model.

The article is organized as follows: Section 2 presents an analysis of the topology of electrical power systems and how this is affected by the OTS; then, in Section 3, the problem is posed, and the proposed methodology is shown. In Section 4, all the results obtained are analyzed and discussed, comparing the different electrical variables under different case studies, and finally, in Section 5 the main conclusions of the research are shown.

2. Power System Topology

The topology of electrical power systems is defined by the links generated via the TL, that is, which nodes are interconnected through them. The operational planning of power systems is based on different aspects, such as cost minimization, loss minimization, and the maximization of the stability and reliability of the EPS.

In the scientific literature, there are several studies on OTS, which directly affects the topology of the power system; these studies have focused on deciphering a path through which power flows without affecting the normal operating conditions of the EPS, with the main difficulties being the overload generated in the TL, changes in the voltage profiles of the nodes, the degradation of reliability and, evidently, effects on the stability limits of the power system.

2.1. Optimal Transmission Line Switching (OTS)

The OTS describes the redistribution of power flows in the SEP, helping to decongest the system. Switching allows adaptation to various scenarios where power flow reconfiguration significantly increases reliability. The OTS allows the system’s energy to be distributed with the knowledge that these modifications cause the system parameters to vary due to the disconnection of several elements that contributed to the original design. The OTS solves several problems; its purpose is to keep the EPS stable while isolating the failures or contingencies to which the system is exposed; it also helps the energy balance remain stable [23].

By making the different modifications to the lines, congestion in the system is reduced, and its support is seen in the useful life of the electrical components [4], with a reduction in operating costs and an increase in the capacity to operate at higher ranges than those established as the nominal range of the system. OTS is highlighted as a progressive solution to mitigating losses generated via contingencies while optimizing the efficiency of electrical networks by controlling short-circuit currents and overloads. All of this is achieved without incurring additional costs in the [24] operation.

The different models of the OTS from the same perspective are based on DC optimal power flows (OPF-DC), whose objective is the analysis of active power where an efficient economic dispatch is anticipated that ensures the supply of demand, disregarding losses since it does not consider in its analysis the different effects of reactive power and voltage. But it creates an impact on the study of loadability, voltage angles in transmission lines, and generation costs. Where their OTS objectives do not apply, leading to load shedding since the established technical limits are not violated. Different scenarios arise where losses are generated in the transmission lines, such as islands’ formation due to the topology variation [4,11,25].

The OTS model proposes a DC power flow with control over switching in the EPS, aiming to minimize costs; Equation (1), with transmission line switching, considers different constraints that limit the power flow in the system. Equations (2) and (3) also consider the maximum and minimum susceptance. Equation (4) establishes the power balance at the nodes, where the input power must equal the sum of the loads, with the maximum or minimum generation capacity taken into account, depending on the connection state in Equations (5) and (6). Constraint (7) limits the total number of switches in all transmission lines, considering the admittance (8), and the connection state (9) [26].

Objective function:

$$minZ=\sum _{g=1}^{N_G}{C}_g{P}_g$$

(1)

Restricted to:

$$P_{i-j}-B_{i-j}(\delta i-\delta j)\le (1-{\zeta }_{i-j})M$$

(2)

$$P_{i-j}-B_{i-j}(\delta i-\delta j)⩾(1-{\zeta }_{i-j})M$$

(3)

$$\sum _{gϵ{\mathsf{\Omega }}_G^i}{P}_g+L{S}_i-L_i=\sum _{gϵ{\mathsf{\Omega }}_l^i}{P}_{i-j}:{\lambda }_i\to iϵ{\mathsf{\Omega }}_B$$

(4)

$$-P_g^{max}{\zeta }_{i-j}⩽P_{i-j}⩽P_g^{max}{\zeta }_{i-j}\to i-jϵ{\mathsf{\Omega }}_l$$

(5)

$$P{g}_i^{min}\le Pg\le P{g}_i^{max}$$

(6)

$$\sum _{i-j}(1+{\zeta }_i)⩽N_{sw}\to i-jϵ{\mathsf{\Omega }}_l$$

(7)

$$B_{i-j}={\displaystyle \frac{1}{X_{i-j}}}$$

(8)

$${\zeta }_{i-j}ϵ\left(0,1\right)$$

(9)

The OTS analyzed using a different approach aims to carry out a study that covers various variables, combining the optimal power flow, reactive losses, and the effects on voltage. Addressing the problem of optimal switching represents a significant challenge from a mathematical perspective, given that the analysis encompasses nonlinearities and is not convex, in addition to the fact that its variables can assume discrete values [27]. Although diverse approaches provide optimal solutions for switching transmission lines to achieve economic dispatch, these solutions are not entirely global; however, they contribute to mitigating the impacts of adverse events [6,27,28].

The various approaches used in the scientific community to carry out optimal switching on transmission lines within electrical power systems have proven capable of generating multiple solutions, which vary, depending on the specific problem to be addressed. Furthermore, the complexity of this analysis process is influenced by the set of restrictions incorporated into the study objective. Therefore, below is a diagram that summarizes the different approaches developed to date in this area of research; see Figure 1.

2.2. Optimum AC Power Flow

The analysis of power flows represents a fundamental stage in studying the electrical power system (EPS). Its central purpose is to evaluate the various operating conditions of a system, calculating the voltage magnitude and the corresponding angle for a specific load. This process involves considering numerous variables related to the network and generation. Through various types of analysis, strategies can be determined for planned expansion, operational planning, and real-time control of systems. Power flows are applied in operations and developing efficient designs with lower losses [29].

Creating an effective model for the system requires a thorough understanding of the various parameters encompassing the entire system. In analyzing power flow, it is essential to consider the admittance matrix, which stores very important information about the impedances of the transmission lines. By incorporating these data into the model, a complete and accurate representation of the electrical system interactions is achieved, facilitating effective decision-making in the planning, operation, and control phases [29].

When addressing the reduction in AC power flow (OPF-AC) losses, we focus on optimizing energy distribution in the electrical system to minimize losses during transmission as much as possible. This objective is achieved by adjusting critical variables in the system, such as transmission line configurations and power generation, to achieve more efficient operation and, consequently, reduce power losses. In loss minimization, advanced techniques such as mixed-integer nonlinear programming (MINLP) and other optimization methods are used to determine ideal operating conditions that significantly reduce these losses. This approach seeks to improve the efficiency of the electrical system, reducing excess energy and optimizing its performance [29].

The fundamental purpose of the Optimal Power Flow in Alternating Current (OPF-AC) approach is to minimize the costs associated with the operation of the electrical power system (EPS). This objective is defined through the objective function (10), which is subject to critical constraints to ensure system balance. Constraints (11) and (12) establish the balanced relationship between the generated active and reactive power, the system flow, and the load. Additionally, Equations (13) and (14) outline the power flow in the transmission lines in terms of active and reactive power. Furthermore, constraints (15) and (16) define the power limits for generators in terms of active and reactive power. Equation (17) sets the voltage limits at each node, while (18) establishes the angular difference limits (19) in each branch [30].

The efficient capacity of transmission lines to transport electrical energy within acceptable operational parameters is reflected in constraint (20). This constraint is crucial in defining the load limits to ensure efficient and operationally viable energy transport in the system [12].

Objective funtion:

$$minZ=\sum _{g=1}^{N_G}{C}_g{P}_g$$

(10)

Restricted to

$$\sum _{i=1}^{n_{bus}}\sum _{j=1}^{n_{bus}}{P}_{i-j}={P_g}_i-{P_D}_i;\forall i,j\in N_B$$

(11)

$$\sum _{i=1}^{n_{bus}}\sum _{j=1}^{n_{bus}}{Q}_{i-j}={Q_g}_i-{Q_D}_i;\forall i,j\in N_B$$

(12)

$$P_{i-j}=V_i^2{G}_{i-j}-V_i{V}_j\left[G_{i-j}cos\left({\delta }_{i-j}\right)-B_{i-j}sin\left({\delta }_{i-j}\right)\right]$$

(13)

$$Q_{i-j}=V_i^2{B}_{i-j}+V_i{V}_j\left[G_{i-j}sin\left({\delta }_{i-j}\right)-B_{i-j}cos\left({\delta }_{i-j}\right)\right]$$

(14)

$${P_g}_i^{min}\le {P_g}_i\le {P_g}_i^{max}$$

(15)

$${Q_g}_i^{min}\le {Q_g}_i\le {Q_g}_i^{max}$$

(16)

$$V_i^{min}\le V_i\le V_i^{max}$$

(17)

$${\delta }_i^{min}\le {\delta }_i\le {\delta }_i^{max}$$

(18)

$${\delta }_{i-j}={\delta }_i-{\delta }_j$$

(19)

$$-SIL\le S_{i-j}\le SIL$$

(20)

3. Problem and Methodology

One of the essential functions of the transmission system is to guarantee the continuous provision of electrical energy to end users. Achieving this objective and maintaining operation through transmission line switching (OTS) involves performing a comprehensive analysis that covers aspects such as voltage, frequency, power balance, and angles. This approach aims to minimize losses in transmission lines. Given that various scenarios inevitably arise during and after the switching of transmission lines, such as increased power flow or overloaded lines, it is important to address the issue of system losses in these processes. Seeking solutions that analyze voltage levels, power flows, balance, and losses, specifically focusing on predefined parameters, will contribute to establishing stability limits [4,23].

Previous research has demonstrated changes in topologies through AC power flow analysis [31], highlighting increased efficiency when actual system constraints, such as grid decongestion and the prevention of islanding, are considered [17]. By characterizing the scenarios of the new topology, the most vulnerable lines and those contributing to a reduction in losses can be identified. In this analysis, restructuring with corrective control can provide significant results for the transmission system [23].

A method is proposed that will evaluate, as an objective function, the active power losses in the transmission lines in electrical power systems through the corrective control of the topology to find an optimal solution for loss minimization. The objective function will be routed with active and reactive power balance restrictions, the operability limits of voltage, power, angle, and capacity, and the connection status of the line. The proposed method will use the IEEE-14 bus system; different application cases in the EPS will be evaluated, from a base case, such as the optimal AC power flow (OPF-AC), to cases of random contingencies.

3.1. Study Cases

In the first case study (called the base case), the IEEE 14-bus model of an OPF-AC was examined without considering the minimization of losses due to transmission line switching, and the compensations of the original model was not considered.

The second case study analyzed the same test model, but transmission line switching was applied to minimize losses this time. The results obtained in the OPF-AC were also analyzed.

In the third case study, the system was evaluated under N − 1 contingencies, which implies the system’s behavior when a transmission line is disconnected and how switching transmission lines will minimize losses.

Finally, as the fourth and last case, the IEEE 14-bus test system was examined under random contingencies and, at the same time, how the OTS mitigates the impact by reducing active power losses in the EPS transmission lines was also investigated. To solve the optimization problem for the minimization of active power losses with OTS, the GAMS software release 27.3.0 was used to verify the proposed methodology; the data of the IEEE 14-bus system were entered, such as the analysis conditions, the objective function, and the different restrictions. The data obtained in each results table were validated using the Digsilent software 2022 SP2.

3.2. Proposed Methodology

The proposed methodology minimizes losses in transmission lines (TLs) through optimal transmission switching (OTS); see Algorithm 1. Therefore, we defined that, in the TL, active power is transferred from node i to node j (23), and vice versa (24); the real part is the apparent power (22) (the PI model of the transfer lines). When an energy analysis is performed at the instant when node i is in balance, and similarly for node j, the equation for minimizing losses in the transmission lines in the EPS is mathematically derived (21) [1].

Objective funtion

$$minPloss=\sum _{i=1}^{N_B}\sum _{k=j}^{N_B}{G}_{i-j}(V{i}^2+V{j}^2-2ViVjcos\left({\delta }_{i-j}\right))$$

(21)

Restricted to:

$$S_{ij}=V_i{I}_{i-j}^{*}$$

(22)

$$P_{i-j}=V_i^2{G}_{i-j}-V_i{V}_j\left[G_{i-j}cos\left({\delta }_{i-j}\right)-B_{i-j}sin\left({\delta }_{i-j}\right)\right]$$

(23)

$$P_{j-i}=V_i^2{G}_{i-j}-V_i{V}_j\left[G_{i-j}cos\left({\delta }_{i-j}\right)+B_{i-j}sin\left({\delta }_{i-j}\right)\right]$$

(24)

Algorithm 1 OTS with OPF-AC to minimize losses in TLs.

Step: 1 Input IEEE 14-bus model data EPS Parameterization Generators, transformers, lines, connectivity matrix, loads Step: 2 OPF-AC Equations (10)–(20) Save results Base Case (case 1) Step: 3 Loss Minimization with OTS O.F.:       $minPloss={\sum }_{i=1}^{N_B}{\sum }_{k=j}^{N_B}{G}_{i-j}(V_i^2+V_j^2-2V_i{V}_{j}cos\left({\delta }_{ij}\right))$ s.t.:       $P_{Gi}-P_{Di}={\sum }_{i=1}^{NB}\left[-V_i^2{G}_{i-j}-V_i{V}_j\left[G_{i-j}cos\left({\delta }_{i-j}\right)+B_{i-j}sin\left({\delta }_{i-j}\right)\right]\right]$       $Q_{Gi}-Q_{Di}={\sum }_{i=1}^{NB}\left[-V_i^2{B}_{i-j}-V_i{V}_j\left[G_{i-j}cos\left({\delta }_{i-j}\right)-B_{i-j}sin\left({\delta }_{i-j}\right)\right]\right]$       $-Q_g^{max}{\zeta }_{ij}⩽Q_{ij}⩽P_g^{max}{\zeta }_{ij}\to ijϵ{\mathsf{\Omega }}_l$       $\left(2\right)$–$\left(9\right), \left(11\right)$–$\left(20\right)$ Save results case 2 Step: 4 Loss Minimization with OTS under N − 1 and Random Contingency for  i = 1:n; n  $ϵ$   TL        $\mathbf{for}j=1:n$           $\mathbf{if}{C}_{ij}=1$                  $C_{ij}=0$                   $OPF-AC+OTS$                   $\mathbf{O}.\mathbf{F}.:$                   $minPloss={\sum }_{i=1}^{N_B}{\sum }_{k=j}^{N_B}{G}_{i-j}(V{i}^2+V{j}^2-2ViVjcos\left({\delta }_{ij}\right))$                  $\mathbf{s}.\mathbf{t}.:$                  $P_{Gi}-P_{Di}={\sum }_{i=1}^{NB}\left[-V_i^2{G}_{i-j}-V_i{V}_j\left[G_{i-j}cos\left({\delta }_{i-j}\right)+B_{i-j}sin\left({\delta }_{i-j}\right)\right]\right]$                  $Q_{Gi}-Q_{Di}={\sum }_{i=1}^{NB}\left[-V_i^2{B}_{i-j}-V_i{V}_j\left[G_{i-j}cos\left({\delta }_{i-j}\right)-B_{i-j}sin\left({\delta }_{i-j}\right)\right]\right]$                  $Q_g^{max}{\zeta }_{ij}⩽Q_{ij}⩽P_g^{max}{\zeta }_{ij}\to ijϵ{\mathsf{\Omega }}_l$                  $\left(2\right)$–$\left(9\right), \left(11\right)$–$\left(20\right)$                  $C_{ij}=1$                   $k=k+1$           $\mathbf{end}\mathbf{if}$        $\mathbf{end}\mathbf{for}$ $\mathbf{end}\mathbf{for}$ for  k = 1:m; m  $ϵ$   Contingencies        if $minPloss>k$                  $minPloss=k$                   $TL=k$        $\mathbf{end}\mathbf{if}$ end for Save results cases 3 and 4 Step: 5 Show results       $P_g,Q_g,V_{i-j},{\zeta }_{i-j},N_{sw},Ploss$

Algorithm 1 shows the process for changing the EPS topology, considering OPF-AC in electrical systems after contingencies. Step 1 includes the input of the EPS connectivity variables and parameters. The main variables and parameters required are as follows: the characteristics of the generators, lines, transformers, and loads. In Step 2, the behavior of the EPS after the application of the OTS based on OPF-AC is determined, minimizing the losses, with which the initial reconfiguration is obtained, which is the starting point for the application of the contingencies in Step 3, in which N − 1 contingencies are generated randomly. It is identified whether the system needs a new reconfiguration to minimize losses, and finally, in Step 4, the results of the proposed model for each switching and contingency scenario are shown.

Steps 2, 3, and 4 of Algorithm 1 were analyzed using GAMS and verified using a Digsilent Power Factory. In GAMS, the traditional techniques for OTS and the new proposal were implemented, considering the random generation of N − 1 contingencies, and these scenarios were simulated using the Digsilent Power Factory, thus verifying the behavior of the power system under these operating conditions. It should be emphasized that the Digsilent Power Factory does not have OTS within its algorithm, so it was necessary to generate a DPL script so that the same simulation condition as the optimization proposal was achieved.

4. Results Analysis

For data analysis, an Acer Aspire computer with an Intel(R) Core(TM) i5-8300H CPU @ 2.30GHz 2.30 GHz processor was used (Intel, Santa Clara, CA, USA), with 16.0 GB of RAM and a 64-bit operating system installed. The optimization of loss minimization in transmission lines based on corrective control was carried out in the GAMS software, version 27.3, where a DICOPT solver that uses MINLP was used. Additionally, to verify and validate each of the results obtained, it was simulated in the Digsilent Power Factory, version 2022 SP2.

4.1. EPS Reconfiguration

To begin the analysis of the results, the connection characteristics of the IEEE 14 BUS test system are detailed below. Its base case presents an AC power flow (OPF-AC) without line disconnection. The connection status of the EPS can be seen in Figure 2 and Figure 3; the connection status of the system is denoted with visual detail; in this case, if there is a connection, it is green, and if there is no connection between bars, it is gray.

In case 2, an analysis of loss minimization in the transmission lines was described through OTS. The optimal solution was found by disconnecting lines 3–4, represented in black in Figure 2 and Figure 3. Subsequently, an AC power flow (OPF-AC) was executed. In case 3, the system was evaluated by introducing the N − 1 contingency, shown in red for lines 10–11. After the OTS analysis, it was determined that the optimal disconnection to minimize losses was lines 12–13, highlighted in black in Figure 2 and Figure 3, followed by the execution of an AC power flow (OPF-AC). In case 04, the system was tested with a random contingency that coincides with lines 6–11, identified in red in Figure 2 and Figure 3. After the OTS analysis was performed, it was determined that the most appropriate disconnections to minimize losses were lines 1–5 and 2–3, identified in black. Subsequently, an AC power flow (OPF-AC) was applied, as in all the case studies.

Once the study cases to be analyzed were identified, the most significant magnitudes were evaluated in the IEEE 14-bus test system. These magnitudes were compared with the results obtained using the Digsilent Power Factory software 2022 SP2 to validate the developed Algorithm 1.

4.2. Voltage Profiles

The voltage profiles obtained in the four case studies conducted using the GAMS software release 27.3.0 are very similar to those obtained using the Digsilent software 2022 SP2, as shown in

Table 1. Voltage profile.

	Base Case		Case 2		Case 3		Case 4
BUS	GAMS	PF	GAMS	PF	GAMS	PF	GAMS	PF
1	1.06	1.06	1.06	1.06	1.06	1.06	1.06	1.06
2	1.04	1.04	1.061	1.05	1.061	1.05	1.061	1.05
3	1.03	1.03	1.052	1.05	1.051	1.05	1.051	1.05
4	1.03	1.03	1.034	1.04	1.033	1.03	1.033	1.03
5	1.03	1.03	1.035	1.04	1.036	1.04	1.037	1.04
6	1.02	1.02	0.959	0.92	0.987	0.99	0.991	0.99
7	1.03	1.03	0.992	0.98	0.981	0.98	0.979	0.98
8	1.07	1.07	1.001	1.00	0.979	0.98	0.981	0.98
9	1.01	1.00	0.965	0.97	0.957	0.96	0.952	0.95
10	1.00	1.00	0.960	0.97	0.949	0.95	0.939	0.94
11	1.01	1.01	0.955	0.96	0.980	0.98	0.933	0.94
12	1.01	1.01	0.936	0.94	0.963	0.96	0.967	0.97
13	1.00	1.00	0.932	0.94	0.956	0.96	0.959	0.96
14	0.99	0.98	0.930	0.94	0.36	0.40	0.935	0.94

. It is worth mentioning that the voltage magnitude has specific setting parameters within a range where the electrical system operates under stable conditions; for the angle, the limits are from 0.6 to −0.6 rad, while its magnitude is within a range of 0.9 to 1.1 [p.u.] [1].

The voltage measurements at the buses of the evaluated electrical system in cases 2, 3, and 4 show higher levels than the base case. It can be observed in Figure 4 that, at buses 6 to 14, the voltage levels are below those of the base case. These results are evaluated with the AC power flows. In cases 2, 3, and 4, a preliminary analysis is carried out, considering the objective function (21) for loss minimization in the transmission lines through OTS; additionally, N − 1 and random contingencies are considered in cases 3 and 4.

The voltage magnitudes on each bus in cases 2, 3, and 4 maintain a similar behavior since they are under transmission line switching (OTS), where the connection or disconnection analysis defines an optimal solution. The voltages from bars 1 to 5 are very close in magnitude. It should be noted that, due to the disconnection of lines, there is a decrease in voltage measurements in the bars where there is no generation.

4.3. Active Power

Table 2. Comparison of real power dispatch between GAMS and Power Factory.

	Base Case		Case 2		Case 3		Case 4
G	Gams	PF	Gams	PF	Gams	PF	Gams	PF
1	1.430	1.424	0.000	0.000	0.061	0.060	0.064	0.065
2	0.200	0.200	1.400	1.401	1.400	1.401	1.400	1.400
3	0.610	0.614	0.000	0.000	0.000	0.000	0.000	0.000
4	0.200	0.200	0.751	0.800	0.435	0.440	0.412	0.411
5	0.200	0.200	0.171	0.172	0.431	0.430	0.434	0.444
TOTAL	2.64	2.64	2.322	2.373	2.327	2.331	2.31	2.32

presents the dispatch of each generator in the studied cases. When each generator is analyzed, it can be noted that Generator 1 has a dispatch of 1.43 [p.u.], representing a significant difference compared to cases 2, 3, and 4, where OTS has already been implemented in the test system. In case 2, the power dispatch of Generator 1 is not necessary to meet the demand and maintain system balance. The results obtained with the algorithm in GAMS were verified with Digsilent Power Factory software, which concludes that the proposed algorithm in this study is reliable.

Generator 2 dispatched the most power when OTS is implemented in cases 2, 3, and 4, supplying 1.4 [p.u.] to the system, unlike the base case, where it contributed 0.20 [p.u.]. Generator 3 only dispatched power in the base case, which is a notable point for analysis, as the disconnection of a line in the test system reduces losses, making the contribution of this unit unnecessary. On the other hand, generator 4 dispatched more power in case 2. Its dispatch was similar in cases 3 and 4, while its power contribution was minimal in the base case. Similarly, generator 5 dispatches more power in cases 3 and 4 compared to the base case and case 2.

Figure 5 presents a bar graph showing each case study’s active power results. It is highlighted that, despite disconnections due to contingencies or application of the OTS, the system maintains a constant power dispatch. Therefore, a stable operation of the EPS is indicated by minimizing losses in the transmission lines.

4.4. Reactive Power

In the base case, Generator 1 does not dispatch reactive power. However, when OTS is implemented in cases 2, 3, and 4, it supplies reactive power to the EPS, reaching a value of 0.100 [p.u.]. Generator 2 dispatches 0.072 [p.u.] of reactive power in the base case, while, in the cases where OTS is applied, this value significantly increases to 0.500 [p.u.]. On the other hand, Generator 3 maintains a similar level of reactive power dispatch in cases 2, 3, and 4, witha value of 0.195 [p.u.] in the base case. Regarding Generator 4, its reactive power dispatch varies compared to cases 2 and 3 due to the new topological configuration resulting from the optimization. Finally, Generator 5 shows different levels of reactive power dispatch in each case, depending on the best option for loss reduction, see

Table 3. Comparison of reactive power dispatch between GAMS and Power Factory.

	Base Case		Case 2		Case 3		Case 4
G	Gams	PF	Gams	PF	Gams	PF	Gams	PF
1	0.000	0.000	0.100	0.100	0.100	0.100	0.100	0.100
2	0.072	0.073	0.500	0.500	0.500	0.500	0.500	0.500
3	0.195	0.195	0.400	0.400	0.400	0.400	0.400	0.400
4	0.198	0.198	0.108	0.108	0.240	0.241	0.240	0.240
5	0.240	0.240	0.052	0.051	0.010	0.010	0.029	0.030
TOTAL	0.705	0.706	1.160	1.159	1.250	1.251	1.269	1.27

.

In Figure 6, it can be seen that the reactive power dispatch in the system increases when the lines are disconnected in each of the proposed study cases. It is essential to highlight that, in situations where more than two lines are disconnected due to contingencies, a greater dispatch of reactive power is observed, and this is due to the new topological configuration and its stable operation level to minimize losses. Analyzing the total reactive power for each case allows us to understand that disconnecting the transmission lines injects more reactive power into the system to reduce losses and keep the EPS operational.

As can be seen, the proposed OTS manages to redistribute the dispatch of the generators, taking advantage of the transmission lines available in each case study and making optimal use of the technical characteristics of the EPS.

4.5. Losses in Transmission Lines

Two scenarios were analyzed in the active power losses: where the OTS enters and where the optimization does not. The loss percentages for the same transmission line vary, depending on its characteristics and the architecture it adopts with the switching. This occurs due to the load on the connected nodes, which varies due to the change in topology, the parallel susceptance of the lines, and the operating conditions in which the system adapts so as not to leave the established normal limits.

4.5.1. Base-Case Losses

The active power losses in the transmission lines can be observed in

Table 4. Comparison of real power losses in TL between GAMS and Power Factory.

	Base Case		Case 2		Case 3		Case 4
TL	GAMS	PF	GAMS	PF	GAMS	PF	GAMS	PF
1–2	0.016	0.016	9.1 × ${10}^{-24}$	9.1 × ${10}^{-24}$	1.2 × ${10}^{-24}$	1.0 × ${10}^{-24}$	1.5 × ${10}^{-25}$	1.0 × ${10}^{-25}$
1–5	0.011	0.011	3.4 × ${10}^{-23}$	3.2 × ${10}^{-23}$	4.5 × ${10}^{-24}$	4.4 × ${10}^{-24}$	9.2 × ${10}^{-25}$	9.1 × ${10}^{-25}$
2–3	0.005	0.046	4.4 × ${10}^{-24}$	4.3 × ${10}^{-24}$	3.4 × ${10}^{-26}$	3.1 × ${10}^{-26}$	5.5 × ${10}^{-25}$	5.2 × ${10}^{-25}$
2–4	0.006	0.006	1.2 × ${10}^{-23}$	1.1 × ${10}^{-23}$	2.3 × ${10}^{-24}$	2.0 × ${10}^{-24}$	7.0 × ${10}^{-23}$	7.0 × ${10}^{-23}$
2–5	0.003	0.003	7.1 × ${10}^{-23}$	7.1 × ${10}^{-23}$	9.2 × ${10}^{-24}$	9.0 × ${10}^{-24}$	1.7 × ${10}^{-24}$	1.7 × ${10}^{-24}$
3–4	0.003	0.003	1.6 × ${10}^{-24}$	1.6 × ${10}^{-24}$	1.8 × ${10}^{-24}$	1.1 × ${10}^{-24}$	5.9 × ${10}^{-23}$	5.9 × ${10}^{-23}$
4–5	0.002	0.002	5.9 × ${10}^{-24}$	4.9 × ${10}^{-24}$	8.6 × ${10}^{-23}$	8.1 × ${10}^{-23}$	3.9 × ${10}^{-22}$	3.3 × ${10}^{-22}$
6–11	7.17 × ${10}^{-4}$	7.13 × ${10}^{-4}$	1.1 × ${10}^{-24}$	1.0 × ${10}^{-24}$	1.2 × ${10}^{-20}$	1.1 × ${10}^{-20}$	0.000	0.000
6–12	8.06 × ${10}^{-2}$	8.07 × ${10}^{-4}$	6.4 × ${10}^{-20}$	6.1 × ${10}^{-20}$	4.7 × ${10}^{-20}$	4.8 × ${10}^{-20}$	6.4 × ${10}^{-20}$	6.4 × ${10}^{-20}$
6–13	0.002	0.002	1.6 × ${10}^{-19}$	1.5 × ${10}^{-19}$	1.6 × ${10}^{-19}$	1.3 × ${10}^{-19}$	1.6 × ${10}^{-19}$	1.3 × ${10}^{-19}$
9–10	1.13 × ${10}^{-4}$	1.14 × ${10}^{-4}$	3.6 × ${10}^{-20}$	3.0 × ${10}^{-20}$	3.4 × ${10}^{-20}$	3.1 × ${10}^{-20}$	6.6 × ${10}^{-20}$	6.2 × ${10}^{-20}$
9–14	0.0001	0.0001	1.9 × ${10}^{-19}$	1.0 × ${10}^{-19}$	2.1 × ${10}^{-19}$	2.0 × ${10}^{-19}$	1.9 × ${10}^{-19}$	1.8 × ${10}^{-19}$
10–11	1.88 × ${10}^{-4}$	1.86 × ${10}^{-4}$	5.2 × ${10}^{-23}$	5.2 × ${10}^{-23}$	0.000	0.000	1.2 × ${10}^{-20}$	1.0 × ${10}^{-20}$
12–13	7.73 × ${10}^{-5}$	7.7 × ${10}^{-4}$	9.9 × ${10}^{-22}$	9.5 × ${10}^{-22}$	6.0 × ${10}^{-21}$	5.9 × ${10}^{-21}$	1.0 × ${10}^{-21}$	1.0 × ${10}^{-21}$
13–14	6.86 × ${10}^{-4}$	6.83 × ${10}^{-4}$	1.5 × ${10}^{-20}$	1.2 × ${10}^{-20}$	8.8 × ${10}^{-21}$	8.2 × ${10}^{-21}$	1.6 × ${10}^{-20}$	1.4 × ${10}^{-20}$
TOTAL	4.77 × ${10}^{-2}$	4.88 × ${10}^{-2}$	4.6 × ${10}^{-19}$	2.9 × ${10}^{-19}$	4.8 × ${10}^{-19}$	4.4 × ${10}^{-19}$	5.0 × ${10}^{-19}$	4.6 × ${10}^{-19}$

, which shows the values obtained from the GAMS and Digsilent Power Factory software. A minimal error difference in the results can be observed, validating the method of the implemented algorithm. There is a significant percentage difference in losses between the base case and cases 2, 3, and 4, where OTS has already been implemented to minimize transmission-line losses.

The active power losses in the base case, as shown in Figure 7, indicate that lines (1–2), (1–5), (2–3), (2–4), and (2–5) present the highest losses compared to the other lines in the EPS. Given that there is no disconnection of any transmission line, a reduction in losses can be observed in lines (6–12), (6–13), (9–10), (9–14), (10–11), (12–13), (6–11), (4–5), and (13–14). In line (3–4), the losses are lower compared to the rest of the lines in the first case study.

4.5.2. Losses for Cases 2, 3, and 4

When these cases are analyzed, it is important to consider that, in case 2, the loss in lines 3–4 is evaluated during OTS. Still, when the OPF-AC is executed, that line is disconnected due to a modification in the system topology to minimize losses, as illustrated in Figure 8. In case 3, the system is examined under the N − 1 contingency, which involves the disconnection of lines 10–11 before the OTS is executed, as indicated in Table 4.

However, OTS proposes the disconnection of lines 12–13 as the best option for the system, as observed in Figure 2 and Figure 3. In case 4, a random contingency occurs in the transmission lines, specifically in lines 6–11, as detailed in Table 4. Once the fault is identified, OTS is carried out, resulting in a reconfiguration of the EPS with lines 1–5 and 2–3 disconnected. Despite the modifications in the system configuration in all the analyzed cases, this test model continues to operate within the allowed levels.

The losses in cases 2, 3, and 4 show a notable reduction compared to the base case, where OTS is not yet applied. It is observed in Figure 8 that transmission lines (6–12), (6–13), (9–10), and (9–14) present the highest losses in the three cases studied with OTS. When the different cases are analyzed, it is evident that losses are more pronounced during random contingencies. Regarding transmission lines (1–2), (1–5), (2–3), (2–4), (2–5), (3–4), and (4–5), it is observed that the losses are minimal when a change in the system topology is made through OTS, or rather, they have less impact.

When comparing the total losses obtained in GAMS with those calculated in Digsilent in Table 4, the following error margins are observed. The base case has a difference of 0.0011 [p.u.]. In case 2, the difference is 1.68 × ${10}^{-19}$ [p.u.]. In case 3, the difference is 4.40 × ${10}^{-20}$ [p.u.], while in case 4, it is 4.30 × ${10}^{-20}$ [p.u.]. These differences are minimal and acceptable in the study. It is important to emphasize that the optimal power flow performed in the Digsilent Power Factory software was based on the same parameter configuration used in the GAMS software, including generators, transformers, lines, and loads, thereby validating the method used for optimizing TL losses through transmission line switching (OTS).

4.6. Generation Costs

Table 5. Generation cost [USD/H].

	Base Case		Case 2		Case 3		Case 4
G	Gams	PF	Gams	PF	Gams	PF	Gams	PF
TOTAL	3415.895	3415.912	9932.442	9932.481	10,184.682	10,184.690	10,188.192	10,188.195

presents the total generation costs for the test systems after the OTS was implemented. A significant increase in total costs from the base case to the cases where OTS was applied was observed. This increase was progressive, with an increment of 6516.547 USD/H between the base case and case 2, 6768.787 USD/H between case 1 and case 3, and 10,188.192 USD/H between the base case and case 4. These results indicate that the total costs of economic dispatch increase due to the modifications made to the EPS structure.

The results obtained from both the GAMS and Digsilent software show great proximity, indicating a minimal margin of error. In the base case, this margin is only 0.017 USD/H, while in case 2, it is 0.039 USD/H; in case 3, it is 0.062 USD/H, and in case 4, it is 0.003 USD/H. Although the transmission line losses are reduced, this benefit significantly increases generation costs compared to the base case where OTS was not applied.

5. Conclusions

Different authors have studied the OTS, starting from those proposed by Professor Emily Fisher in 2008, in which the objective function was considered to be minimizing the operating costs of the EPS and having the optimal commutation of transmission lines under different operating states. The present work proposes a novel way to modify the objective function. It now considers minimizing losses of the electrical power system, which is the basis for the optimal switching of transmission lines. In addition to an operating condition in a normal state, it was proposed to consider contingencies randomly to verify whether the algorithm responds or not to the optimal computation of transmission lines, resulting in optimization in the face of emerging operating states.

The impact of OTS on the parameters in an OPF-AC was analyzed, considering that economic dispatch costs increase when TLs are taken out of service. Comparing the base case with cases 2, 3, and 4 reveals that total costs increase as active energy losses in the TLs are reduced. It should be understood that this happens because the analysis prioritizes loss reduction over generation-cost reduction. The study focused on the transmission lines and their characteristics to minimize losses.

Reducing the reactive power dispatch in the system does not necessarily equate to a loss reduction. Reactive power in these proposed case studies helps maintain operational parameters within stable limits. Reaction dispatch is needed when the EPS architecture is modified to minimize transmission line losses. OTS must perform strategic dispatches that improve system efficiency and reduce losses by optimizing power flow. Reducing reactive power dispatch can affect voltage stability, potentially causing losses if voltage profiles fall below acceptable limits.

As Table 4 and Figure 8 show from the proposed scenarios, a notable reduction in transmission line losses was observed. It is essential to highlight that active power (Table 2) remained almost constant in all cases, with a slight decrease of 0.32 [p.u.] compared to the scenario where OTS was not implemented. On the other hand, analyzing reactive power dispatch shows a significant value increase. From the base case to case 2, an increase of 0.455 [p.u.] was recorded, followed by a rise of 0.545 [p.u.] from the base case to case 3, and finally an increase of 0.564 [p.u.] from the base case to case 4. These results indicate that, as the number of contingencies increases, it is necessary to increase the reactive power dispatch in order to mitigate losses and maintain the stability of the electrical system.

After the disconnection, the load on the nodes caused variations in operational conditions due to modifications in the EPS topology. The results shown in Figure 8 confirm that loss values vary according to which line was disconnected and its proximity to others, with those closer lines experiencing higher loadability. These characteristics are evident only in cases where an analysis was previously conducted using the OTS algorithm. It was verified that modifying the system topology can either increase or decrease transmission-line losses. Therefore, OTS performs an optimal analysis to establish a minimal system impact, maintaining operations within permissible limits and achieving a 99% reduction in TL losses compared to the base case.