Optimal Management of High-Voltage Line Congestions Using Power Source Redispatching

Abstract

The increasing integration of renewable energy sources (RESs) reduces dependence on conventional generators, thereby minimizing the negative environmental impact of fossil fuels. The distributed location of RESs also affects the voltage profiles (voltage values in network nodes) and reduces power losses. The growing number of RESs connected to the network increases the total installed power in the sources in the power system. This contributes to the periodic excess of generated power. It creates the need to limit generation in conventional power plants and to switch off some RESs. This article proposes an original methodology for optimally managing overloads of high-voltage power lines. The combination of the power flow tracking method and metaheuristic optimization allows for the effective elimination of line overloads. The aim of the calculations is to find the optimal power distribution in the selected sources, which provide minimal power limitation. As a result, this means a minimal reduction in the total generation in RESs. In this way, the effect of eliminating line overloads is achieved at the lowest possible cost of power redispatching. On the basis of the IEEE 118 bus test network, computational cases are considered, which are examples of emergency states.

1. Introduction

The rapid development of renewable energy involves various technical threats. A large number of connected and planned renewable energy sources cause balance problems in the network and overloads of power lines, mainly in the high-voltage network. In practice, it is required that the power network is adapted not only to work in normal conditions but also in emergency situations (e.g., unexpected line or transformer outages). In emergency situations, there is a high risk of overloading network elements. If such lines occur in the area under consideration, they can be overloaded. These lines very often constitute so-called “bottlenecks” in network capacity. The problem of overloading existing lines is more noticeable in a network with high RES saturation. The task of the network operator is then to respond appropriately in order to eliminate the threat that has occurred. However, this requires the appropriate instruments, such as effective software and algorithms that facilitate and improve network operation management and support the making of appropriate decisions. This article proposes a methodology for the optimal redispatching of the power of sources, in the sense of minimal generation limitation in RESs. In order to simultaneously maintain the power balance in the power system, the power loss resulting from the reduction of power in RESs should be compensated by increasing generation in selected conventional sources. Power loss compensation, in order to maintain balance, can be distributed in various ways to selected conventional sources, which is described later in the article. The problem of overloading the power lines considered in the article often appears in practice during the performance of expert opinions. This is one of the reasons for issuing refusals of connection conditions for RESs in Poland. The number of refusals to connect to the grid is very high. An example is the year when it amounted to 7448. A certain solution to these problems is the EU Regulation [1] introduced in 2019, which allows for the redispatching of power. In Poland, an additional regulation [2] was introduced that indicates the need to modernize approximately 200 high-voltage power lines, which, to the greatest extent, limit the capacity of the grid.

The organization of the paper is as follows. Section 1 provides an introduction to the research topic under consideration. Section 2 contains a literature review. Section 3 describes the calculation methodology. Section 4 contains a description of the test network and the calculations and a discussion of the obtained results. Section 5 contains a summary and conclusions.

2. Literature Review

An increasingly common problem is the need to limit the power of RESs. These limitations are caused by various factor–balance issues related to too-high solar generation, too-high wind generation, or line overloads. Other factors that determine the occurrence of network problems are changes in the network structure related to operational switching, the occurrence of emergency states, or even the construction of new lines or the installation of new transformers [3]. Precise data on energy curtailment in Poland are presented in the paper [4]. In Europe, apart from Poland, wind energy limitations are increasingly common in Ireland [5] and Germany [6]. In the world, however, problems with the integration of all wind sources with the system and, if necessary, the need to limit them occur in China [7] and the United States [8]. The issue of limiting the power of sources has recently gained great importance in Poland due to the overloading of high-voltage lines caused by the connection of an increasing number of RESs [9]. In 2023 alone, 7448 refusals to connect facilities to the grid with a planned capacity of over 83 GW were issued in Poland, which constituted 99.3% of applications for the connection of renewable energy sources [10,11]. The need to periodically limit power in RESs also occurs in other countries, e.g., in Great Britain or Germany, which are the leaders in Europe in terms of total installed power in RESs [12,13]. This is due to the fact that, in a given area, a common problem is too many power plants with a relatively high total capacity, which contributes to the occurrence of current overloads [14,15,16]. For this reason, both in the literature and in everyday use by operators, one can increasingly encounter attempts to solve this problem by using methods based on so-called congestion management [17,18,19,20,21,22], redispatching [23,24,25,26,27,28,29,30,31,32], or curtailment power in RESs [33,34,35], as well as methods based on artificial intelligence and optimization methods [36,37,38,39,40,41,42] or the reconfiguration of the network operation [43,44,45,46,47]. Redispatching is a change in the previously established generation size as a result of reduced demand for electricity. The use of the redispatching method is possible thanks to the regulation of the European Parliament and the Council of the European Union of 2019 [1]. The issue of examining the impact of connected RESs on the occurring overloads in high-voltage lines was addressed in article [48]. For this purpose, the authors used the power flow tracking method and the DC method. In study [49], the authors used a method based on the optimal selection of sources in which the power was limited. The authors’ methodology allowed them to avoid a situation in which the power was limited in a source that did not directly affect the overloaded line. Huang [25]. also took up a similar direction of their research. He and his team also proposed a methodology based on the optimal selection of limited sources, and they checked the accuracy of their research on the IEEE 118 test network. The authors of [23] used the technique of minimizing overloads using historical data. They conducted their research on a power system with a high saturation of renewable sources, and the obtained results prove that the proposed methodology can significantly contribute to reducing current overloads in transmission lines. As mentioned earlier, optimization methods are increasingly used to balance the power system. For example, in [40], the Naruto and Kohler team proposed the use of the Genetic Algorithm (GA) to support operators in optimizing the limitation of wind sources with the optimal consideration of regulatory aspects occurring in given countries, as well as the economic aspects. Also in study [50], by Alves and his team, the focus was on the optimal limitation of wind generation using the metaheuristic method of Particle Swarm Optimization (PSO). According to the authors, the obtained results of their research allowed for achieving almost ideal solutions in the scope of limitation in when the method was implemented in large power systems. In Miranda and his team’s [51] research on the energy system problems used the Evolutionary Particle Swarm Optimization (EPSO) method, which combines both the advantages of Evolutionary Algorithms and the advantages of PSO. In [41], the authors proposed the use of the metaheuristic method Flower Pollination Algorithm (FPA). According to the authors, this method allows for finding the optimal point of generation limitation at a given moment. Metaheuristic optimization methods were also applied in the work [32], where the Ant Lion Optimization (ALO) method was used. The same method was used by Mouassa and his team [52] in order to optimally solve the problem of power in the system. In [37], the PSO method was used again, while in [53], the teaching–learning based optimization (TLBO) method was used. The authors of these works conducted research on closed large IEEE test networks, and the results they obtained proved that both of these methods can be successfully used in the power system to limit current overloads. One of the available options for balancing the energy system of a given country is international exchange with another country. The issue of the proper balancing of the network in various energy systems was discussed in detail in the work where the authors focused on various methods of maintaining the network balance [54]. In study [55], one can learn about the techniques of balancing the system used when there is a high saturation of wind energy in Germany. The issue of maintaining a proper balance in a system with a high saturation of renewable energy sources was also focused on in [56], where the authors proposed their own proprietary method based on deep reinforcement learning (DRL). The authors’ methodology was tested on a small closed IEEE network, and the obtained results were extremely interesting. An extremely interesting solution to the problem of optimal power limitation in the system was proposed by the authors of [31], who used fuzzy linear programming for this purpose. The authors tested the proposed methodology on a closed 118-node IEEE network, and the obtained results prove the effectiveness of their solution. The paper [57] discusses the calculation of power system parameters using the grey wolf optimization (GWO) algorithm. This algorithm was implemented by the author to a dozen test functions, and the total calculation times and costs performed using the GWO algorithm were lower than those obtained using other commonly used optimization algorithms. The algorithm proposed by the author is both fast and reliable for power systems. Optimization techniques were also used in the article [58]. The estimation of system parameters is a key factor in analyzing power flows, planning the extension of an electricity system, or ensuring its stability. The authors of this paper proposed the use of hybrid moth–flame optimization (MFO) with particle swarm optimization to estimate transmission line parameters based on different scenarios and mathematical validations. Information on source scheduling in the power system can be found in articles [59,60]. For example, in the [60] article, the authors focused on proposing hybrid policy-based reinforcement learning (HPRL) to realize optimal power system operation.

There is a lack of research in the literature on identifying the RESs that are responsible for the resulting overloads of power lines in a high-voltage network saturated with distributed sources. Knowledge on this subject will allow network operators to effectively relieve the overloaded lines. Moreover, this will be an optimal action. Therefore, it is necessary to include the optimization task in this process. Due to the specificity of the issue (the possibility of a divergent iterative process, the need to take into account emergency states and other specific requirements of network operators), the use of classic optimization methods may prove ineffective. Using metaheuristics for this purpose therefore seems to be justified. The authors found a gap in the literature and decided to fill it with their own research, resulting not only from theory but primarily from practical experience. Of course, this is not the only proposal for solving the analyzed problem. Other methods can also be used that are based on different correlation coefficients or methods based on artificial intelligence, e.g., machine learning. In order to use them, many different optimal states of the network must be considered in advance in order to teach the machine. This is another research area that will be considered in the future.

3. Description of the Research Methodology

The considered research problem concerns the elimination of power line overloads by means of an optimal change in the power distribution in RESs (minimal power reduction in RESs) while maintaining the power balance in the network. The assumption is that, at the current value of power generated in the considered RESs (which is assumed as the maximum possible generation at a given moment of time), such an optimal power distribution in these sources is sought that will ensure the elimination of the resulting overloads of power lines. This means that the obtained power values in RESs (as a result of solving the optimization problem) will always be lower than their current generation capabilities. The necessary reduction of power in RESs will be at the same time minimal in a given operating state, and thus, the total power in RESs will be maximal from the point of view of eliminating line overloads and maintaining the power balance in the power system. The maximum generation at a given moment of time is expected by investors. This is also beneficial from the point of view of network operators, who will use power redispatching to a lesser extent and, thus, incur lower costs of this action. Therefore, those RESs that contribute to congestion to the least extent will be rewarded. During the optimization process, their generation will be minimally reduced. A greater power reduction will occur in those RESs that contribute to the occurrence of congestion to a greater extent. It should also be noted that each change in generation resulting from changes in atmospheric conditions (sunlight in the case of photovoltaic farms and wind speed in the case of wind farms) will involve resolving the optimization task and searching for the optimal vector of power generated in the RESs under the given conditions.

The general algorithm of the proposed calculation methodology is presented in Figure 1.

The optimal change of power generated in both the RESs and the sources responsible for maintaining the power balance is referred to as power redispatching. This is a periodic action, used after other possibilities have been exhausted, because it involves the need for financial compensation for investors. The possibility of redispatching the power of sources is enabled by the EU regulation [1], which appeared in 2019. Previous methods of limiting power in RESs consisted either of switching off sources indicated by network operators or of proportionally limiting their generated power. These were not optimal actions. This is a research gap in the literature. Works devoted to this topic focused mainly on optimizing the operation of RES sources without the prior selection of those that are most responsible for line congestion. This article proposes an original methodology aimed at minimizing power reduction in RESs. This means maximizing the total power generated in the analyzed RESs. Generally speaking, the main assumption is to eliminate line congestion with minimal generation limitation in the indicated sources.

Generally speaking, the calculations are carried out in two stages. After reading the test network model and indicating the sources responsible for maintaining the power balance, in the first step, the sources responsible for the occurring line overloads are selected. The method of tracking active power flows is used for this purpose. The essence of the method is to determine the vector of nodal flow P and its distribution matrix A_u, which relates nodal flows to specific powers of P_G sources, according to the following equation [61]:

$$A_{\mathrm{u}}\cdot P=P_{\mathrm{G}}$$

(1)

As a result of the inversion, nodal powers are determined, according to the following relationship:

$$P=A_{\mathrm{u}}^{-1}{P}_{\mathrm{G}}$$

(2)

Power flowing through branch i–l, can be determined from the following relation:

$$P_{il}\approx {\displaystyle \frac{P_{il}}{P_i^{\Rightarrow }}}\cdot {\displaystyle \sum _{k\in N}{a^{\prime }}_{\mathrm{u}ik}\cdot P_{\mathrm{G}k}}={\displaystyle \sum _{k\in N}\left(u_{il,k}\cdot P_{\mathrm{G}k}\right)}=u_{il,1}\cdot P_{\mathrm{G}1}+u_{il,2}\cdot P_{\mathrm{G}2}+\dots +u_{il,N}\cdot P_{\mathrm{G}\mathrm{N}}$$

(3)

Based on this method, the coefficient u_il,k is determined, which is the share of the source connected in node k in the load of branch i–l. The coefficient u_il,k is determined from the following relationship [61]:

$$u_{il,k}={\displaystyle \frac{P_{il}}{P_i^{\Rightarrow }}}\cdot {a^{\prime }}_{\mathrm{u}ik}$$

(4)

where $P_i^{\Rightarrow }$ means active power flowing through node i, and ${a^{\prime }}_{\mathrm{u}ik}$ means the expressions of the matrix $A_{\mathrm{u}}^{-1}$, which is the inverse matrix of the matrix A_u, the expressions of which a_uik are determined from the following formula [61]:

$$a_{uik}=\left\{\begin{array}{ll}1,\hfill & i=k\hfill \\ -{\displaystyle \frac{\left|P_{ki}\right|}{P_k^{\Rightarrow }}},\hfill & P_k^{\Rightarrow }\ne 0\hfill \\ 0,\hfill & \mathrm{in} \mathrm{other} \mathrm{cases}\hfill \end{array}\right.$$

(5)

where P_ki is the active power in branch i–k (taken from node k), and $P_k^{\Rightarrow }$ is the active power flowing through node k. This quantity means the sum of power flows towards node k.

For further calculations, the sources whose u_il,k participation coefficients are the largest are selected.

In the second step, optimization calculations are performed using one of the metaheuristic optimization methods. The objective functions (F_obj1(x), F_obj2(x)) are proposed to maximize the total power in RESs (the number of which is N), according to the following formulas:

$$F_{\mathrm{obj}1}\left(x\right)={\displaystyle \sum _{j=1}^N{P}_{\mathrm{G}j}}=P_{\mathrm{G}1}+P_{\mathrm{G}2}+\dots +P_{\mathrm{GN}}$$

(6)

$$F_{\mathrm{obj}2}\left(x\right)={\displaystyle \sum _{j=1}^N{w}_j\cdot P_{\mathrm{G}j}}=w_1\cdot P_{\mathrm{G}1}+w_2\cdot P_{\mathrm{G}2}+\dots +w_N\cdot P_{\mathrm{GN}}$$

(7)

where x is a vector of decision variables that contains the values of active powers (P_Gj) generated in the RESs, and w_j is weights whose values are given later in the article (in Section 4.2).

This form of the objective function ensures the maximization of the total power generated in RESs, while eliminating power line overloads. As a result, this means a minimal reduction of the current total generation in RESs. The optimal power distribution is sought, in the sources indicated in the first step, which ensures minimal power limitation. Thus, the load shedding effect is achieved at the lowest possible cost of power redispatching.

The optimization task is, therefore, to maximize the objective function by finding the optimal value of the vector x, as follows:

$$F_{\mathrm{obj}}\left(x\right)\Rightarrow \max$$

(8)

while meeting the equality and inequality constraints according to the following formulas:

$$\begin{array}{l}{P}_{\mathrm{G}i}-P_{\mathrm{L}i}-f_{\mathrm{P}}(\mathit{U}\mathit{,}\mathit{\delta })=0\\ Q_{\mathrm{G}i}-Q_{\mathrm{L}i}-f_{\mathrm{Q}}(\mathit{U}\mathit{,}\mathit{\delta })=0\\ P_{\mathrm{Gmin}}\le P_{\mathrm{G}}\le P_{\mathrm{Gmax}}\\ Q_{\mathrm{Gmin}}\le Q_{\mathrm{G}}\le Q_{\mathrm{Gmax}}\\ U_{\min }\le U\le U_{\max }\\ I\le I_{\max }\\ S_{\mathrm{T}}\le S_{\mathrm{nT}}\end{array}$$

(9)

During the calculations, both equality and inequality constraints are taken into account. The equality constraints assume the maintenance of the power balance in the network and fulfill the basic nodal equations when determining the power flows. These equations take into account the active and reactive powers generated (P_Gi, Q_Gi) and consumed (P_Li, Q_Li), as well as the voltages and their angles (U, δ), which are the result of solving the power flow problem (according to the functions f_P(U, δ) and f_Q(U, δ)), using the Newton–Raphson method. The inequality constraints are the minimum and maximum values of voltages in nodes (U_min, U_max), the minimum and maximum values of power generated in sources (P_Gmin, P_Gmax, Q_Gmin, Q_Gmax), and the permissible load capacities of lines and transformers (I < I_max, S_T < S_nt).

To solve the optimization problem, the metaheuristic Algorithm of the Innovative Gunner (AIG) [62] was used. To determine new solutions of vector x, in the next k-iteration, the AIG algorithm uses the following relationship [62]:

$$x_l^{\left(k+1\right)}=x_l^{\left(k\right)}\cdot g_l\left(\xi \right)$$

(10)

where $x_l^{\left(k+1\right)}$ denotes the value of the variable x_l of vector x in iteration (k + 1), $x_l^{\left(k\right)}$ denotes the value of the variable x_l of vector x in iteration (k), and $g_l\left(\xi \right)$ denotes special multipliers, the number of which (in the AIG algorithm) is 2. These are special functions that have the form g_l1(ξ₁) = cosα when α < 0 and g_l1(ξ₁) = (cosα)⁻¹, when α > 0 and g_l2(ξ2) = cosβ, when β < 0, and g_l2(ξ₂) = (cosβ)⁻¹ when β > 0. α and β are the so-called correction angles, drawn from the variable interval (−α_max, α_max) and (−β_max, β_max) using the normal distribution. Therefore, the generation of new solutions is carried out using the following relationship [62]:

$$x_l^{\left(k+1\right)}=x_l^{\left(k\right)}\cdot g_{l1}\left({\xi }_1\right)\cdot g_{l2}\left({\xi }_2\right)$$

(11)

The block diagram of the algorithm is shown in Figure 2.

The maximum adopted number of iterations is 1000. The size of the AIG algorithm population corresponds to the number of decision variables. In each iteration, the power flow problem is solved. As mentioned earlier, during the optimization process, the power in RES is reduced in order to eliminate the resulting line overloads. The resulting loss of generated power ΔP_G in the system should be supplemented by an increase in generation in conventional sources (their number is L). This is conducted by dividing the power deficit ΔP_G into conventional sources. These sources are assigned appropriate p_fi (participation factor) coefficients. For each source k that takes part in covering the power imbalance, the participation coefficient w_k is determined according to the following equation:

$$w_k={\displaystyle \frac{p_{\mathrm{f}k}}{{\displaystyle \sum _{i=1}^L{p}_{\mathrm{f}i}}}}$$

(12)

Taking into account the power loss ΔP_G and the w_k coefficients, the changes in power in the sources responsible for the balance are determined according to the following equation:

$$\Delta P_{\mathrm{G}k}=w_k\cdot \Delta P_{\mathrm{G}}$$

(13)

Different methods of determining the p_fi coefficients can be considered. This article uses the following five methods to distribute the power imbalance:

• Case 1 (C1)–the power deficit is covered only by the source connected in the balancing node theoretical case;
• Case 2 (C2)–the distribution of the power deficit is determined in relation to the maximum power of the sources responsible for maintaining the balance;
• Case 3 (C3)–the distribution of the power deficit is determined in relation to the difference between the maximum and current power of the sources responsible for maintaining the balance;
• Case 4 (C4)–the distribution of the power deficit is determined using the same coefficients for each source responsible for maintaining the balance;
• Case 5 (C5)–power deficit distribution is determined by different coefficients for each source responsible for maintaining the balance, which is determined arbitrarily.

Maintaining the power balance in the power system in practice is usually covered by conventional power plants. This can be performed in different ways, as described by cases C2, C3, C4, and C5. Of the methods mentioned, cases C2 and C3 are the most realistic. Therefore, case C1 is only a theoretical case that has been added for comparison with other methods. The proposed methodology can be used by network operators. The time to obtain the result in not very extensive networks is several minutes. In extensive networks (large in size), the number of decision variables and constraints can be significantly greater. Therefore, the computation time can be longer. The speed of the computational process also depends on the power and efficiency of the computing unit. In dispatch centers, computing units are much more efficient and faster than standard computers used for research at universities. The costs of implementing such a solution should not be high. The proposed methodology can be easily implemented through integration with the existing power grid management system.

4. Calculation Results and Discussion

4.1. Test Network Description

In this article, a modified IEEE 118 test network was used, which is presented in Figure 3 [63]. The network has a closed structure. In this network, the voltage levels were changed to 400 kV, 220 kV, and 110 kV. The line cross-sections, line current-carrying capacity, total load, and generation were also changed. The purpose of these changes was to adapt the network parameters to those occurring in the Polish power grid. The vast majority of lines in the Polish power system were designed for 40 °C. Most of the lines in the IEEE 118 test network were adapted to the same temperature, but some of them were adapted to a higher operating temperature, i.e., 60 °C or up to 80 °C. These lines are additionally marked in Figure 3. For better clarity of the drawing, different colors were used to indicate voltage levels and their corresponding wire cross-sections. In the 110 kV network, typical wire cross-sections were used, i.e., 120 mm², 185 mm², and 240 mm². In the 220 kV network, wire cross-sections of 525 mm² were used, while in the 400 kV network, they were 2 × 525 mm². Red ink indicates 400 kV lines. Green indicates 220 kV lines, while the other colors used correspond to 110 kV lines. RESs were also connected in selected network nodes. The balancing node was marked with number 10.

4.2. Computational Case

The selected operating state of the test network from Figure 3 was considered. The emergency state was analyzed as a shutdown of the 400 kV transmission line connecting nodes 8-30. As a result of the emergency shutdown of this line, five 110 kV lines were overloaded. These are the lines connecting nodes 100-103 (line overload of 35%), 17-113 (line overload of 26%), 23-24 (line overload of 10%), 94-95 (line overload of 7%), and 16-17 (line overload of 3%). These are typical values of current overloads that occur in real networks when performing computational analyses. Sometimes, overloads are observed on lines located far from the source connection point. This state is illustrated in Figure 3.

For the emergency case of network operation, shown in Figure 3, the matrix A_u, the individual expressions of which are determined from relation (5), and the matrix inverse to it $A_{\mathrm{u}}^{-1}$ are illustrated in Figure 4. Figure 4a,c shows the filling density of both matrices. Figure 4b,d additionally illustrates the values of the individual terms of both distribution matrices.

In such a situation, the network operator should take appropriate steps to eliminate the threat. One of the remedial actions is the proposed methodology described in point 3. According to this methodology, in the first step, all sources responsible for the load of the previously indicated 110 kV lines were designated. For this purpose, the active power flow tracking method was used, which resulted in determining the values of the u_il,_k coefficients using Equation (4). Additionally, the power values flowing through individual lines from the selected RESs were calculated. These results are presented in

Table 1. Share of selected RES in the line load between nodes 16 and 17.

Line	P16-17	Line Sources Affecting the Line
From–To	MW	G-17	G-24	G-25	G-26	G-27	G-31	G-72	G-113
16-17	72.38	Power generated by sources, MW
		37.507	49.602	99.674	60.259	9.212	13.871	49.813	69.719
		Source coefficients uil,k according to Equation (1)
		0.365	0.008	0.156	0.310	0.002	0.133	0.004	0.316
		Power flowing through the line, originating from a given source, MW
		13.695	0.381	15.551	18.679	0.022	1.847	0.187	22.036
		Power contribution of the source to the line load 16-17 [%]
		18.92	0.53	21.48	25.80	0.03	2.55	0.26	30.44

,

Table 2. Share of selected RES in the line load between nodes 17 and 113.

Line	P17-113	Line Sources Affecting the Line
From–To	MW	G-113
17-113	56.26	Power generated by the source, MW
		67.719
		Source coefficients uil,k according to Equation (1)
		0.807
		Power flowing through the line from a given source, MW
		56.256
		Power contribution of the source to the line load 17-113 [%]
		100

,

Table 3. Share of selected RES in the line load between nodes 23 and 24.

Line	P23-24	Line Sources Affecting the Line
From–To	MW	G-24	G-72
23-24	62.40	Power generated by the source, MW
		49.602	49.813
		Source coefficients uil,k according to Equation (1)
		0.844	0.412
		Power flowing through the line from a given source, MW
		41.864	20.539
		Power contribution of the source to the line load 23-24 [%]
		67.09	32.91

,

Table 4. Share of selected RES in the line load between nodes 94 and 95.

Line	P94-95	Line Sources Affecting the Line
From–To	MW	G-91	G-99	G-100	G-103	G-105	G-110	G-111	G-112	G-125
94-95	43.64	Power generated by the source, MW
		48.867	91.456	59.878	67.025	9.576	11.760	21.964	10.621	95.815
		Source coefficients uil,k according to Equation (1)
		0.047	0.033	0.223	0.177	0.010	0.074	0.074	0.074	0.101
		Power flowing through the line, originating from a given source, MW
		2.305	3.045	13.354	11.873	0.096	0.871	1.627	0.787	9.686
		Power contribution of the source to the line load 94-95 [%]
		5.28	6.98	30.60	27.20	0.22	2	3.73	1.80	22.19

and

Table 5. Share of selected RES in the line load between nodes 100 and 103.

Line	P100-103	Line Sources Affecting the Line
From–To	MW	G-103	G-110	G-111	G-112
100-103	53.33	Power generated by the source, MW
		67.025	11.760	21.964	10.621
		Source coefficients uil,k according to Equation (1)
		0.689	0.273	0.273	0.273
		Power flowing through the line from a given source, MW
		46.209	3.215	6.006	2.904
		Power contribution of the source to the line load 100-103 [%]
		79.21	5.51	10.30	4.98

. Each of these tables contains separate results for each overloaded line. In each table, the RESs (conventional sources that participate in maintaining the power balance and therefore cannot be considered as decision variables in the optimization process) that have the greatest impact on the load of the 110 kV line are marked in blue. These sources are the decision variables in the optimization process. Debates on whether all RESs should be used to limit power (even those with a very low participation factor) or only those that have the greatest impact on overloading a given line were conducted in article [14] from the literature list. In the case of power reduction in sources with a small contribution to line overload, they have no measurable significance for its effective load relief. Taking into account only the sources with the greatest share in the power flow on a given line allows for minimising the number of RESs covered by power reduction. This reduces the size of the task due to the smaller number of decision variables, which allows for a shorter calculation time and allows us to limit the number of signals necessary to be sent to the sources during the power redispatching process. This increases the effectiveness and efficiency of the entire process. The optimization is carried out in the next part. The effectiveness of the proposed methodology is also proven.

As mentioned earlier, in the second step, the optimization task described in point 3 is solved using relations (6), (7), and (8). The decision variables are only those RESs that have the greatest impact on the overloaded lines. According to the results in Table 1, Table 2, Table 3, Table 4 and Table 5, these are sources G-113, G-24, and G-103. In the case under consideration, the detailed form of the objective function is as follows:

$$F_{\mathrm{obj}1}\left(x\right)={\displaystyle \sum _{j=1}^3{P}_{\mathrm{G}j}}=P_{\mathrm{G}\text{-}113}+P_{\mathrm{G}\text{-}24}+P_{\mathrm{G}\text{-}103}$$

(14)

It is worth mentioning that, in the case of the 110 kV line of relation 16-17, the highest u_il,k coefficient is possessed by source G-17 (u16-17,17 = 0.365), not G-113 (u_16-17,17 = 0.365). However, due to the fact that source G-113 generates much greater power (P_g = 69.719 MW) than source G-27 (P_g = 37.507 MW) it has a greater impact on the load on the line between nodes 16 and 17. This is proven by the value of power flowing through this line (22.036 MW), which comes from the G-113 source. The power from the G-27 source, flowing through the 16-17 line, is 13.695 MW, which is significantly lower. Attention should also be paid to the lines between nodes 94 and 95. The biggest part in its load is taken by the G-100 source (u_94-95,100 = 0.223 in Table 4). Since this is a conventional source, taking part in maintaining the power balance, the renewable energy source with the largest share, i.e., G-103 (u_94-95,103 = 0.177 in Table 4), was taken for optimization. As mentioned earlier, conventional sources do not take part in the optimization process, but they provide the power balance in the system. These sources, together with their p_fk coefficients, are presented in

Table 6. List of sources responsible for maintaining the power balance with p_fk coefficients for variants C1, C2, C3, and C4.

	G-99	G-49	G-59	G-65	G-116	G-100	G-125	G-38	G-8	G-26	G-64
C2	600	200	300	800	800	200	300	400	1000	600	400
C3	508.3	139.6	255	719.4	719.4	102.8	204.1	359.7	899.3	539.6	359.7
C4	10	10	10	10	10	10	10	10	10	10	10
C5	91.7	60.4	45	80.6	80.6	97.2	95.9	40.3	100.7	60.4	40.3

. The values of w_k coefficients, calculated for different cases using Equation (12), are presented in Table 6. In case 1 (C1), the source connected to node 10 was responsible for maintaining the power balance. The remaining cases are presented in Table 6 and

Table 7. List of sources responsible for maintaining the power balance along with the w_k coefficients determined using Equation (7) for variants C1, C2, C3, and C4.

	G-99	G-49	G-59	G-65	G-116	G-100	G-125	G-38	G-8	G-26	G-64
C2	0.107	0.036	0.054	0.143	0.143	0.036	0.054	0.071	0.179	0.107	0.071
C3	0.106	0.029	0.053	0.15	0.15	0.021	0.042	0.075	0.187	0.112	0.075
C4	0.091	0.091	0.091	0.091	0.091	0.091	0.091	0.091	0.091	0.091	0.091
C5	0.116	0.076	0.057	0.102	0.102	0.123	0.121	0.051	0.127	0.076	0.051

.

In the optimization calculations, a vector of decision variables x (power values in previously selected RESs) was sought to eliminate the threats in the form of overloaded lines.

Table 8. List of RESs along with their generated power P_g and optimal power values P_Gopt, which were determined in the optimization process.

RES	PG, MW	Calculation Cases/PGopt, MW
		C1	C2	C3	C4	C5
G-24	50	50	38	39	37	38
G-113	70	42	36	36	35	36
G-103	90	48	47	48	36	25
Sum of power ΣPG, MW	210	140	121	123	108	99
Power reduction ΣPG-ΣPGopt, MW		70	89	87	102	111

presents the results for five different cases (C1, C2, C3, C4, and C5), which were described earlier. It should be noted that, in case 1, bus 10 was assumed as the balancing node.

Based on the obtained results, it can be stated that the minimum power reduction takes place in case 1 (C1) when assuming balancing bus 10. The balancing node is assumed to be the node located at the end of the network. However, this is a theoretical case that does not occur in practice. If the power loss caused by the optimal power reduction in RESs is covered by conventional sources, then the most favorable is case 3 (C3). It is the closest to reality. In reality, conventional sources are responsible for the power balance in the power system. Case 2 (C2) is also favorable, but in this operating state, it seems to be less favorable than C3. In practice, therefore, both cases can be considered, depending on the considered failure state. Cases C4 and C5 are less favorable because the assumed power distribution factors are assumed arbitrarily, i.e., they have the same values (C4) or values resulting from the current generation (C5).

In most of the calculation cases, the active constraints in the optimization problem were the permissible current capacities of 110 kV power lines between nodes 23 and 24, 94 and 95, and 17 and 113. The results obtained are illustrated in Figure 5.

Figure 6 shows the course of the best values of the objective function in each iteration for the calculation case C3. The results obtained using the AIG algorithm were compared with the results obtained additionally using the Cuckoo Search (CS) algorithm and Particle Swarm Optimization.

The results (objective function values) obtained using the three metaheuristic algorithms are similar as shown in Figure 6. The values of decision variables (power generated in the selected RES sources) are also comparable and are presented in Table 8. Based on Figure 6, it can be seen that the algorithm finds a solution after approximately 800 iterations.

In the next step, calculations are performed once again for the best case (C3), assuming different weight values in the objective function (7), according to the following equation:

$$F_{\mathrm{obj}2}\left(x\right)={\displaystyle \sum _{j=1}^3{w}_j\cdot P_{\mathrm{G}j}}=w_1\cdot P_{\mathrm{G}\text{-}113}+w_2\cdot P_{\mathrm{G}\text{-}24}+w_3\cdot P_{\mathrm{G}\text{-}103}$$

(15)

The weights are determined based on the amount of congestion of individual power lines. In general, it can be said that the operator may decide to make the value of the power limitation in RESs dependent on the degree of line congestion or another indicator. In this way, this procedure seems to be more objective and expected from the point of view of investors. This means that source G-113 has the greatest influence on the overload of relation lines 16-17 (I_{%overload_16-17} = 3%) and 17-113 (I_{%overload_17-113} = 26%), source G-24 has the greatest influence on the overload of relation lines 23-24 (I_{%overload_23-24} = 10%), and source G-103 has the greatest influence on the overload of relation lines 94-95 (I_{%overload_94-95} = 7%) and 100-103 (I_{%overload_100-103} = 35%). Thus, the contribution (c) of each source to the line congestion can be determined as follows:

$$\begin{array}{l}{c}_{113}={\displaystyle \frac{I_{\%\mathrm{overload}\_16\text{-}17}+I_{\%\mathrm{overload}\_17\text{-}113}}{I_{\%\mathrm{overload}\_16\text{-}17}+I_{\%\mathrm{overload}\_17\text{-}113}+I_{\%\mathrm{overload}\_23\text{-}24}+I_{\%\mathrm{overload}\_94\text{-}95}+I_{\%\mathrm{overload}\_100\text{-}103}}}\\ c_{113}={\displaystyle \frac{3+26}{3+26+10+7+35}}=0.358\\ c_{24}={\displaystyle \frac{I_{\%\mathrm{overload}\_23\text{-}24}}{I_{\%\mathrm{overload}\_16\text{-}17}+I_{\%\mathrm{overload}\_17\text{-}113}+I_{\%\mathrm{overload}\_23\text{-}24}+I_{\%\mathrm{overload}\_94\text{-}95}+I_{\%\mathrm{overload}\_100\text{-}103}}}\\ c_{24}={\displaystyle \frac{10}{3+26+10+7+35}}=0.123\\ c_{103}={\displaystyle \frac{I_{\%\mathrm{overload}\_94\text{-}95}+I_{\%\mathrm{overload}\_100\text{-}103}}{I_{\%\mathrm{overload}\_16\text{-}17}+I_{\%\mathrm{overload}\_17\text{-}113}+I_{\%\mathrm{overload}\_23\text{-}24}+I_{\%\mathrm{overload}\_94\text{-}95}+I_{\%\mathrm{overload}\_100\text{-}103}}}\\ c_{103}={\displaystyle \frac{7+35}{3+26+10+7+35}}=0.519\end{array}$$

(16)

Based on the obtained values, it can be stated that the smallest share in line congestion is from source G-24 and the largest is from G-103. Therefore, generation in source G-24 should be limited to the smallest extent, and in source G-103, it should be limited to the greatest extent. Therefore, source G-103 should have the smallest weight, and source G-24 should have the largest. Various weight values were analyzed, and it turns out that only the weight value (w₃) for source G-103 equal to 0.06 and lower ensures a smaller reduction of power in source G-24 compared to the results obtained previously. For example, for weight values equal to w₁ = 0.38, w₂ = 0.56, and w₃ = 0.06, the optimal power values are P₂₄ = 40 MW, P₁₁₃ = 36 MW, and P₁₀₃ = 38 MW. For weights equal to w₁ = 0.39, w₂ = 0.56, and w₃ = 0.05, the optimal power values are P₂₄ = 45 MW, P₁₁₃ = 36 MW, and P₁₀₃ = 0 MW. This means that, in the analyzed case, the optimization results are more favorable for the G-24 source only after the G-103 source is switched off. The optimal power value in source G113 does not change, which means the maximum value from the point of view of eliminating line congestion is between nodes 16 and17 and 17 and 113. If the weight value (w₂) for the G-24 source is less than 0.16 (np. w₁ = 0.75, w₂ = 0.15, w₃ = 0.1), then the optimal power values are P₂₄ = 0 MW, P₁₁₃ = 45 MW, and P₁₀₃ = 44 MW. This means that the optimal power value for the G113 source is higher but at the cost of switching off the G-24 source. In general, at G-24 weight values lower than 0.19 the optimal G-113 power starts to increase, and at values lower than 0.16, it reaches its maximum value (with G-24 disabled). For example, for w₁ = 0.72, w₂ = 0.18, and w₃ = 0.1, the optimal power values are P₂₄ = 35 MW, P₁₁₃ = 37 MW, and P₁₀₃ = 48 MW. Increasing the weight for G-103 (w₃) does not increase its generation in relation to the value obtained using the objective function F_obj1. Of course, each case should be analyzed individually. Priorities should be established after a thorough analysis of the given situation.

The calculation time depends on the number of decision variables, the number of iterations, the size of the population and the size of the model of the network under analysis. The result time for the IEEE 118 bus network is approximately 2 min. Therefore, it is reasonable to look for other methods to reduce this time. The calculations were performed on a computer with a 13th Gen Intel Core i9 3.0 GHz processor, 64 GB RAM. As previously mentioned, in future studies, the authors intend to use methods that will also enable the elimination of risks in real mode (online). It should also be assumed that the national dispatching center, where the calculations according to the proposed method would be performed, has a much higher data processing capacity than the ordinary computer used in this study to support research and office work.

5. Conclusions

One of the main technical problems resulting from the increasing number of RESs connected to the power grid is line overload and problems with maintaining the power balance. They are particularly troublesome in conditions favorable to high generation. The relatively large dispersion of RESs causes the occurrence of areas with high saturation of these sources in practice. The advantage of power generated in these areas contributes to its flows towards other areas. In emergency states, consisting of unforeseen line or transformer outages, other branches may be overloaded. The task of the network operator then consists of an appropriate response in order to eliminate the threat. The high-voltage network has a grid structure (closed). It is difficult in such a network to intuitively indicate those elements that have the greatest impact on the occurring overloads. Appropriate methods and algorithms that enable making the right decision become helpful in such a case. This article addresses this issue. It proposes a methodology that allows for eliminating branch overloads by redispatching source power. It will allow operators to manage the network more efficiently and make appropriate decisions more easily. The novelty of the proposed methodology consists of combining the method of tracking power flows with metaheuristic optimization, taking into account different methods of balancing power in the power system in emergency conditions. The method of tracking active power flows makes it possible to identify those sources that are most responsible for line overloads. Using metaheuristic optimization, it is possible to find the optimal distribution of power in the RES to eliminate overloads. In addition, different ways have been used to maintain the power balance in the system by varying the power generated at conventional power plants. This is performed to minimize the reduction of power in the RESs selected in advance in order to minimize the economic impact of this procedure (redispatch). From several calculation variants considered, the operator has the option of choosing the best one. The authors’ future work will be focused on sensitivity analysis by using different correlation coefficients between the power generated in the RESs and the power flows in the lines. Another promising approach seems to be the use of operational procedures such as network reconfiguration. In fact, in order to make online power limiting decisions, it is necessary to shorten the time it takes to obtain the computational results. The disadvantage of metaheuristic optimization is the relatively long computational time. Therefore, in the future, the authors also intend to reach for methods based on artificial intelligence, e.g., machine learning. However, in order for the machine to be optimally selected, various cases must be analyzed. After considering many different cases and obtaining results from optimization, it will be possible to use them as training data for the machine. The optimization results will then be the input data for the machine. After training, it will be possible to say that the machine works optimally. It will be possible to use it online. The trained machine should provide a solution practically immediately, in real time.

Each possible action, supported by its effectiveness, is worth attention, especially if it contributes to the effective elimination of threats.