Elimination of Line Overloads in a Power System Saturated with Renewable Energy Sources

Abstract

The increasing number of renewable energy sources (RESs) connected to power grids contributes to the emergence of not only balancing problems but also technical ones, such as the overloading of power lines. If renewable sources with a high generation level are planned to be connected in the area under consideration, then a large number of significant overloads should be expected, especially during contingency analysis. As a rule, high-voltage networks have a mesh topology, which is why the concept of using advanced mathematical algorithms was developed, with the help of which the resulting threats can be eliminated. This article presents a proposal for a new method of eliminating line overloads and determining the currently available nodal generation levels. Its innovation is a new method of eliminating problems related to the capacity of power grids. The high efficiency of the method results from the appropriately defined response of properly selected RES sources to the state of network congestion. The problem under consideration is illustrated with the example of a modified IEEE 118-bus test network. In order to eliminate line overloads, the article proposes a two-stage approach. In the first step, the sources that are most responsible for the occurring overloads are determined. In the second step, a metaheuristic algorithm is used to solve a nonlinear optimisation problem with constraints. This task involves reducing the power generated in the sources selected in the previous step in such a way that the resulting line overloads are eliminated, and, at the same time, the total value of the curtailed generation is minimal.

1. Literature Review

The subject of this article concerns problems related to the overloading of existing power lines as a result of the growing number of renewable energy sources connected to the grid as well as energy storage facilities. When talking about energy storage, we should refer to states in which energy is discharged, i.e., fed into the grid. There are various works in the literature covering the subject under consideration. It is often referred to as congestion management and redispatching. An example of an overview, list, and description can be found in works [1,2,3,4,5]. Some of them concern only problems related to overcurrents in conditions of high-RES saturation. Others, however, raise the problem of determining the available transfer capability (ATC). The problem of overloading lines and transformers is then treated as one of the limitations.

Different types of methods are proposed for the analysis of the problem in question. Among others, nonlinear classical and metaheuristic optimisation with constraints [6,7,8,9], power flow tracking [10,11], selected machine learning methods [12], and optimisation supported by machine learning [13,14] are used. The DC method is also used in combination with linear programming [15,16], as well as fuzzy logic, fuzzy reasoning, and expert systems [17,18,19]. For some time, the so-called redispatching method has been used in order to reduce the power in RESs. The possibility of redistributing the power of such sources, with the idea of, among others, eliminating overloads, is provided by Regulation 2019/943 of the European Parliament and of the EU Council [20]. From the point of view of network operators, redispatching is treated reluctantly due to the relatively high costs of such actions.

Generally speaking, the subject of powerline overloads appears in many works that deal with various topics, and the problem of overcurrent is either the main issue or has been included as a limitation of transmission capacity.

The method used by the authors of this article is original and not found in other works. The authors also dealt with this problem in [16,21,22,23], where a slightly different methodology was used. The problem of overloading has been solved with a simplified approach, namely, a combination of the classic method of tracking power flows and linear optimisation with constraints. This method is called the iterative tracking of active power flows method, a detailed description of which can be found in [16]. The applied approach is characterised by the short amount of time it takes to obtain a solution and the simplicity of its implementation. Such a solution is also associated with some simplifications and inaccuracies, such as not directly taking into account reactive power flows. Reactive power is then indirectly taken into account via cyclical calculations in several iterations. It should be noted that reactive power sometimes significantly affects the values of currents in power lines, which is why the approach proposed in this article is much better and more effective.

In the doctoral thesis in [16], the DC method of determining power flows in combination with linear optimisation with constraints was also considered. This approach can be used for estimations rather than for exact analyses. Inaccurate results may lead to inappropriate conclusions. In practice, this is unacceptable, as it has a direct impact on the operational security of the power system.

As already mentioned in the literature, there are many works on the elimination of overloads [6,24,25,26,27] and analyses related to determining the available transmission capacities of power grids [7,8,28,29,30,31], as well as transmission capacities between areas, e.g., [32,33,34,35,36,37].

Various optimisation methods have been used to solve the problems of overloading power lines, which can be found in [6,7,8,31,38]. It should be noted that metaheuristic optimisation is used more and more often due to its effectiveness, simplicity of implementation, flexibility, universality, and avoidance of local optima. Various objective functions are used, such as minimising the costs of electricity generation [15], minimising the difference between the allowable and actual power that flows through a given line [39], maximising the total power generated by RES [23,23,40], minimising power curtailment in RES [16,21,22], etc.

Eliminating branch congestion is also performed through power redispatching operations. As was written earlier, the possibility of redistributing the capacity of RE sources is possible thanks to regulation [20]. Similar analyses were performed, among others, in works [41,42,43,44,45,46].

An interesting approach to eliminating overcurrents is described in articles [10,11]. Using the method of tracking active power flows, a proportional curtailment of generation in sources was made on the basis of designated shares. Articles [47,48] propose methods based on sensitivity analyses and the power flow of entropy. In [49], the authors describe a method of reducing overloads by taking into account the variability of generation and load in the algorithm of classic line unloading.

Another group of works is articles in which network reconfiguration is considered in order to eliminate possible threats. One can distinguish works [50,51], in which distribution networks are analysed. The reconfiguration of a closed-loop network was considered in articles [52,53,54,55,56,57,58]. In [59], the authors used a method involving the optimal division of the power network based on analysing the N-1 criterion.

The topics related to the available transmission capacity or the available capacity of a given area also include the problem of line overloading. A detailed review of the literature on this topic can be found, among other things, in [60]. In [61], a probabilistic approach was used to determine the ATC. A method based on taking into account the capacity benefit margin (CBM) for determining the ATC was presented. The authors claim that this approach significantly affects ATC values. A different approach was proposed in [62]. In two computational stages, the problem was reduced to linear programming, and available optimisation methods were used. The issues of security and the reliable operation of the power system from the point of view of the ATC were considered in [63]. In articles [64,65], the ATC was determined online, taking into account random generation in RES.

There is a constant need to search for new methods that effectively solve real problems in the power industry. Modern power systems are becoming more and more complex. On the other hand, correctly estimated network models are more often available, which makes it possible to manage their operation in an online mode [66,67]. Threats and dangers require a quick reaction in order to eliminate them. Progressive digitisation and computerisation cause new problems that have not been observed before. Taking this into account, it becomes reasonable to reach for more and more advanced mathematical methods and their development, as well as to search for new solutions that will guarantee the safe and reliable operation of the power system while also reducing costs.

This article is organised in such a way that the Section 1 is a review of the literature. The Section 2 presents an introduction to the subject of the research and a justification for the use of advanced methods and algorithms to solve the problem of line and transformer overloads. The Section 3 describes the calculation methods. The Section 4 describes a modified IEEE 118-bus test network. Calculation results showing the effectiveness of the proposed method are included in the Section 5. A discussion of the results and conclusions is included in the Section 6.

2. Technical Considerations Regarding the Capacity of the Power Grid in Poland

The problem of power lines and transformers overloading is becoming more and more visible along with an increase in the number of distributed sources connected to the grid. It would seem that the large dispersion and varied power of these installations should not cause overloading in the branches. However, it often happens that, in a given area, the number of power plants and their total capacities are relatively large, which contributes to the occurrence of many exceedances. This problem has been observed in Poland for a long time, where, in the past, a large part of overhead power lines was designed for low operating temperatures of 40 °C. Taking into account the fact that some lines are made of conductors with small cross-sections (120 mm²), there are many situations in which overloads occur. Despite the progressive modernisation of the network, these measures are not keeping up with the increase in the number of connected RES and energy storage facilities.

The increase in demand for electricity in subsequent years and the number of connected energy storage facilities are not adequate to handle the increase in the number of distributed sources. This means that the total generation can sometimes exceed the total network load. In practice, when such situations are observed, especially in states of minimum power demand, it is necessary to reduce generation. Another problem is the random distribution of power generated in RES and the high dynamics of changes in power generated over time.

At the stage of planning the connection of new consumers and sources, grid operators very carefully examine the potential for the occurrence of overload threats, often with extremely pessimistic assumptions, and the system has a large margin of safety. However, it may happen that, for the existing network infrastructure, in the event of several random events (e.g., hot weather, unexpected increases in demand for active and reactive power, forest fires under the line, the emergency shutdown of another line), an overload condition will occur, resulting in no reaction, or worse, the downtime of another line can lead to cascading outages and a widespread system failure. The essence of the considerations presented in this article is the solution to a multidimensional optimisation problem that changes the distribution of generation in such a way as to lead to the elimination of congestion while minimising the costs of such activities. Formally speaking, the optimisation procedure involves maximising the power generated in selected RESs (in other words, minimising the total reduced power) while also meeting the load capacity conditions of all lines and maintaining an active power balance. One of the bottlenecks blocking the development of renewable sources in Poland (and other countries with poorly developed and outdated energy infrastructure) is the insufficient capacity of the 110 kV network. Adjusting the cross-sections of the wires, and with them, the supporting structures of power transmissions from wind farms and photovoltaic farms, is associated with high costs. Therefore, it seems logical to take actions aimed at developing algorithms, control systems, and methods of controlling RES so as to reduce the value of generated power in extreme conditions and adjust it to the current transmission capacity of the network. The total effect of power curtailment should be minimised, and the transmission capacity of the network should be fully used. This approach will mitigate and rationalise the requirements related to the necessary modernisation of the power grid. Thus, the costs of these treatments will be much lower.

As mentioned earlier, in the past, 110 kV power lines in Poland were made with ACSR (Aluminium-Conductor Steel-Reinforced) conductors with cross-sections of 120 mm², 185 mm², and 240 mm². The assumed operating temperatures are 40 °C, 60 °C, or 80 °C. The capacities of these lines in the summer are as follows:

• 120 mm²—39 MW (40 °C), 66 MW (60 °C), 78 MW (80 °C);
• 185 mm²—51 MW (40 °C), 86 MW (60 °C), 102 MW (80 °C);
• 240 mm²—62 MW (40 °C), 104 MW (60 °C), 122 MW (80 °C).

New 110 kV lines are usually made with ACSR/TW (Trapezoidal Aluminium-Conductor Steel-Reinforced) conductors with a cross-section of 240 mm² or 300 mm² and an operating temperature of 80 °C.

In Poland, 220 kV power lines are made with 525 mm² conductors. The assumed operating temperatures are 40 °C, 60 °C, and 80 °C. The summer capacities of these are as follows:

• 525 mm²—196 MW (40 °C), 333 MW (60 °C), 392 MW (80 °C).

In Poland, 400 kV power lines are made with 2 × 525 mm² bundle conductors. The assumed operating temperatures are 40 °C, 60 °C, and 80 °C. The summer capacities of these are as follows:

• 2 × 525 mm²—713 MW (40 °C), 1212 MW (60 °C), 1427 MW (80 °C).

The new 400 kV lines are made with 3 × 350 mm² and 3 × 468 mm² conductors. The assumed operating temperature is 80 °C. The summer capacities of these are as follows:

• 350 mm²—1683 MW (80 °C);
• 468 mm²—1983 MW (80 °C).

Currently, there is a visible tendency to upgrade lines in cases justified by overloading as a consequence of the saturation of the system with RE sources. These upgrades, aimed at increasing the current-carrying capacity, usually involve the following:

• Increasing operating temperatures from 40 °C to 60 °C or even 80 °C, which is most often accomplished by adjusting the strain of the line wires or raising the selected poles;
• The replacement of traditional conductors with segment conductors;
• The replacement of conductors with HTLS (High-Temperature, Low-Sag) conductors;
• The reconstruction of the considered line into a line with conductors with a larger cross-section or a double-circuit line.

As mentioned earlier, intensive upgrade efforts to increase the capacity of power lines are not keeping up with the increase in the number of connected RESs and energy storage facilities. Therefore, there is a need to use advanced methods that will effectively eliminate the resulting threats and will be an alternative to costly construction and modernisation work. These methods mainly involve changing the distribution of power generated in RESs in a given area of the network in such a way that all overloads are eliminated. Often, a change in power distribution is associated with its curtailment in previously selected sources, which have the greatest impact on the existing overloads. However, one should look at states in which the total power curtailment in the selected power plants is minimal, and such a method is presented below.

3. Description of the Proposed Two-Step Method

3.1. Algorithm Concept

The algorithm used consists of two-stage calculations aimed at identifying generation units for redispatching while also eliminating overloads in grid elements. In the first step, a power flow tracking method is used to select those sources that have a real and significant impact on the overloaded lines. This situation is shown in Figure 1.

Many sources are connected to the modelled network. It is assumed that three lines, marked in red, are overloaded. Based on the power flow tracking method, four sources (marked in green) were selected to actually affect the occurring overloads. As a result of this procedure, the number of decision variables significantly decreases because overloading is affected only by these four sources. It is extremely important for the optimisation to be effective because the size of the task increases significantly with extensive power networks. In networks with a very large number of nodes (e.g., several thousands of nodes), the total number of potential sources (constituting decision variables) that could be optimised is significant (theoretically, all sources could be taken into account). The number of constraints to be considered would also be considerable. Therefore, in the second step, to carry out the power redispatching operation, the optimisation method was used, albeit with a small number of variables. This approach allows for a significant reduction in the calculation time in situations in which all sources should be taken into account, even those that do not affect overloads. This would extend the calculation time and reduce the efficiency of the method. In general, the solution boils down to the selection of sources that actually affect the overloading of power lines. Then, the sensitivity of overloading lines to the generation change in the considered source is determined. It is possible to determine the actual value of power coming from selected power plants that flows through a given line.

The following sections present the theoretical basis of both algorithms used.

3.2. Power Flow Tracking Method

The power flow tracking method has been described in detail in [68,69,70,71,72]. This method makes it possible to “track” which part of the power generated by a given source flows through a selected power line. The originator of this method—Prof. Bialek [68]—compared its effect to dyeing water in streams with different colours and identifying the share of dye in the flow at the mouth of the river. It should be noted that in the electrotechnical application of the tracing method, the analysed line does not have to have a direct connection to the node to which the source is connected. The described situation is presented in Figure 2 (the selected source at node k and the line connecting nodes j–l are marked in red).

Generally speaking, the power flow tracking method involves determining the vector of nodal flows, P_w, and the distribution matrix, A_u, which relates nodal flows to specific power generated by P_G sources. This relationship can be expressed by the following equation:

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

(1)

The A_u distribution matrix is determined based on the knowledge of the power flows. The individual terms of this matrix are determined from the following relationship:

$$a_{\mathrm{u}ij}=\{\begin{array}{l}1, i=j\\ -\frac{\left|P_{ji}^{}\right|}{P_j^{\Rightarrow }}, P_j^{\Rightarrow }\ne 0\\ 0, \mathrm{in} \mathrm{other} \mathrm{cases}\end{array}$$

(2)

where

P_ij—power value in branch i–j (taken from node j).

$P_j^{\Rightarrow }$—power value flowing through node j.

In order to link the nodal flows with the power generated in individual sources, the A_u matrix inversion operation should be performed, according to the following relationship:

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

(3)

Taking into account the above relationships and detailed considerations described in the literature [16,68,69], the following equation is obtained, which determines the active power flowing through the i-l line. This power is the sum of the products of the share of individual sources and the values of power generated in them:

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

(4)

where

$u_{il,k}=\frac{P_{il}}{P_i^{\Rightarrow }}\cdot a_{\mathrm{u}ik}$—coefficient determining the use of branch i-l by the source located in node k (share factor),

$P_{\mathrm{G}k}$—active power generated by the source connected to node k.

As mentioned above, the total power flow in the i-l branch is the sum of the products of the power generated in the selected sources and their share factors. Detailed considerations of this subject have been presented in the literature [16,68,69].

If the “tracking” operation is applied to all sources in the network, then each of them can be assigned a share factor and exclude those for which the value is zero. In the next step, the reduced number of sources will reduce the number of decision variables in the optimisation process, and thus shorten the calculation time and increase the effectiveness of the optimisation described in the next section.

3.3. Description of the Optimisation Task

In order to save and solve a complete optimisation task, all the operating conditions of the transmission network and the constraints associated with them must be taken into account. It becomes necessary to introduce notations for vectors x (the vector of decision variables), y (the vector of independent variables), and z (the vector of dependent variables), which are described below.

The optimisation task involves maximising the power in the sources selected in the first step using the power flow tracking method described in Section 3.1. This means minimising the power curtailment in these power plants while also meeting the required restrictions. Therefore, the generation curtailment is minimal, and the transmission capacity of the lines is fully used. This task can be defined in the following general form:

$$F_{\mathrm{obj}}\left(x,y,z\right)\to \min$$

(5)

with equality

$$g\left(x,y,z\right)=0$$

(6)

and inequality constraints

$$h\left(x,y,z\right)\ge 0$$

(7)

Considering that the whole procedure takes place in two steps, this task is called a Special Optimal Power Flow (SOPF) task.

3.3.1. Formulation of the Objective Function and Constraints

It was assumed that the objective function, F_obj, will be the maximum value of the sum of the generated power of a separate group of RESs in accordance with the following relationship:

$$F_{\mathrm{obj}}\left(x,y, z\right)={\displaystyle \sum _{j=1}^N{x}_j}$$

(8)

where

x = [P_G1,…, P_Gj]—the vector of active power generated by the considered RES generation, for which the sum of the power is to be maximized (the vector of decision variables),

y = [P_L1,…, P_Li, Q_L1,…, Q_Li, P_Gn1,…, P_Gnn]—the vector of nodal loads and uncontrolled generation (the vector of independent variables),

z = [U₁,…U_i, δ₁,…δ_i, P_Gr1,…, P_Grk]—the phasor vector of node voltage and power generated in nodes ensuring system balancing (the vector of dependent variables).

The constraints of the decision variables (vector x) are the minimum and maximum generation limits of the sources.

The constraints of the dependent variables (vector z) are as follows:

• Permissible voltages in nodes (generally maintained in a range from 0.9 U_n to 1.1 U_n);
• Minimum and maximum generation limits of sources ensuring system balance.

Equality constraints, $g\left(x,y,z\right)$, include

• Nodal power flow equations;
• The active power balance of the considered system.

Inequality constraints, $h\left(x,y,z\right)$, are as follows:

• Permissible load capacity of power lines;
• Allowable transformer loads;
• An acceptable exchange balance with neighbouring areas.

The determination of state vector z for the known values of the x and y vectors is one of the basic computational problems included in the computer analysis of power systems. This problem, known as load flow analysis (LF), is described in textbooks such as [73,74].

Many algorithms and computer programs have been developed to solve this problem. Generally speaking, the components of the unknown vector, z (state variables), are determined from a system of nonlinear equations in the form of

$$0=f_{\mathrm{LF}}\left(x,y,z\right)$$

(9)

Each equation is a formula for a node in the analysed network so that the power balance of this node is equal to zero.

3.3.2. Description of the Metaheuristic Optimisation Algorithm

An original metaheuristic algorithm was used to test the proposed method. It should be emphasised that classical optimisation algorithms are not suitable for this type of task. They do not ensure the achievement of the global optimum. They can get stuck in a local optimum. In the case of a divergent computational process, they do not guarantee the ultimate success, while in the case of heuristics, the network data can be reloaded and the computation continued. The best recently found solution is then not lost because the vector of solutions is remembered at every stage of the calculation. The classical method would then have to be interrupted and the calculations started again.

The original, proprietary AIG metaheuristic optimization algorithm (Algorithm of the Innovative Gunner) [23,75] was used for the calculations. A block diagram illustrating the operation of the AIG algorithm is shown in Figure 3 [23].

The inspiration for the AIG algorithm was based on the selection of artillery parameters such that the next shot would precisely hit the target. However, the applied adjustment of the gun’s setting parameters is different compared with what is defined by the classical theory of artillery, hence the word “innovative...” in the name of the method, referring to the gunner. The innovativeness of the algorithm results from the fact that, in each step of the iterative process, the correction of the previous solution is applied by specially selected multipliers. This is a fundamental difference from most methods, in which the corrective element is added to the previous solution in the next step.

The AIG algorithm is an example of an exhaustive, forceful search algorithm. It is based on an assumed population number. In the beginning, the entire domain of functions is explored. In the next steps, the search area is narrowed down to refine the solution sought. The computation ends when the maximum number of iterations is reached. In the military interpretation, the algorithm corresponds to a massive amount of fire on enemy territory, while narrowing the search field leads to the precise identification of the target (i.e., the command point). Research has shown that, for some applications, such searches work faster than other more sophisticated algorithms and well-known algorithms (e.g., the particle swarm optimization algorithm, the cuckoo search algorithm, and the genetic algorithm).

The AIG algorithm differs from other algorithms In its innovative way of generating new solutions: $x_l^{\left(k+1\right)}$. In most methods found in the literature, e.g., [23,76,77,78,79,80,81,82,83], regardless of their specificity, in the next iteration step, an action is performed involving the “additive” modification of the solution, $x_l^{\left(k\right)}$, from the previous iteration:

$$x_l^{\left(k+1\right)}=x_l^{\left(k\right)}+\Delta x_l^{\left(k\right)}$$

(10)

In the case of the AIG algorithm, the components of the decision vector, x, are subject to “multiplicative” modifications in subsequent iterations, which can be described by the following relationship [23,75]:

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

(11)

where k is the next iteration, and functions $g_l\left(\xi \right)=g_{l1}\left({\xi }_1\right)\cdot \dots \cdot g_{lp}\left({\xi }_p\right)$ and $\Delta x_l^{\left(k\right)}$ are a symbolic notation and characteristic of the heuristic method used.

The AIG algorithm uses two correction multipliers (p = 2) and functions g₁ and g₂ of the same specific form, i.e., [23,75],

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

(12)

The functions $g\left({\xi }_1\right)$ and $g\left({\xi }_2\right)$ are in the form of the cosα^- function and its inverse, (cosα)⁻¹, while α and β are corrective angles drawn from the variable intervals, $\left(-{\alpha }_{\max },{\alpha }_{\max }\right)$ and $\left(-{\beta }_{\max },{\beta }_{\max }\right)$, via the normal distribution.

The calculations were performed using a script written in the MATLAB environment. Power flow calculations were performed in PowerWorld Simulator, version 22. The connection between the two programs is possible thanks to the SimAuto add-on (included in PowerWorld), which also acts as an interchangeable calculation engine, enabling the exchange of data between different applications.

4. Description of the Test Network

The calculations were performed on the IEEE 118-bus test network [84]. The network has been adapted to the parameters of the network in Poland. The nominal voltage levels of the nodes were changed to 400 kV, 220 kV, and 110 kV; the number of generators was limited; and the data of lines and transformers were adopted as typical in the Polish power system. A diagram of the modified IEEE 118-bus test network is shown in Figure 4. It was assumed that 35 sources are connected to the network, marked in black in Figure 4. The RE sources planned for connection, the number of which is 17, are marked green. Sources connected to nodes 8, 10, 26, 54, 61, 65, 66, 69, 80, 92, 99, and 116 were used for grid power balancing.

Different grid voltage levels and different cross-sections of power line conductors are marked with different colours in Figure 4. A red colour means 400 kV lines, green stands for 220 kV lines, other lines are a 110 kV network. Most lines are designed for 40 °C. There are also lines with a higher operating temperature, which is also shown in Figure 4.

5. Results of Test Analysis

As an example of analyses related to the connection of RES installations, an emergency state was assumed involving one 400 kV power line—situated between nodes 26–30—being switched off, which is shown in Figure 5.

The selected N-1 state is characterised by the overloading of five 110 kV lines:

• Lines (74–75)—load, 128%;
• Lines (71–72)—load, 123%;
• Lines (17–31)—load, 113%;
• Lines (17–113)—load, 110%;
• Lines (15–19)—load, 108%.

As mentioned earlier, the calculations were divided into two stages: the selection of sources affecting the observed line overloads and the optimal, total power curtailment in these power plants.

Calculations in the first stage

In the first step, the power flow tracking method was used to determine those sources that affect power flows in overloaded lines. These sources are listed in

Table 1. List of overloaded lines with percentage shares of individual sources in the power flow.

Overloaded Line	Power Flowing through the Line, MW	Percentage Share of a Given Source in the Power Flow of a Given Line, %
		G-19	G-24	G-25	G-27	G-31	G-70	G-72	G-73	G-74	G-113
19–15	66.2	81.5	-	18.5	-	-	-	-	-	-	-
31–17	69	-	-	14	16	70	-	-	-	-	-
113–17	115.8	-	-	22	21	23	-	-	-	-	34
72–71	75.5	-	-	-	-	-	-	100	-	-	-
74–75	77.8	-	2	0.3	-	-	2.5	12.5	4	82.7	-

along with their percentage share in the power flow through a given line. Based on the results, it can be concluded that the total number of sources on which power flows in overloading lines depend is 10. It should also be noted that, for some of them, this share is small, but there are also those that significantly affect the load line. These sources are marked in red in Table 1. One source is marked for each overloaded line. The sources selected have the largest percentage share in the power flow of a given overloaded line. This situation is shown in Figure 5.

The line switched off in the emergency state is marked in bold red dashed line. Overloaded lines are marked with a bold purple line. Sources affecting the loads of overloaded lines are surrounded by a black dashed line. As mentioned earlier, their number is 10. Those that have the greatest impact on line loads are additionally marked in yellow. Their number is five, and they are also listed in Table 1.

Calculations in the second stage

In the second step, optimisation calculations were performed for four cases:

• The first case involved the fact that the values of power generated in all 10 selected sources from Table 1, which affect the identified line overloads, were taken into account as decision variables. The calculations assumed that some of the selected sources may both decrease and increase their current generation (sources marked in blue in

  Table 2. Optimal values (AIG/PSO) of power generated in 10 selected sources (sources marked in blue can both decrease and increase their current generation).

  Lp.	Generator	Generator Node Voltage, kV	PGmax, MW	PG before Optimization, MW	PG after Optimization, MW	Curtailment ΔPG, MW
  1	G-19	110	100	100	95/95	−5/−5
  2	G-24	110	100	20	28/34	+8/+14
  3	G-25	110	100	100	100/100	−0/0
  4	G-27	110	100	100	48/54	−52/−46
  5	G-31	110	100	100	100/99	−0/−1
  6	G-70	110	100	60	98/97	+38/+37
  7	G-72	110	100	100	78/60	−22/−40
  8	G-73	110	100	30	50/59	+20/+29
  9	G-74	110	100	100	65/65	−35/−35
  10	G-113	110	100	40	52/51	+12/+11
  Total			1000	750	714/714	−36/−36

  ).
• The second case was that the powers of all 10 selected sources from Table 1, which affected the identified line overloads, were also taken into account as decision variables. However, the calculations assumed that all of the selected sources could only reduce their current generation (

  Table 3. Optimal values (AIG/PSO) of curtailed power generated in 10 selected sources (all sources can only reduce their current generation).

  Lp.	Generator	Generator Node Voltage, kV	PGmax, MW	PG before Optimization, MW	PG after Optimization, MW	Curtailment ΔPG, MW
  1	G-19	110	100	100	99/100	−1/0
  2	G-24	110	20	20	20/19	0/−1
  3	G-25	110	100	100	100/100	0/0
  4	G-27	110	100	100	98/91	−2/−9
  5	G-31	110	100	100	96/100	−4/0
  6	G-70	110	60	60	60/60	0/0
  7	G-72	110	100	100	66/68	−34/−32
  8	G-73	110	30	30	30/30	0/0
  9	G-74	110	100	100	78/78	−22/−22
  10	G-113	110	40	40	40/40	0/0
  Total			750	750	687/686	−63/−64

  ).
• The third case involved the fact that only those sources (five items) that had the greatest impact on the load of each of the overloaded lines were taken into account. The calculations assume that one of the selected sources can both decrease and increase its current generation (the source marked in blue in

  Table 4. Optimal values (AIG/PSO) of reduced power generated in 5 selected sources (source marked in blue can both decrease and increase its current generation).

  Lp.	Generator	Generator Node Voltage, kV	PGmax, MW	PG before Optimization, MW	PG after Optimization, MW	Curtailment ΔPG, MW
  1	G-19	110	100	100	99/99	−1/−1
  2	G-31	110	100	100	94/93	−6/−7
  3	G-72	110	100	100	65/65	−35/−35
  4	G-74	110	100	100	78/78	−22/−22
  5	G-113	110	100	40	41/42	+1/+2
  Total			500	440	377/377	−63/−63

  ).
• The fourth case involved the fact that only those sources (five items) that had the greatest impact on the load of each of the overloaded lines were taken into account. However, the calculations assumed that all of the selected sources could only reduce their current generation (

  Table 5. Optimal values (AIG/PSO) of reduced power generated in 5 selected sources (all sources can only reduce their current generation).

  Lp.	Generator	Generator Node Voltage, kV	PGmax, MW	PG before Optimization, MW	PG after Optimization, MW	Curtailment ΔPG, MW
  1	G-19	110	100	100	99/99	−1/−1
  2	G-31	110	100	100	93/95	−7/−5
  3	G-72	110	100	100	66/66	−34/−34
  4	G-74	110	100	100	78/78	−22/−22
  5	G-113	110	40	40	40/37	0/−3
  Total			440	440	376/375	−64/−65

  ).

The results of the optimisation calculations obtained using the AIG algorithm are presented in Table 2, Table 3, Table 4 and Table 5. To check the reliability of the results obtained with the AIG method, calculations were also made using the particle swarm optimization (PSO) method. The particle swarm method is a heuristic method with high efficiency and is, in a way, a classic of metaheuristics. The results obtained with the PSO method are marked in red in the tables. The calculations assumed 20 “shots” (20 particles), and the maximum number of iterations was equal to 500. Correction angles were drawn using the normal distribution in the range <−90°÷+90°>.

Based on the obtained results presented in Table 2, Table 3, Table 4 and Table 5, it follows that if the sources have the ability to both reduce and increase the generated power, then the total power curtailment is smaller compared with when there is only the possibility of reducing the generated power.

Figure 6 shows the variability of the best objective function values in the optimisation process using AIG and PSO (for comparison). The obtained results confirm the high efficiency of the AIG algorithm. In practically every case, we find a solution relatively quickly. Thus, this algorithm can be successfully used to solve the problem of minimising power curtailment in sources as a particularly important optimisation task regarding the safe and reliable operation of the power system.

The performed two-stage calculations showed the effectiveness of the proposed method. In the first stage, 10 sources were selected that affect power flows in overloaded lines, which limited the number of decision variables to 10, with the total number of sources being 52. In the first two cases, in the second stage (the optimised power curtailment), power regulation was taken into account in all 10 selected sources. In the third and fourth cases, it was shown that taking into account only one source in the optimisation process, which has the greatest impact on the overloads of each of the five lines, limits the number of decision variables to five. The optimisation process nevertheless guarantees the elimination of the identified current exceedances. During the optimisation process, if the possibility of both increasing and reducing the generated power (a RES with energy storage) is taken into account, then the total power curtailment is higher by several megawatts compared with the analysis performed using the 10 sources considered. If the sources can only curtail their generation (specifically, in the RES without energy storage), then the total necessary generation curtailment in the systems with 10 sources and 5 sources is comparable. Most of the limitations on the solution turned out to be the carrying capacity of the 110 kV line, which was subject to the overload presented earlier.

Table 6. Loads of overloaded 110 kV lines after optimisation (after unloading).

No.	Line	Line Load, %
	(From-To)	Case 1	Case 2	Case 3	Case 4
1	(74–75)	99.4/99.9	99.9/99.9	100/99.9	100/99.7
2	(71–72)	93.5/87.3	98.7/100	98.5/98.3	99.4/98.8
3	(17–31)	93.8/94.3	100/100	99.9/100	99.6/100
4	(17–113)	100/100	99.1/98.5	100/99.8	99.2/97.6
5	(15–19)	100/100	100/100	99.8/99.7	99.9/100

shows the load on the 110 kV line after optimisation. As a result of the change in generated power, the load on the overloaded lines decreased to 100% and below 100% of their permissible carrying capacity, and, therefore, the goal of the method was achieved, with generation curtailment at a level of 5% to 14%.

The proposed method can be implemented in practice both as a support for dispatchers managing the operation of the power system, as well as an application included in the EMS (Energy Management System) that performs automatic redispatching. The selection of the number of decision variables (all sources selected in the first stage or limiting their number only to those dominant in the identified overloads) depends on technical conditions (the speed of online calculations) and economic conditions (costs of redispatching and balancing generation curtailment).

As part of testing the effectiveness of the proposed method, an alternative analysis was performed using the arbitrary power curtailment method on the selected sources (five and ten) to check whether the intuitive approach would provide similar results to the optimisation procedure using the AIG algorithm. Calculations were made for two cases. In the first case, in the next iteration, the power in each source was curtailed by the same value, equal to 1 MW. The second case involved proportionally curtailing power in relation to the power generated before the commencement of the curtailment procedure. It was assumed that the total power curtailment in each iteration was 5 MW. It turned out that arbitrary (intuitive) generation curtailment eliminating line overloads requires a greater total generation limitation than the curtailment optimised using the AIG algorithm. For some sources, the power even had to be reset. The results obtained from the intuitive method are presented in

Table 7. The effect of using an alternative (intuitive) method of reducing the power generated in 10 selected sources.

Generator	PGmax, MW	PG before Curtailment, MW	PG after Curtailment		ΔPG, MW
			Equal Steps, MW	Proportional Steps, MW	Equal Steps, MW	Proportional Steps, MW
G-19	100	100	71	73.4	−29	−26.6
G-24	100	20	0	16.2	−20	3.8
G-25	100	100	71	73.4	−29	−26.6
G-27	100	100	71	73.4	−29	−26.6
G-31	100	100	71	73.4	−29	−26.6
G-70	100	60	31	44.8	−29	15.2
G-72	100	100	71	73.4	−29	−26.6
G-73	100	30	1	22.4	−29	−7.6
G-74	100	100	71	73.4	−29	−26.6
G-113	100	40	11	28.6	−29	−11.4
Razem	1000	750	469	552.4	−281	−135.28

and

Table 8. The effect of using an alternative (intuitive) method of reducing power generated in 5 selected sources.

Generator	PGmax, MW	PG before Curtailment, MW	PG after Curtailment		ΔPG, MW
			Equal Steps, MW	Proportional Steps, MW	Equal Steps, MW	Proportional Steps, MW
G-19	100	100	70	69.2	−30	−30.8
G-31	100	100	70	69.2	−30	−30.8
G-72	100	100	70	69.2	−30	−30.8
G-74	100	100	70	69.2	−30	−30.8
G-113	100	40	10	27.4	−30	−12.6
Razem	500	440	290	304.2	−150	−135.8

. Based on the presented results, it can be concluded that the method of eliminating overloads in power curtailment using the same value is not effective because the total generation curtailment is significant and much greater than in the proposed two-stage method. This method is inefficient because it contributes to too much generation curtailment in the sources. Slightly better results can be obtained during power curtailment based on proportional generation decreases according to the following principles: high initial power, high curtailment; low initial power, low curtailment. The total generation limitation is much smaller than in the previous case but still significantly greater than the effects of the proposed two-stage method.

The limitations that had the greatest impact on the obtained solution turned out to be the previously identified overloads of the 110 kV line. As a result of intuitive power curtailment (Table 7 and Table 8), their load was reduced to the following values:

Case with 10 selected sources

• Line (74–75)—equal curtailment steps—72.6%; proportional curtailment—86.7%;
• Line (71–72)—equal curtailment steps—99.5%; proportional curtailment—99.6%;
• Line (17–31)—equal curtailment steps—46.8%; proportional curtailment—59.1%;
• Line (17–113)—equal curtailment steps—41.2%; proportional curtailment—60.8%;
• Line (15–19)—equal curtailment steps—61.8%; proportional curtailment—68.0%.

Case with five selected sources

• Line (74–75)—equal curtailment steps—90.7%; proportional curtailment—90.0%;
• Line (71–72)—equal curtailment steps—99.4%; proportional curtailment—99.7%;
• Line (17–31)—equal curtailment steps—77%; proportional curtailment—80.3%;
• Line (17–113)—equal curtailment steps—68%; proportional curtailment—79.7%;
• Line (15–19)—equal curtailment steps—74%; proportional curtailment—74.1%.

As can be seen, controlling the congestion of lines (71–72) with engineering (intuitive) methods is associated with the deep relief of the remaining four lines, and thus a significant generation curtailment in the redispatched sources (from 18% to 38%). Optimisation based on the AIG algorithm copes much better with the specificity of the closed network, and the value of the necessary generation curtailment is much smaller.

6. Discussion and Conclusions

The article analyses the problem of overloading power lines as a result of connecting more and more renewable energy sources. This problem is now becoming important not only from a technical point of view but also due to economic conditions. The possibilities of redistributing power offered by the Regulation [20] mean that system operators will bear the costs of such actions, which are aimed at eliminating line overloads. However, they are inevitable in the era of an increasing number of distributed energy sources connected to the power grid.

This article presents an original method consisting of two stages. It turns out that the combination of the power flow tracking method with the metaheuristic optimisation algorithm shortens the calculation time and increases the efficiency of optimisation calculations. Selecting only those sources that actually affect power flows through overloaded lines in the first step guarantees an optimal solution to the analysed problem, improves the effectiveness of the method, and also contributes to lower redispatching costs incurred by grid operators. The novelty of the proposed method is changing the power distribution in the appropriate RES indicated earlier in order to effectively eliminate the identified line overloads. An appropriate reaction from the operator during threats to the operation of the network contributes to the improvement of security and increases the ability to control the power system. The determination of the optimal power distribution in a properly selected RES is possible by combining the advantages of two methods: power flow tracking and heuristic optimization.

The conducted analysis shows that, in the event of overloading network elements in the National Power System, their complete liquidation is possible. Operator activities should be supported by appropriate software using computational capabilities in the field of tracking power flows and optimising the operation of the power system. This optimisation should determine changes in the distribution of power generation in conventional units and RESs, but in critical situations, it should also cover the grid structure.

The performed calculations proved that the change in power distribution in the previously selected sources in the analysed power grid can be determined by solving a complex problem of nonlinear optimization with constraints. Such a solution does not contain the subjective approach of the operator but the result of an objectively operated algorithm that selects the best actions in the global sense. The proposed method may be helpful for operators in determining the current capacity of the power grid.

The proposed method does not exhaust all possibilities, but it can be the impetus for further research and necessary changes in the field of determining the optimal distribution of generated power and coping with line congestion conditions.

The next step towards dynamising the solution of the problem under consideration may be the use of methods based on artificial intelligence, that is, methods that have the ability to solve real but nonstandard problems resulting from the requirements of network operators. There are various techniques based on artificial intelligence (e.g., machine learning) that can be used in analyses in the field of power engineering. In future work, the authors also intend to use selected machine learning techniques to further shorten the time it takes to obtain optimal redispatching parameters. Such a system could also be successfully used in practice. However, it would be necessary to analyse many different operating states for the decision machine to be properly trained.