Applying an Optimization Algorithm Based on the Cauchy Distribution for Active and Reactive Power Management with Batteries in Energy Distribution Systems

Objective functions

The primary aim of an EDM for coordinating BESUs and DERs is to optimize key performance metrics for distribution companies, encompassing economic, technical, environmental, or social indicators or a combination thereof¹⁸. This study focuses on minimizing two main objectives: the expected daily energy losses (Dloss) and the daily CO₂ emissions (DCO₂ ), as defined by Eqs. (1) and (2) ¹⁷.

Here, Re {·} extracts the real part of the argument; H and N are the sets of analyzed time periods and network nodes, respectively; Vk,h and Vm,h denote the complex voltage at nodes k and m during time h; Yk,m represents the complex nodal admittance between nodes k and m; ∆ _h signifies the time discretization interval; and and indicate the complex power output of the conventional and renewable sources connected to node k at time h, with and being their corresponding CO₂ emission coefficients.

Notably, power generation using fossil fuels at medium-voltage levels is rare in urban distribution networks. Nonetheless, these systems contribute to equivalent CO₂ emissions by connecting to national power grids via transmission systems. Such grids often include mixed-generation matrices composed of renewable and fossil fuel sources ⁽¹⁹⁾. For instance, Colombia’s electricity generation comprises approximately 67 % renewable sources (mainly hydropower), while the remaining 33 % relies on fossil fuels like coal, natural gas, and diesel ⁽²⁰⁾. In this vein, the coefficient models the equivalent emissions linked to distribution network operations ⁽¹⁷⁾.

Model constraints

The challenge of achieving an efficient operation of DERs in distribution networks involves adhering to a series of constraints derived from Kirchhoff’s laws for each network node at every time interval, as well as to constraints related to the operational limits of the equipment used. Our EDM designed for the optimal coordination of DERs in medium-voltage distribution networks must consider the constraints outlined in Eqs. (3) to (18).

Here, S ^d,⋆ _k,h denotes the complex power consumption at node k during time period h. The SoC of the type-b battery connected to bus k at times h and h + 1 is represented by SoC _k,h ^b and SoC _k,h ^b ₊₁, respectively. The parameter/is the charge/discharge efficiency coefficient of the battery. The variables and denote the initial and final SoC values expected for the type-b battery at bus k. The impedance of the branch connecting nodes k and m is expressed by Zkm, with Ikm,h representing the current that flows through this branch at time h. The symbols S _k, ^g _nom and S _k, ^dg _nom specify the maximum apparent power capabilities of the conventional and distributed generators at bus k. The generation availability profile for a distributed source at bus k is denoted by . The maximum power transfer capacity of the power electronic converter associated with the type-b battery at bus k is represented by S _k, ^b _nom. The parameters P _k, ^b _m´ın and P _k, ^b _ma´x set the limits for active power injection or absorption for the battery at bus k. The variables V _m´ın and V _m´ax indicate the minimum and maximum voltage allowable in all network nodes at any given time, while I^ma´x _km defines the maximum current limit for the line connecting nodes k and m. Finally, the set L contains all network branches, and h _m´ın and h _m´ax represent the start and end of the period set H.

The constraints outlined in ^(3)-(18 can be interpreted as follows.

Eq. (3) outlines the power balance at each network node for each time interval, ensuring compliance with Kirchhoff’s current law ^(21). Eq. (4) details the energy storage dynamics of the batteries, modeled through the SoC for each time period ⁽⁴⁾Eq. (5) applies Ohm’s law to each branch in the network, describing the relationship between voltage and current across branch impedances, and Eqs. (6) and (7) specify the initial and final SoC for the batteries at the start and end of the scheduling period, thereby establishing consistent energy management across the timeline ⁶.

Constraints ⁸ and ⁹ impose upper bounds on the power flowing from the substation and ensure that it only provides active power. Constraints ¹⁰ and ¹¹ set the power transfer limits for the DGs and BESUs. Eq. ¹² ensures that the DGs inject power only when renewable resources are available, thus preventing power absorption. Eq. (13) defines the permissible range for active power charging/discharging by the type-b battery at node k during period h. Finally, Box-Type Constraint ¹⁴ ensures that the voltage at all network nodes adheres to regulatory standards for medium-voltage systems ²², and Constraint ¹⁸ guarantees that the current flowing through any network branchremainsbelowthethermalcapacity of the conductor, thus maintaining a safe operation.

To address the challenges posed by the non-convex constraints inherent in the Optimization Model ¹- ¹⁸ , this research employs a novel metaheuristic optimizer: the CbDO. The Set of Constraints ³- ¹⁸ encapsulates the technical requirements for operating a power distribution network with DERs. While many of these constraints are convex and manageable within classical optimization frameworks, critical aspects such as the Power Balance Constraint ³ and the Voltage Regulation Bounds ¹⁴ introduce significant non-convexities. These complexities arise from nonlinear relationships and inequalities, particularly due to the complex product of voltage terms and the structure of box-type constraints involving inequality conditions.

In light of these characteristics, conventional solvers often struggle to find feasible or globally optimal solutions, especially when handling non-convex, multi-modal search spaces. The aforementioned optimizer, as a metaheuristic technique inspired by the Cauchy distribution, excels in exploring such spaces efficiently ²³. Its design allows for a better handling of diverse solution landscapes, leveraging the long-tailed nature of the Cauchy distribution, which promotes extensive exploration and reduces the risk of premature convergence. This is particularly valuable for solving non-convex optimization problems, where local optima may otherwise hinder traditional algorithms.

Thus, employing the CbDO for optimizing complex power system models, such as the one represented in ¹- ^18), provides a robust alternative that aligns with the nature of the problem. This approach facilitates the search for high-quality solutions that adhere to technical constraints while effectively navigating intricate non-convexities. The set of variables related to the Optimization Model ¹- ¹⁸ is presented in Table I.

Table I: Number of variables in the Optimization Model 1- 18.

Fitness function

In metaheuristic optimization, fitness functions are essential for managing constraints through penalty methods ^6,24. For the problem under study, the fitness function Ff is defined as shown in (16):

where αk(x) denotes the penalty applied for violating the kth constraint, and C is the set of constraints considered in the fitness function. The constraints included in calculating the penalty ensure compliance with operating and technical limits, except for the constraints already handled by the follower stage, as is the case with ³. The penalty function for the current limit of the distribution branches, as expressed in ¹⁸, is given below ⁶:

where γi is a positive penalty coefficient.

This penalty function only contributes a non-zero value if any branch current exceeds its thermal limit, thereby ensuring that the solutions remain feasible under real-world operating conditions ²⁶.

Follower stage: power flow solution

With the aim of evaluating the solutions proposed by the CbDO, a power flow solver based on the successive approximations method is utilized to solve the nonlinear power balance constraint ^{3) (6}. The iterative power flow update formula is presented in ¹⁸.

where is the updated vector of demand-node voltages for time h, and Y⁻ _d,d ¹ is the inverse of the admittance matrix relating the demand nodes. The term S ^⋆ _d,h represents the net power at these nodes, including the contributions of BESUs and RES.

The iterative process converges when the stopping criterion in ¹⁹ is met.

where ε is a user-defined tolerance, typically set at 1 × 10−10. Once convergence is achieved, the power injected by the conventional source is calculated as per ²⁰.

This final calculation ensures that the power dispatch observes the network constraints, allowing for an accurate evaluation of the Objective Function ¹.

Leader stage: CbDO

The CbDO is a new math-inspired metaheuristic algorithm designed to explore and exploit the solution space through the properties of the Cauchy distribution ²³. Below, we outline its application for the active and reactive power dispatch problem involving BESUs. The SoC of the BESUs was chosen as the set of decision variables for the algorithm. The encoding for these variables is illustrated in Eq. (21), showing the SoC values for two BESUs, one situated at bus k and the other at bus m ⁶.

Once the SoC variables for the BESUs have been established, the CbDO determines the corresponding active and reactive power injection into the network. This step ensures compliance with all the operating constraints specified in the Optimization Model ¹- ¹⁸ for the proper functioning of the BESUs.

Computational validation against literature reports

A comparative analysis against a recent metaheuristic approach demonstrated that the proposed CbDO method can solve the problem regarding the effective coordination of BESUs and solar generation systems. For the sake of comparison, we conducted two analyses of the proposed optimization approach:

1. A distribution network analysis considering a unitary power factor for the BESUs and PV generators, in order to compare our approach against the continuous genetic algorithm (CGA) and the parallel versions of the particle swarm optimizer (PPSO) and the vortex search algorithm (PVSA). Here,

   

   was defined as 0,16438 kg/kWh, and

   

   was assumed to be zero since solar generation, in its operation stage, is regarded as a green energy technology.

2. An evaluation of the effectiveness of the proposed SDP relaxation in managing DERs in medium-voltage distribution networks with a variable power factor, considering three simulation scenarios. The first scenario (S1) involved the daily dispatch of PV sources operating with a unitary power factor and BESUs with a variable power factor. In the second scenario (S2), the PV sources were dispatched on a daily basis with a variable power factor, while the BESUs maintained a unitary power factor. Finally, in the third scenario (S3), all DERs operated with a variable power factor. These scenarios were designed to provide a thorough assessment of the approach under different operating conditions.

Unitary power factor análisis

Applying the CbDO to solve the EDM developed for dispatching BESUs and PVs in distribution networks while considering a unitary power factor yielded the results presented in Table III.

Table III: Comparison of results regarding both objective functions analyzed

Based on these numerical results, the following can be stated:

1. The CbDO performs better than the other metaheuristic methods (CGA, PPSO, and PVSA) in relation to both objective functions. Specifically, regarding the expected daily energy losses, CbDO reports a reduction of 3.2000 kWh/day compared to PVSA. As for CO₂ emissions, the CbDO shows a decrease of 0.3489 kg/day with respect to PVSA. Although these improvements may appear modest, they underscore CbDO’s ability to effectively explore the solution space and avoid local optima, wherein other metaheuristics might exhibit some limitations.

2. In comparison with the benchmark case (i.e., grid operation with PVs but without BESUs), the CbDO shows significant reductions in both metrics. The daily energy losses are lowered by approximately 4.4261%, while the CO₂ emissions are reduced by about 0.1840% This confirms the potential benefits of integrating BESUs into distribution networks for optimal energy management, as it yields notable improvements in terms of technical efficiency and environmental sustainability.

Variable power factor análisis

To validate the effectiveness of the proposed optimization approach regarding a variable power factor operation of DERs, the objective function related to energy losses was analyzed in comparison with the results reported in ³¹, which employed an SDP approach.

In this analysis, the benchmark case corresponds to the numerical results reported in Table IV, which pertain to the unitary power factor operation of the PVs and BESS. Here, the CbDO results are labeled as S₀.

Table IV: Comparative analysis considering a variable power factor operation for minimizing energy losses

This comparison aids in evaluating the performance of the proposed CbDO and the SDP approach under various operating scenarios regarding DERs with variable power factor capabilities. Here, the energy losses function serves as the primary metric for assessing the effectiveness of the aforementioned optimization methods. For context, S0 represents the benchmark case where the PVs and BESUs operate with a unitary power factor.

• Benchmark comparison (S0). The baseline S0 scenario, wherein the PVs and BESUs are dispatched withaunitarypowerfactor,providestheinitial energylosses. According totheresults, the SDP approach outperforms the CbDO in this case, with a value of 2373.4053 kWh/day vs. our proposal’s 2374.6028 kWh/day. The marginal difference between these results suggests that, while both optimization methods are effective, the SDP approach has a slight edge in this specific configuration.

• Variable power factor for the BESUs (S1). In scenario S1, the BESUs are operated with a variable power factor, while the PVs continue to function at a unitary power factor. Here, both the SDP approach and CbDO demonstrate significant improvements in reducing energy losses compared to the benchmark case. The SDP approach achieves energy losses of 1470.3694 kWh/day, while CbDO yields 1472.1725 kWh/day. This substantial reduction vs. the S0 scenario suggests that allowing BESUs to operate with a variable power factor greatly enhances network efficiency. The close difference between the analyzed methods confirms our proposal’s comparable performance.

• Variable power factor for the PVs (S2). Scenario S2 explores the impact of operating PVs with a variable power factor while the BESUs maintain a unitary power factor. In this scenario, the SDP approach yields energy losses of 1297.2639 kWh/day, while the CbDO reports a slightly higher value (1298.9856 kWh/day). These results highlight the fact that incorporating a variable power factor operation for PVs contributes to further reducing energy losses, suggesting that flexibility in DERoperation is crucial for improving distribution system performance.

• Variable power factor operation (S3). Scenario S3, where all DERs are dispatched with a variable power factor, reports the most significant improvement in energy losses minimization. The SDP approach achieves the lowest energy losses (1266.7221 kWh/day), followed closely by CbDO (1267.8945 kWh/day). This scenario confirms that a variable power factor operation for all DERs provides the best overall performance. The minimal difference between the results for the SDP and CbDO methods further underscores the efficacy of our proposal as a reliable optimization tool that closely approaches the more complex SDP method.

In summary, the analysis in Table IV reveals that the proposed CbDO provides competitive results across all scenarios, closely matching the performance of the SDP solution. The most significant energy losses reductions occur when operating both the PVs and the BESUs with a variable power factor(S3). While the SDP approach slightly outperforms CbDO in each scenario, the differences are small enough to validate our proposal as a practical and efficient alternative for energy losses minimization in medium-voltage distribution networks with DERs. This validation emphasizes the importance of employing a flexible power factor in DER operation to optimize network efficiency