Active power losses in transmission lines are caused by the flow of current through the ohmic resistance inherent to each line. They, therefore, depend on the load connected, the associated voltages, and the line resistance. Whereby the resistance increases linearly with the line length.

Renewable energy sources (RES) are constructed in locations that provide high returns for the owner companies and where the geography facilitates their development. Therefore, the locations where the power plants are built are not necessarily where the power is consumed.

In the Nordic grid, for example, many essential hydropower plants are located in the north of Norway and Sweden or the southwest of Norway (Energinet and Kraftnät, 2021). At the same time, the main load centers are situated in the southeast of Norway or in the south of Sweden (Energinet and Kraftnät, 2021). This leads to power flows over long distances to the consumers, resulting in considerable losses even at high voltages.

This challenge is not confined to the Nordic grid; it also applies to the Great Britain transmission system operator, National Grid ESO. According to a report (Transmission, 2019), it is believed that the primary cause of future active losses in its grid is geographically distributed generation.

The fundamental issue that arises from this is that the network operator must purchase the active power losses at the spot market price for the associated hour (Statnett, 2024), which ultimately leads to higher costs for the end consumer (i.e., society).

One solution to reduce losses is to upgrade the existing equipment. However, this leads to considerable investment that often cannot be justified by considering active power losses exclusively; furthermore, local improvements may reduce losses at a particular location while also leading to an overall increase in losses due to changes in power flows (Transmission, 2019). Another measure, explored in this paper, is optimizing reactive power procurement and dispatch to minimize losses by properly coordinating reactive power sources.

Reactive power is unavoidable when transmitting active power, and its inappropriate handling can lead to serious security problems, including stability issues and reduced transfer capacity (El-Samahy et al., 2008). However, reactive power has some specific technical characteristics. The most dominant aspects relevant to this paper are that it does not require fuel for generation, and it cannot be transferred over long distances (Wolgast et al., 2022). Determining the price of reactive power is therefore non-trivial; however, since reactive power is vital for system operation, it also has an associated value that must be determined appropriately. Furthermore, a reactive power price set too low or too high can lead to false incentives and, ultimately, to critical technical issues (Rabiee et al., 2009).

When dealing with optimal reactive power procurement and dispatch, it is essential to determine the temporal specifications, such as when the optimization is executed and based on what data. For example (El-Samahy et al., 2008), proposes procuring the optimal reactive power in a first step on a seasonal basis and later dispatching it shortly before the actual application. Zhang and Ren (2005), on the other hand, follows a more real-time dispatch approach, where it is assumed that the loading conditions are approximately constant over a specific time interval (e.g., 1 h). Time-critical aspects of reactive power redispatch using optimal power flow (OPF) were emphasized and investigated in a specifically tailored real-time laboratory setup in Martín et al. (2024).

This article implements the dispatch optimization algorithm in the period between the closed day-ahead market and real-time application, and therefore refers to it as reactive power pre-dispatch.

Besides the temporal aspects of the reactive power procurement and dispatch approach, the modeling and optimization methods used are also relevant. Many authors model the process using a two-step or bi-level optimization approach, as seen for example in the works (El-Samahy et al., 2008; Bhattacharya and Zhong, 2001; Almeida and Senna, 2011; Almeida et al., 2016). In one level or the first step, they calculate the dual variables that they use later in the upper-level or second step as a price indicator. El-Samahy et al. (2008), Bhattacharya and Zhong (2001) use the duals directly to determine the value of reactive power. While (Bhattacharya and Zhong, 2001) examines sensitivity concerning active power losses, (El-Samahy et al., 2008), also considers security aspects in its optimization process. In the second optimization step, the calculated duals are used to maximize a societal advantage function (SAF). The authors in Almeida and Senna (2011), Almeida et al. (2016), on the other hand, utilize the duals of the follower problem as active power price sensitivities, which are then applied in the leader problem to minimize the opportunity costs. Dual variables also play a role in Feng et al. (2024), where the authors analyze three different scenarios for minimizing active power losses with reactive power support and map these scenarios to dual variable configurations. A stochastic two-stage model was proposed by the authors of Jiang et al. (2022). They present a day-ahead market mechanism for reactive power ancillary services and propose a modified version of the Vickrey-Clark-Groves mechanism, specifically designed for reactive power services in systems with high RES penetration.

Besides generators, the grid owner itself typically has equipment for controlling reactive power flows. The switching action of such reactive power control devices, such as capacitor banks, leads to costs, for example, a reduction in the device’s lifespan. Therefore, Zhang and Ren (2005) proposes to find a trade-off between active power losses and these switching costs. Such costs for reactive power-controlled devices are also considered in the work of Lamont and Fu (1999). The objective of Lamont and Fu (1999) is to dispatch the reactive power of generators as well as the reactive power-controlled equipment to minimize real power losses. The optimal power flow problem is iteratively solved, with the prices for the various reactive sources being calculated and adjusted in each iteration. These prices comprise explicit costs (capital and operating costs) and opportunity costs, utilizing a triangular relationship and probability distribution. The use of the triangular relationship proposed by Lamont and Fu (1999) to determine the value of reactive power was followed by the authors of De and Goswami (2014). Furthermore, De and Goswami (2014) proposed three different formulations of an OPF for reactive power procurement and compared them with two classic formulation approaches, one of which takes into account the L index (first published in (Kessel and Glavitsch, 1986)) to determine the proximity to voltage instability, and the other minimizes system losses. Hao (2003) proposes that generation companies should be obligated to provide reactive power free of charge in proportion to their active power output (an approach already implemented in countries such as Norway Statnett (2022)). In addition, Hao (2003) suggests that any provision of reactive power beyond this proportional obligation should be financially compensated.

This paper adopts a similar principle to that proposed by Hao (2003), but instead considers compensation for any deviation from the initially scheduled reactive power set point. Furthermore, to determine both the optimal change in reactive power set point and the corresponding reactive power price, a strategy hereafter called pre-dispatch is formulated based on a bi-level optimization problem. At the lower level, the objective is to minimize costs related to active power losses, along with costs for supplemental reactive power injections from the generators, which is a relatively standard approach. However, this paper does not utilize the dual variables of the lower-level; instead, it varies the unknown reactive power price within a specific cost interval and calculates the power flow solution for each price separately. The authors chose this method because it enables the analysis of the Pareto front of the two conflicting objectives in the upper-level (leader problem). Consequently, this helps to find the best trade-off between the two objectives in the upper-level problem.

In addition to the modeling and temporal aspects of optimal reactive power dispatch and procurement, the choice of solving strategy often plays an essential role. Several of the optimization models mentioned involve AC power flow equations, resulting in nonlinear and nonconvex optimization problems. If no binary variables are included, techniques such as those used in Lamont and Fu (1999) can be applied. These techniques allow for an iterative solution by linearizing the problem at each step. Another method that can be utilized is semidefinite programming, as discussed in Davoodi et al. (2019). However, a common challenge with conventional solvers is that they may converge to a local minimum (Kumar et al., 2023). To address this issue, many authors apply metaheuristic methods, which can help overcome these limitations and potentially converge to the global minimum. For example, Zhang and Ren (2005), uses a cataclysmic genetic algorithm; (De and Goswami, 2014), on the other hand, uses an artificial bee colony algorithm (Cabezas Soldevilla et al., 2019); uses a particle swarm technique; and the authors of Salimin et al. (2024) present a hybrid algorithm named integrated accelerated clonal evolutionary programming. Enhanced differential evolutionary algorithms are proposed in Kumar et al. (2023), Kar et al. (2023). The performance of the algorithms is compared to other metaheuristics using two statistical analysis methods: the Wilcoxon signed-rank test and the Friedman-Nemenyi statistical test (Kar et al., 2024). proposes a modified whale optimization algorithm for solving the OPF-based optimization problem for minimizing active power losses using FACTS devices.

This paper aims to develop a method for optimal pre-dispatch of reactive power, rather than comparing different solvers with each other. A conventional interior-point method was used here as an example. However, the code is publicly available, including the used power system model (Baltensperger, 2025), and solvers can be modified using the Pyomo environment.

The key objective of the proposed method is to reduce the costs for society by lowering active power losses and fairly procuring reactive power from generator companies. For doing so, the paper proposes an optimal pre-dispatch method based on a two-level optimization formulation. The method serves as an intermediary step between day-ahead scheduling and real-time application, enabling the TSO to procure extra reactive power from (GenCos) in order to minimize active power losses. The cash-flow diagram of Figure 1 shows, in the most fundamental way, the objective of the method. The left side illustrates the situation without the proposed reactive power pre-dispatch step, whereas the right side shows the problem with the pre-dispatch step. Without pre-dispatch (left), the TSO pays $ P loss,0 for additional active power support of the GenCos due to system losses. With the pre-dispatch step (right), the TSO pays an extra amount of money for supplemental reactive power $ Q Cos Gen (red arrow), intended to reduce the cost for active power losses $ P loss,1 (blue arrow). Compared to the situation without the pre-dispatch step, reducing active power losses leads to extra income for GenCos since reactive power is now remunerated ( $ Q Cos Gen ). However, it also reduces revenues from active power support as written in (Equation 1). $ P Rev.Red. GenCos = Δ $ P 0,1 = $ P loss,0 − $ P loss,1 (1) $ Tot.Cost TSO,0 = $ P loss,0 , $ Tot.Cost TSO,1 = $ P loss,1 + $ Q Cos Gen (2) $ Saving TSO = Δ $ P 0,1 − $ Q Cos Gen = $ Tot.Cost TSO,0 − $ Tot.Cost TSO,1 (3) From a TSO point of view, the total costs include not only the expenses for active power losses but also the extra costs associated with reactive power, as it is written in (Equation 2). Therefore, the savings for the TSO can be defined as the difference in total costs between the situation without and with the pre-dispatch step as written in (Equation 3).

When discussing optimal pre-dispatch, it is necessary to address the meaning of ‘‘optimal’ specifically. With ‘optimal’ pre-dispatch, the objective is to determine a new price for supplemental reactive power, represented as c g * , that reduces the active power losses and maximizes the savings for the TSO ( $ Saving TSO ) and reactive power income of the GenCos ( $ Q Cos Gen ) . These two objectives are conflicting as written in (Equation 3).

In the context of the mentioned pre-dispatch strategy, this paper addresses two key research questions:

What is the reasonable economic value of reactive power when considering active power losses?

Besides the objective function, the follower optimization model used in this paper is a classical OPF written in a general form in (8). u c g , x c g = arg min u , x C Ahead Day P loss,1 ⏞ $ P loss,1 u , x , c g + c g ∑ k ∈ G | Δ Q G , k | + ∑ k ∈ W | Δ Q W , k | ⏟ | Δ Q Cos Gen | ⏞ $ Q Cos Gen u , x , c g s.t. g u , x ≤ 0 , h u , x = 0 (8)

The objective function contains the costs for active power losses in the transmission lines $ P loss,1 and the income of supplemental reactive power of the GenCos willing to participate in the pre-dispatch service $ Q Cos Gen . At this point, clarifying the meaning of some variables is essential. | Δ Q Cos Gen | represents the supplemental reactive power provided by the GenCos that compensates for active power losses. The sets G and W consist of all synchronous machines and wind turbines of the GenCos that participate in the pre-dispatch service.

The vector x contains all states, such as nodal voltage magnitudes at load buses, and u is the vector with all control variables, for example, generator terminal voltages. In the given problem, C Ahead Day is the active power price (e.g., the day-ahead price) used to consider the losses in terms of costs. c g is the decision variable of the leader problem and represents the price for supplemental reactive power. The constraints g ( u , x ) represent the set of inequalities such as line-flow limits, and h ( u , x ) are, for example, power flow equations.

The modeling aspects of the reactive power sources considered in the OPF are explained below. However, a more detailed description of the fundamental aspects of conventional OPFs, such as written in (Equation 8), is not provided as it is commonly known. For more information on this topic, the reader is referred to the relevant literature such as Conejo and Baringo (2018).

The leader problem model, as stated in (Equation 4), aims to maximize the TSO’s savings ( $ Saving TSO ) and the GenCos supplemental reactive power income ( $ Q Cos Gen ). Therefore, it is modeled as a multi-objective problem and is solved here using the Tchebycheff scalarization method as described, for example, in Pardalos et al. (2017). The aspects relevant to understanding the definitions, approaches, and scalarization method are briefly explained in this section.

The fundamental idea of the Tchebycheff scalarization method is first to define a so-called utopia point. In literature, the utopia point is defined as a point slightly better than the best achievable value (i.e., z i < min c g ∈ C g f i ( c g ) ) (Pardalos et al., 2017; Eichfelder, 2008). To be consistent with the literature, the two utopia points are defined here as written in (Equations 16) and (17), where ϵ is set to 1.001 to make sure that the utopia point is not on the solution of the vector-valued objective function: z 1 = max c g ∈ C g $ Q Cos Gen u c g , x c g , c g ⋅ ϵ (16) z 2 = max c g ∈ C g $ Saving TSO u c g , x c g , c g ⋅ ϵ (17)

The point ( z 1 , z 2 ) is fictitious because the two goals are in conflict and cannot therefore be reached together. The set C g includes all considered prices and will be explained in more detail in Section 2.3.

Taking into account the defined utopia point, the best possible compromise of both functions can be found using the calculation, according to Tchebycheff, written in (Equation 18): min c g ∈ C g max w 1 $ Q Cos Gen u c g , x c g , c g − z 1 , w 2 $ Saving TSO u c g , x c g , c g − z 2 (18)

This paper sets weights w 1 and w 2 equally to avoid any preference between the objectives.

This section is devoted to the method used to solve the presented multi-objective bi-level problem. First, the effect of the two special prices ( c g = 0 ) and ( c g = ∞ ) on the follower optimization problem is examined.

Equation 19 shows the follower problem for the case ( c g = 0 ) . It is apriori clear that the OPF would minimize the losses optimally since supplemental reactive support of GenCos is free. Therefore, it can be concluded that ( P loss,1 ≤ P loss,0 ) and ( $ Q Cos Gen = 0 ) . lim c g → 0 arg min u , x C Ahead Day P loss,1 ⏟ ≤ P loss,0 + c g ⏟ 0 Δ Q Cos Gen ⏟ ≥ 0 , s.t. g u , x ≤ 0 , h u , x = 0 (19)

The second extreme case is the price ( c g = ∞ ) , represented in (Equation 20). In this case, the OPF would choose the opposite since any extra reactive power would lead to extremely high prices. The OPF now chooses ( | Δ Q Cos Gen | = 0 ) , which would lead to ( P loss,0 = P loss,1 ) and ( $ Q Cos Gen = 0 ). Therefore, in both cases, ( c g = 0 ), and ( c g = ∞ ) would lead to no extra income for the GenCos, which will be suboptimal for the leader problem. lim c g → ∞ arg min u , x C Ahead Day P loss,1 ⏟ P loss,0 + c g ⏟ ∞ Δ Q Cos Gen ⏟ 0 , s.t. g u , x ≤ 0 , h u , x = 0 (20)

Consequently, it is essential to find the interval between these two boundaries where the cost for additional reactive power $ Q Cos Gen is larger than zero. Moreover, finding the largest cost-value $ Q Cos Gen will be relevant for the utopia point z 1 .

The proposed procedure for solving the problem is shown in the flowchart in Figure 4. The first step in the pre-dispatch block is to initialize five list variables C ̃ g , U ̃ , X ̃ , $ ̃ Q Cos Gen and $ ̃ Saving TSO . The second step is to solve the follower OPF problem using the parameters Q G 0 , P G 0 , and the reactive power price c g . The resulting control and state vectors u c g , x c g are stored in the list U ̃ and X ̃ , and the reactive power price is updated by incrementally increasing the previous price by δ . This process is repeated until | Δ Q Cos Gen | becomes zero. The set C g includes all price values for which 0 < | Δ Q Cos Gen | .

The objective function values of the leader problem are computed for all previously obtained solutions at each price in C g , and these values are stored in the two lists $ ̃ Q Cos Gen and $ ̃ Saving TSO . Finally, the best trade-off in the two-objective function of the leader problem is computed using the Tchebycheff scaling method.