Stochastic Operation of BESS and MVDC Link in Distribution Networks Under Uncertainty

Abstract

This study introduces a stochastic optimization framework designed to effectively manage power flows in flexible medium-voltage DC (MVDC) link systems within distribution networks (DNs). The proposed approach operates in coordination with a battery energy storage system (BESS) to enhance the overall efficiency and reliability of the power distribution. Given the inherent uncertain characteristics associated with forecasting errors in photovoltaic (PV) generation and load demand, the study employs a distributionally robust chance-constrained optimization technique to mitigate the potential operational risks. To achieve a cooperative and optimized control strategy for MVDC link systems and BESS, the proposed method incorporates a stochastic relaxation of the reliability constraints on bus voltages. By strategically adjusting the conservativeness of these constraints, the proposed framework seeks to maximize the cost-effectiveness of DN operations. The numerical simulations demonstrate that relaxing the strict reliability constraints enables the distribution system operator to optimize the electricity imports more economically, thereby improving the overall financial performance while maintaining system reliability. Through case studies, we showed that the proposed method improves the operational cost by up to 44.7% while maintaining 96.83% bus voltage reliability under PV and load power output uncertainty.

1. Introduction

As the international drive toward carbon neutrality proliferates, the integration of distributed generators (DGs) based on renewable energy sources (RESs) into distribution networks (DNs) has been increasing steadily [1,2,3,4,5,6,7]. This trend introduces significant challenges, due to the inherent intermittency and variability of RESs. Such fluctuations can lead to unpredictable negative impacts on future DNs, including voltage instability and load imbalances, as documented in prior studies [2,8,9,10,11]. In response to these challenges, the present study investigates the utilization of flexible medium-voltage DC (MVDC) link facilities to mitigate the voltage instability and load imbalances that occur across different DNs [12,13,14,15,16,17]. Specifically, these studies implement back-to-back voltage source converter systems (BTB-VSCs) within medium-voltage DNs as an interconnection mechanism to flexibly link different DNs [18,19,20,21,22]. The previous literature has employed various terminologies—such as Soft-Open Point, Power Flow Controller, and MVDC link—to describe the BTB-VSC configuration in DNs; yet, these terms fundamentally refer to the same system architecture.

Recognizing the limitation of the DC system, which is primarily limited to controlling only power flows between DNs, the study proposes a coordinated operational strategy that integrates battery energy storage systems (BESSs) into the network infrastructure. By leveraging the dual capabilities of BESSs for charging and discharging, the proposed strategy aims not only to correct the power imbalances across multiple DNs but also to enhance the overall operational performance for the distribution system operator (DSO). Such improvements are expected to enhance the economic performance by increasing the energy sales revenue and reducing the power importing cost.

Although prior research has advanced in improving DN operational efficiency through DC link integration, these studies have exhibited notable limitations; for example, the approaches in [23,24] employed robust optimization techniques that focused on worst-case scenarios arising from RES forecast errors, thereby achieving a minimal rate of constraint violations. However, such conservative methodologies may unexpectedly degrade operational efficiency, particularly in minimizing operational costs.

Alternatively, the research presented in [25,26] adopted stochastic optimization frameworks that modeled uncertainties using probabilistic functions, which allowed for more flexible DN operations. Despite this flexibility, the requirement to predefine the mathematical form of the probability distribution limits the practical applicability of these models.

To overcome these shortcomings, the proposed study employs a distributionally robust chance constrained optimization (DRCCO) method. This approach is designed to maintain robust performance in stochastic optimization without necessitating prior knowledge of the exact probability distributions of uncertainty variables.

Unlike traditional robust optimization, which is overly conservative, or scenario-based methods that require predefined distributions and a large number of samples, the DRCCO framework offers a flexible trade-off between reliability and cost without relying on strict distributional assumptions. Additionally, compared to the scenario-based approach in [27], which uses metaheuristic algorithms and normal distribution assumptions, our method is solver-friendly, distribution-free, and better suited for practical MVDC-based DN operations.

Within this framework, the optimal power flow control between DNs is achieved via the MVDC link, while the coordinated integration of BESSs further increases the operational benefits for the DSO. The validity and effectiveness of this integrated strategy are demonstrated through a case study based on the modified IEEE 33-bus test system. Additionally, the study conducts an in-depth analysis of several key parameters that influence the trade-off performance of the DRCCO model, thereby providing comprehensive insights into the optimization process and its practical implications for future DNs. Compared to previous research, the main contributions of this study are summarized as follows:

• In this work, we propose a day-ahead optimal scheduling strategy for MVDC links that aims to reduce the energy procurement costs for DSOs by coordinating with BESS under forecast uncertainties. The proposed method incorporates the DRCCO framework to optimize power allocation and ensure reliable operation across interconnected distribution networks.
• Furthermore, we conduct detailed case studies to investigate the impact of critical DRCCO parameters—namely, the ambiguity set radius and the confidence level—on the operational cost, voltage reliability, and energy loss. These analyses offer valuable insights for DSOs seeking to balance economic efficiency and system robustness.

The remainder of this paper is structured as follows. Section 2 introduces the MVDC link and presents the mathematical formulation of the proposed DRCCO problem. Section 3 discusses the simulation results and performance evaluation. Finally, Section 4 concludes the paper by summarizing the key findings.

2. Problem Formulation

2.1. Flexible MVDC Link System

As illustrated in Figure 1, the system under consideration consists of a single DC link shared by two VSCs, which are interconnected in a BTB configuration through a relatively short DC transmission line. Each VSC is responsible for establishing a connection with a distinct DN, thereby enabling controlled power exchange between the networks. The steady-state operational characteristics of this system can be mathematically formulated as follows [28,29].

$${\sum }_{k=1}^2(P_{conv.t}^k+P_{conv.loss.t}^k)=0 ,$$

(1)

$$P_{conv.loss.t}^k=C_{conv.loss}\sqrt{{\left(P_{conv.t}^k\right)}^2+{\left(Q_{conv.t}^k\right)}^2},$$

(2)

$$\sqrt{{\left(P_{conv.t}^k\right)}^2+{\left(Q_{conv.t}^k\right)}^2}\le S_{conv},$$

(3)

where, $P_{conv.t}^k$ and $P_{conv.loss.t}^k$ represent the active power output of k-th converter in the MVDC link. $C_{conv.loss}$ is the loss coefficient of the converter, which is generally set to 0.02. $Q_{conv.t}^k$ and $S_{conv}$ are the reactive power output and rated capacity of k-th converter, respectively.

In this formulation, Equation (1) represents the fundamental principle of power distribution within the MVDC link, ensuring that the total power exchanged through the DC link is balanced. The power losses within the converters are captured in (2). Additionally, Equation (3) represents constraints on the maximum power output of the converters, thereby ensuring operational feasibility within predefined technical limits.

To facilitate computational efficiency and tractability, the nonlinear constraint in (2) can be relaxed into a second-order cone inequality, as shown in (4). The validity of this relaxation has been extensively studied in the prior research [23,24], demonstrating its effectiveness in accurately approximating converter losses while preserving the solution feasibility.

$$\sqrt{{\left(P_{conv.t}^k\right)}^2+{\left(Q_{conv.t}^k\right)}^2}\le {\displaystyle \frac{P_{conv.loss.t}^k}{C_{conv.loss}}}$$

(4)

2.2. Battery Energy Storage System Model

The operational model of a BESS within DNs is formulated to account for its ability to perform both active power charging and discharging while enabling independent control of active and reactive power outputs. The mathematical representation of this model is structured as follows [30,31].

$$\sqrt{{\left(P_{bess.t}^k\right)}^2+{\left(Q_{bess.t}^k\right)}^2}\le S_{bess},$$

(5)

$$P_{bess.loss.t}^k=C_{bess.loss}\sqrt{{\left(P_{bess.t}^k\right)}^2+{\left(Q_{bess.t}^k\right)}^2},$$

(6)

$${SOC}_{(t+1)}^k={SOC}_{\left(t\right)}^k-\left(P_{bess.t}^k+P_{bess.loss.t}^k\right)∆t,$$

(7)

$$SO{C}_{min}\le {SOC}_{\left(t\right)}^k\le SO{C}_{max},$$

(8)

where, Equation (5) imposes constraints on the maximum output capacity of the BESS. $P_{bess.t}^k$ and $Q_{bess.t}^k$ represent the active and reactive power outputs, respectively, of the BESS installed in the k-th DN, with a rated power capacity of $S_{bess}$. The power losses associated with the DC/DC converter in BESS ($P_{bess.loss.t}^k$) are characterized in (6), which account for the conversion inefficiencies using the device loss coefficient ($C_{bess.loss}$). To effectively manage the energy stored within the BESS, Equation (7) establishes constraints on the state-of-charge (SOC), meaning that it operates within an acceptable range to maintain the battery health. Lastly, Equation (8) specifies the allowable SOC range by setting upper and lower bounds, thereby preventing the overcharging or excessive depletion of the BESS.

Similar to the relaxation of the MVDC link losses, Equation (6) can also be approximated to a second-order cone constraint as shown in (9).

$$\sqrt{{\left(P_{bess.t}^k\right)}^2+{\left(Q_{bess.t}^k\right)}^2}\le {\displaystyle \frac{P_{bess.loss.t}^k}{C_{bess.loss}}}$$

(9)

2.3. Load Flow Calculation Model

In this study, the power flow constraints are modeled using the DistFlow method, which provides an accurate representation of power flows in radial DNs [32]. This method is particularly effective for capturing the relationship between bus voltages, power injections, and line flows in distribution systems. The mathematical formulation of the DistFlow-based power flow constraints is presented as follows [32].

$$U_{i.t}^k=U_{j.t}^k-2\left(r_{ij}^k{P}_{ij.t}^k+x_{ij}^k{Q}_{ij.t}^k\right)+\left({\left(r_{ij}^k\right)}^2+{\left(x_{ij}^k\right)}^2\right)l_{ij.t}^k ,$$

(10)

$${\sum }_{i:i\to j}{P}_{ij.t}^k-{\sum }_{i:i\to j}{r}_{ij}^k{l}_{ij.t}^k-{\sum }_{h:j\to h}{P}_{jh.t}^k=P_{load.j.t}^k-P_{pv.j.t}^k-P_{conv.t}^k-P_{bess.j.t}^k,$$

(11)

$${\sum }_{i:i\to j}{Q}_{ij.t}^k-{\sum }_{i:i\to j}{x}_{ij}^k{l}_{ij.t}^k-{\sum }_{h:j\to h}{Q}_{jh.t}^k=Q_{load.j.t}^k-Q_{pv.j.t}^k-Q_{conv.t}^k-Q_{bess.j.t}^k ,$$

(12)

$${\displaystyle \frac{{\left(P_{ij.t}^k\right)}^2+{\left(Q_{ij.t}^k\right)}^2}{U_{i.t}^k}}\le l_{ij.t}^k,$$

(13)

where, the squared bus voltage and line current are defined as $U_{i.t}^k={\left(V_{i.t}^k\right)}^2$ and $l_{i.t}^k={\left(I_{ij.t}^k\right)}^2$. $r_{ij}^k$ and $x_{ij}^k$ denote the resistance and reactance magnitudes of branch (i, j) in k-th DN, respectively.$P_{ij.t}^k$ and $Q_{ij.t}^k$ represent the active and reactive power flows into branch (i, j) at time t in k-th DN, respectively. $P_{pv.j.t}^k$ and $Q_{pv.j.t}^k$ mean the active/reactive power output of the PV generators, $P_{load.j.t}^k$ and $Q_{load.j.t}^k$ also represent the load demands, respectively.

2.4. Uncertainty Variable Model

To effectively capture uncertainties from forecast error when predicting load demand and PV power generation outputs, this study models them as a single aggregated uncertainty variable for each DN connected to the MVDC link. The mathematical representation of this uncertainty modeling is provided as follows.

$${\tilde{P}}_{load.j.t}^k=P_{load.j.t}^k\left(1+{\xi }_{load.t}^k\right),\hspace{0.33em}\forall j\in {\Omega }_{load}^k,$$

(14)

$${\tilde{P}}_{pv.j.t}^k=P_{pv.j.t}^k\left(1+{\xi }_{pv.t}^k\right),\hspace{0.33em}\forall j\in {\Omega }_{pv}^k,$$

(15)

where superscript ($\widetilde{·}$) denotes the value after the error has been applied to the predicted value of the variable. ${\xi }_{pv.t}^k$ denotes the uncertainty variables for PV output forecasts. Specifically, it quantifies the proportional deviation of the actual PV output from the forecasted value at time t, reflecting PV generation uncertainty. ${\xi }_{load.t}^k$ represents the uncertainty variables for load demand forecasts. This also represents the proportional deviation of the actual load demand from the forecasted value at time t, capturing the load forecast uncertainty. Both coefficients range from 0 to 1. $P_{pv.j.t}^k$ and $P_{load.j.t}^k$ represent the day-ahead forecasted values of PV generation and load demand for bus j at time t in the k-th DN. These forecast values are obtained based on historical data and predictive models.

Finally, the bus voltage, which reflects the forecast errors of PV generation and load output, as well as the effects of the outputs from the MVDC link and BESS, can be expressed using sensitivity coefficients as follows.

$${\tilde{V}}_{i.t}^k=V_{i.t}^k+\Delta {\tilde{V}}_{load.i.t}^k+\Delta {\tilde{V}}_{pv.i.t}^k+\Delta V_{conv.i.t}^k+\Delta V_{bess.i.t}^k$$

(16)

$$\Delta {\tilde{V}}_{load.i.t}^k=\sum _{j\in {\mathit{\Omega }}_{load}^k}\left\{{\displaystyle \frac{\mathit{\partial }{V}_i^k}{\mathit{\partial }{P}_j^k}}{P}_{load.j.t}^k+{\displaystyle \frac{\mathit{\partial }{V}_i^k}{\mathit{\partial }{Q}_j^k}}{Q}_{load.j.t}^k\right\}\left({\xi }_{load.t}^k\right)$$

(17)

$$\Delta {\tilde{V}}_{pv.i.t}^k=\sum _{j\in {\mathit{\Omega }}_{pv}^k}\left\{{\displaystyle \frac{\mathit{\partial }{V}_i^k}{\mathit{\partial }{P}_j^k}}{P}_{pv.j.t}^k+{\displaystyle \frac{\mathit{\partial }{V}_i^k}{\mathit{\partial }{Q}_j^k}}{Q}_{pv.j.t}^k\right\}\left({\xi }_{pv.t}^k\right)$$

(18)

$$\Delta V_{conv.i.t}^k={\displaystyle \frac{\mathit{\partial }{V}_i^k}{\mathit{\partial }{P}_j^k}}{P}_{conv.t}^k+{\displaystyle \frac{\mathit{\partial }{V}_i^k}{\mathit{\partial }{Q}_j^k}}{Q}_{conv.t}^k$$

(19)

$$\Delta V_{bess.i.t}^k={\displaystyle \frac{\mathit{\partial }{V}_i^k}{\mathit{\partial }{P}_j^k}}{P}_{bess.t}^k+{\displaystyle \frac{\mathit{\partial }{V}_i^k}{\mathit{\partial }{Q}_j^k}}{Q}_{bess.t}^k$$

(20)

While there are various previous studies related to the calculation methods for voltage-to-power sensitivity, this study adopts the methodology in [33] due to its relatively fast computation speed.

While the sensitivity coefficients were calculated using the method in [33] for computational efficiency, we acknowledge that this approach involves linear approximations. To ensure that the DRCCO reliability guarantees were met, we performed a Monte Carlo-based scenario validation using 3000 additional forecast error samples and full nonlinear power flow calculations, as described in Section 3.

2.5. Distributionally Robust Chance-Constrained Optimization Model

Chance-constrained optimization is an advanced optimization technique that ensures constraints are satisfied with a predefined confidence level in problems involving uncertainty. This approach formulates the optimization problem such that the probability of constraint violations does not exceed a specified threshold, thereby enhancing the reliability of decision-making under uncertainty.

Traditional studies addressing chance-constrained optimization typically assume that the probability distribution functions are precisely known. However, in practical applications, it is often difficult to assert with confidence that uncertainty factors—such as forecast errors in PV generation and load demand—adhere to specific probability distributions, such as Gaussian distribution. This uncertainty in distribution modeling necessitates the use of a methodology to quantify the distance between different probability distributions.

Due to these limitations, the concept of DRCCO has been proposed [34]. As illustrated in Figure 2, an ambiguity set is constructed, which encompasses various potential probability distributions of the uncertain variables. This ambiguity set is typically defined on the empirical distribution function derived from sample data of the uncertainty. It is formulated to include all distributions within a certain radius around the empirical distribution, thereby considering the true distribution (but unknown) in a realistic manner. Among various measures of distributional distance, this study adopts the Wasserstein distance [34] for its robustness and flexibility in addressing distributional uncertainty.

$$d_W\left({\mathbb{Q}}_1,{\mathbb{Q}}_2\right)≔\underset{f\in \mathcal{L}}{sup}\left\{{\int }_{\mathit{\Xi }}f\left(\xi \right){\mathbb{Q}}_1\left(d\xi \right)-{\int }_{\mathit{\Xi }}f\left(\xi \right){\mathbb{Q}}_2\left(d\xi \right)\right\} ,$$

(21)

$${\mathcal{B}}_{\epsilon }\left({\widehat{\mathbb{P}}}_N\right)≔\left\{\mathbb{P}\in \mathcal{M}\left(\mathit{\Xi }\right):\hspace{0.33em}{d}_W\left(\mathbb{P},{\widehat{\mathbb{P}}}_N\right)\hspace{0.33em}\le \hspace{0.33em}\epsilon \right\},$$

(22)

$$\underset{\mathbb{P}\in {\mathcal{B}}_{\epsilon }\left({\widehat{\mathbb{P}}}_N\right)}{inf}\mathbb{P}\left(V_{min}\le {\tilde{V}}_{i.t}^k\le V_{max}\right)\ge 1-{\alpha }_V,$$

(23)

where Equation (21) defines the Wasserstein distance used to measure the difference between two probability distributions (${\mathbb{Q}}_1,{\mathbb{Q}}_2$), while Equation (22) specifies that all distributions within the ambiguity set must lie within a certain radius ($\epsilon$) from a reference distribution (${\widehat{\mathbb{P}}}_N$). $V_{min}$ and $V_{max}$ denote the allowable bus voltage limits, while $(1-{\alpha }_V)$ represents the minimum confidence level for constraint satisfaction. For instance, ${\alpha }_V=0.05$ limits the bus voltage violation probability to 5% under PV and load demand forecast errors. This constraint ensures that the true distribution lies within a Wasserstein ball of radius ε centered at the empirical distribution, enabling the model to protect against sampling errors and distributional shifts while maintaining tractability.

The choice of the Wasserstein distance for defining the ambiguity set was motivated by several factors [34,35,36,37,38]. First, the Wasserstein metric offers greater modeling flexibility in scenarios with limited sample sizes, as it does not require the empirical distribution to be absolutely continuous with respect to the true distribution, unlike other measure such as the Kullback–Leibler divergence [38]. Second, it captures both the shape and support of the distributions, allowing for better characterization of the distributional shifts that may occur under extreme PV and load forecast errors. Finally, the DRCCO problem formulated with the Wasserstein distance can be converted into a tractable convex optimization problem, enabling an efficient solution using standard solvers. Although this study did not perform a direct comparison with other distance metrics, the sensitivity analysis as shown in the following tables and figures in Section 3 illustrates how varying the radius of the Wasserstein ambiguity set impacts the operational cost and voltage reliability.

In (23), both lower and upper constraints are included. Therefore, the inequalities for the left-hand side and right-hand side can be separated and expressed as (24) and (25). In the case of voltage, it can be expressed as a linear combination of the active/reactive power output variables of the MVDC link and BESS, as well as uncertainty variables, similar to (16). Thus, it can be represented by the following Equations:

$$\underset{\mathbb{P}\in {\mathcal{B}}_{\epsilon }\left({\widehat{\mathbb{P}}}_N\right)}{inf}\mathbb{P}\left({\mathit{\ell }}_{V.t}(x,\xi )\le 0\right)\ge 1-{\alpha }_V ,$$

(24)

$${\mathit{\ell }}_{V.t}\left(x,\xi \right)=\underset{k\in N_d,i\in {\mathit{\Omega }}_b^k}{max}\left\{\left({\tilde{V}}_{i.t}^k-V_{max}\right),\left(V_{min}-{\tilde{V}}_{i.t}^k\right)\right\},$$

(25)

$${\mathit{\ell }}_{V.t}\left(x,\xi \right)=\underset{n\in N_p}{max}\left({\mathit{a}}_{\mathit{n}.\mathit{t}}{\mathit{X}}_{conv.t}+{\mathit{b}}_{\mathit{n}.\mathit{t}}{\mathit{\xi }}_{\mathit{t}}+{\mathit{c}}_{\mathit{n}.\mathit{t}}\right),$$

(26)

where the vector ${\mathit{X}}_{\mathit{c}\mathit{o}\mathit{n}\mathit{v}.\mathit{t}}=\left[P_{conv.t}^k,Q_{conv.t}^k,P_{bess.t}^k,Q_{bess.t}^k\right]$ represents the set of decision variables, which includes the active and reactive power outputs of both the MVDC link and the BESS at time t. The vector ${\mathit{a}}_{\mathit{n}.\mathit{t}}$ denotes the voltage sensitivities of bus voltages in the DN, as expressed in (19) and (20). The vector ${\mathit{b}}_{\mathit{n}.\mathit{t}}$ consists of coefficients that correspond to the forecast errors in PV generation and load demand, as detailed in (17) and (18). These coefficients account for the uncertainty associated with the forecasting of generation and demand. Finally, ${\mathit{c}}_{\mathit{n}.\mathit{t}}$ represents the differences between the predicted voltage and the maximum/minimum allowable voltage limits. Specifically, it includes the terms ($\left(V_{i.t}^k-V_{max}\right),\left(V_{min}-V_{i.t}^k\right)$), which quantify the deviation from the upper and lower voltage constraints, respectively.

The implementation of the DRCCO in (24) using standard optimization solvers can be facilitated by applying the results from earlier research [34], which enable the problem to be reformulated in its dual form as follows:

$${\lambda }_t\epsilon +{\displaystyle \frac{1}{N\left({\mathit{\Omega }}_s\right)}}{\sum }_{k\in {\mathit{\Omega }}_s}{\delta }_{t.k}\le {\alpha }_V{\theta }_t ,$$

(27)

$$\left({\mathit{a}}_{n.t}{\mathit{X}}_{conv.t}+{\mathit{b}}_{n.t}{\overline{\mathit{\xi }}}_{\mathit{t}}+{\mathit{c}}_{\mathit{n}.\mathit{t}}\right)+{\mathit{\rho }}_{n.i.t}^T\left(\mathit{d}-\mathit{C}{\overline{\mathit{\xi }}}_{\mathit{t}}\right)+{\mathit{\theta }}_{\mathit{t}}\le {\delta }_{t.k},$$

(28)

$${‖{\mathit{C}}^T{\mathit{\rho }}_{n.i.t}^T-{\mathit{b}}_{n.t}‖}_{\infty }\le {\lambda }_t ,$$

(29)

$${\lambda }_t\ge 0, {\delta }_{t.k}\ge 0, {\mathit{\rho }}_{n.i.t}^T\ge 0,$$

(30)

where the vector ${\mathit{X}}_{\mathit{d}\mathit{u}\mathit{a}\mathit{l}.\mathit{t}}=\left[{\mathit{\rho }}_{\mathit{n}.\mathit{i}.\mathit{t}},{\mathit{\delta }}_{\mathit{t}.\mathit{k}},{\mathit{\theta }}_{\mathit{t}},{\lambda }_t\right]$ denotes the decision variables in the dual problem.

It is worth noting that the relaxation of bus voltage reliability constraints within the DRCCO framework is bounded by a predefined confidence level (e.g., 95%), which quantitatively limits the risk of constraint violation under uncertainty. This does not imply a relaxation of the absolute voltage limits specified in standards such as IEEE 1547 or IEC 61,970 but rather allows for probabilistic operation under high-variability conditions. This approach is consistent with emerging grid codes that allow limited statistically controlled deviations in the presence of renewable generation. Moreover, if necessary, reactive power control through BESS and MVDC converters can provide dynamic voltage support, mitigating any adverse impacts on long-term system stability.

2.6. Proposed Optimization Model

Through the proposed coordinated operation of the MVDC link and BESS, the DSO can focus on reducing operational costs within the DN or maximizing profits derived from power sales to the upstream grid. To achieve this, the objective function is formulated as the cumulative value of electricity procurement costs over the course of a day for the DNs that are interconnected through the MVDC link. Since the operational objective of the DSO is to minimize the expected value of the electricity procurement costs across all DNs interconnected via the MVDC link and BESS, the expected value of the sum of net loads for the two DNs can be formulated as (31) and (32).

$${\tilde{P}}_{nl.t}^k=\sum _{j\in {\mathit{\Omega }}_b^k}\left({\tilde{P}}_{load.j.t}^k-{\tilde{P}}_{pv.j.t}^k\right)+\sum _{(i,j)\in {\mathit{\Omega }}_l^k}\left(r_{ij}^k{\tilde{l}}_{ij}^k\right)-P_{conv.t}^k$$

(31)

$$\mathbb{E}\left[\sum _{k=1}^2{\tilde{P}}_{nl.t}^k\right]=\sum _{k=1}^2\left\{\sum _{j\in {\mathit{\Omega }}_b^k}\left(P_{load.j.t}^k-P_{pv.j.t}^k-P_{bess.j.t}^k\right)+\sum _{(i,j)\in {\mathit{\Omega }}_b^k}\left(r_{ij}^k{l}_{ij}^k\right)+P_{conv.loss.t}^k\right\}$$

(32)

This formulation assumes that the forecast errors for the load demand and PV power generation, as represented in (14) and (15), have an expected value of zero. Consequently, this implies that the expected value of the line power flows, which are influenced by these uncertainties, is also zero. Furthermore, it is also important to note that the sum of the outputs in the MVDC link, along with the converter losses, is zero, as described in (1). This condition is crucial for ensuring power conservation within the MVDC link and simplifies the formulation of the optimization problem by removing any power imbalances in the interconnected DNs.

This study assumes a scenario where two distinct DNs are interconnected through the MVDC link. Based on this assumption, an optimal operation problem is formulated with the objective of minimizing the total electricity procurement cost across both DNs. The problem is structured as follows, considering the relevant operational constraints and the interactions between the MVDC link and BESS in DNs.

$$\begin{gathered} \underset{X_{\mathit{conv}}}{\mathit{min}}\sum _{t=1h}^T{C}_{s.t}\left\{\sum _{(i,j)\in {\mathit{\Omega }}_b^k}\left(r_{ij}^k{l}_{ij}^k\right)+P_{conv.loss.t}^k-P_{bess.t}^k\right\}∆t, \\ s.t.\left(1\right)–\left(32\right) \end{gathered}$$

(33)

The visual representation of the optimization problem proposed in this study is shown in Figure 3. Depending on the user’s operational intent, the ambiguity set is constructed by inputting two key parameters from the DRCCO models: the radius of the ambiguity set and the violation probability, along with sample data of the uncertainty variables.

By utilizing the predicted daily profiles of the PV generation and load, along with hourly electricity cost data, the daily operation model of the DN, the operation model of the MVDC link and BESS, and the objective function are formulated. These components are integrated to construct the final optimization model for the probabilistic operation of the MVDC link and BESS. This approach ensures the coordinative operation between the MVDC link and BESS considering the uncertainty of PV and load outputs in DNs.

Within the architecture of the proposed DRCCO framework in Figure 3, the power flow is managed in a coordinated manner between the interconnected DNs, the MVDC link, and the BESS units. The MVDC link facilitates bidirectional active power transfer between the two DNs, enabling the redistribution of excess generation or load support depending on real-time net load imbalances. Simultaneously, the BESS units in each DN perform active power charging and discharging operations to further mitigate forecast uncertainties and optimize the total procurement cost. During periods of high PV generation, surplus power can be stored in the BESS or exported to the neighboring DN via the MVDC link. Conversely, during peak load hours or when electricity prices are high, stored energy in the BESS can be discharged to reduce grid procurement costs.

The proposed DRCCO framework determines the optimal setpoints for MVDC link power flows and BESS outputs in each time step, ensuring that all operational constraints—such as voltage limits and power balance—are satisfied within the desired reliability level under forecast uncertainties. This integrated power flow coordination between the MVDC link and BESS plays a critical role in enhancing both economic efficiency and voltage stability across the interconnected DNs.

3. Case Studies

To validate the proposed approach in this study, a scenario was assumed where two IEEE 33-bus test systems are interconnected through the MVDC link, as illustrated in Figure 4. The detailed specifications of these test systems are summarized in

Table 1. Detailed specification of test systems.

DN	PV Location	MVDC Link Location	BESS Location
1	8, 10, 24, 28, 33	18	33
2	11, 22, 29	33	18
DN	PV capacity	MVDC Link Capacity	BESS Capacity
1 and 2	700 kVA (Each)	1000 kVA	500 kVA/1500 kWh

. The IEEE 33-bus test system was chosen, as it is a widely used benchmark in DN studies, especially for MVDC link and BESS operation analysis [18,19,20,21,22].

As shown in Figure 4, DN1 has a higher PV penetration compared to DN2, and both the MVDC link and BESS units are placed at the end buses of each feeder to maximize their voltage regulation impact. The daily forecast profiles for PV generation, load demand, and hourly power purchase costs for each test system are shown in Figure 5 and Figure 6 [39]. Figure 5 and Figure 6 illustrate the daily load share and electricity price profiles, respectively. During the day, DN2 accounts for a higher share of the total load, while in the evening—when PV output drops to zero—DN1 becomes dominant. In addition, electricity import prices peak during 8:00–10:00 and 18:00–20:00.

All simulations were performed using MATLAB R2022b. The optimization problems were implemented using the CVX modeling language and solved with the MOSEK solver.

Figure 7 and Figure 8 present the results of determining the optimal power outputs of the MVDC link and the BESS in each DN over a day. As outlined in the optimal operation problem described in (33), this optimization was achieved by minimizing the operational costs while restricting the probability of constraint violations regarding bus voltage maintenance in the two interconnected DNs to a specified value. Using the predicted daily PV generation and load outputs, the proposed DRCCO model was applied to the interconnected DNs (DN1 and DN2). This approach ensured that the bus voltage confidence level remained within the desired probability (95%) through the optimal operation of the BESS and MVDC link.

To form the ambiguity set for uncertainty variables, such as forecast errors in PV generation and load demand, 30 samples were used in this study ($N_S$ = 30).

As observed in Figure 7, the power imbalance caused by the differences in net load between the two DNs can be mitigated to some extent through the operation of the MVDC link, given the differences in their load profiles and PV capacities.

For example, during the 18–24 h period, when the load in DN1 is higher than in DN2, the MVDC link draws active power from DN2 and injects it into DN1, balancing the load between the two networks. When the net loads connected through the MVDC link are summed, the power injected through the MVDC link offsets each other in opposite directions, as shown in (32). This behavior ensures that the MVDC link operates in a way that reduces line losses across the interconnected DNs, thereby indirectly contributing to the minimization of operational costs.

In Figure 8, during periods of low electricity costs (4–6 AM, 2–4 PM), the BESS in each DN performs maximum charging to take advantage of the cheaper electricity. On the other hand, during high-cost periods, such as 8–11 AM and 6–8 PM, the BESS maximizes the discharge of stored power, effectively utilizing opportunities to reduce the overall procurement costs by discharging when electricity prices are higher. This strategy optimizes both the operational efficiency and the economic performance of the interconnected DNs. Furthermore, by controlling the reactive power output, the MVDC link and BESS in each DN play a key role in voltage stabilization, ensuring that the system can deliver maximum power output when necessary.

To further compare and verify the performance of the proposed method, four scenarios were considered:

• Scenario I: Baseline cases without the integration of the MVDC link and BESS.
• Scenario II: Incorporates the MVDC link and BESS, employing a deterministic optimization framework that neglects forecast uncertainties.
• Scenario III: Incorporates the MVDC link and BESS, employing a robust optimization that considers only the maximum and minimum values of forecast uncertainties.
• Scenario IV: Incorporates the MVDC link and BESS, employing the proposed DRCCO method to explicitly account for forecast uncertainties.

Table 2. Comparison of operational performance under different scenarios.

Scenarios	Average Energy Loss Per Day [kWh/day]	Optimal Cost [USD/day]	Bus Voltage Reliability [%]
I	3490.90	132.86	0
II	2237.10	−53.38	75.17
III	2403.20	−33.96	100
IV	2255.37	−49.14	96.83

presents a performance comparison across several quantitative indicators for each scenario. A comparison with Scenario I clearly shows that the coordinated operation of the BESS and MVDC link can transform the optimal costs of the two DNs into negative values, indicating that power sales generate profits. However, Scenario II, which applies a deterministic approach that neglects PV and load forecast uncertainties, shows the lowest optimal cost (USD −53.38/day) but suffers from severely degraded voltage reliability (75.17%). This low reliability arises because the deterministic method does not consider forecast errors, leading to frequent constraint violations in real operation. In contrast, Scenario III, based on robust optimization, guarantees 100% bus voltage reliability by considering worst-case forecast errors but at the expense of higher operational cost (USD −33.96/day). Notably, the proposed DRCCO approach (Scenario IV) effectively balances these two aspects, achieving a high voltage reliability of 96.83% while securing a 44.7% cost improvement compared to the robust method.

Specifically, 3000 additional forecast error scenarios were generated for the predetermined operating points, power flow calculations were performed, and the number of scenarios where constraints were satisfied was counted to calculate the rate.

In contrast, Scenario III, which employs robust optimization, shows no constraint violations (100% reliability); however, its conservative nature leads to the least significant minimization of the objective function. Finally, in Scenario IV—using the proposed DRCCO—the bus voltage reliability is regulated to approximately 96.83%, resulting in an increase in the DSO’s power sales revenue by up to $49.14 per day, which is an 44.7% improvement compared to the robust optimization approach.

Furthermore,

Table 3. Impact of voltage confidence level on operational performance.

Bus Voltage Confidence Level $(1-{\mathit{\alpha }}_{\mathit{V}})$	Average Energy Loss Per Day [kWh/day]	Cost Improvement [%]	Bus Voltage Reliability [%]
95%	2255.37	44.7	96.83
90%	2251.02	47.1	95.40
85%	2247.87	49.8	92.40
80%	2244.49	51.2	90.03

compares the performance based on the size of the bus voltage confidence level ($1-{\alpha }_V$). As the set value for voltage violation increases, the DRCCO model allows for a broader range of constraints. This expansion leads to a lower bus voltage reliability (i.e., higher constraint violation rate); however, it also provides greater flexibility in the outputs of the MVDC link and BESS, thereby resulting in increased power sales revenue.

Lastly,

Table 4. Impact of radius of ambiguity set on operational performance.

Radius of Ambiguity Set $\mathbf{\left(}\mathit{\epsilon }\mathbf{\right)}$	Average Energy Loss Per Day [kWh/day]	Cost Improvement [%]	Bus Voltage Reliability [%]
0.001	2248.21	49.2	91.67
0.002	2250.54	47.1	94.73
0.003	2255.37	44.7	96.83
0.005	2270.79	38.7	98.63

presents a performance comparison based on the radius of the ambiguity set employed in the proposed DRCCO problem. As the radius of the ambiguity set increases relative to the sample data, the candidate set for uncertainty variables is defined more broadly, which leads to more conservative solutions to the optimization problem. As presented in Table 4, an increase in the radius leads to higher average network energy losses and the increase in power sales revenue.

Figure 9 visualizes this trade-off relationship between bus voltage reliability and cost improvement with respect to varying voltage confidence levels and radius of ambiguity sets. This highlights the inherent balance between economic efficiency and operational robustness in the proposed DRCCO-based dispatch strategy.

Consequently, the impact of two key parameters—namely, the confidence level ($1-{\alpha }_V$) and the radius of the ambiguity set ($\epsilon$)—on system performance was evaluated using three primary metrics: operational cost (USD/day), bus voltage reliability (%), and cost improvement (%). Additionally, the average energy loss (kWh/day) was reported to capture the system efficiency. As shown in Table 3 and Table 4, these indicators enable a quantitative understanding of the trade-off between economic benefits and reliability risks under different DRCCO settings.

The selection of the ambiguity set radius (ε) and the voltage confidence level ($1-{\alpha }_V$) was guided by a sensitivity analysis, as presented in Table 3 and Table 4 and Figure 9. By evaluating the trade-off between operational cost and voltage reliability across different parameter settings, we identified that ε = 0.003 and ($1-{\alpha }_V$) = 95% offered a reasonable balance. Further increases in parameter conservativeness beyond this point resulted in marginal reliability improvements but a sharp rise in operational cost.

Figure 10 and

Table 5. Comparison of performance indices under different sample sizes.

Sample Sizes	30	50	100	200	300
Cost Improvement [%]	44.7	42.8	41.0	39.7	38.9
Bus voltage reliability [%]	96.83	96.94	98.07	98.77	98.82
Computation time [s]	53.05	90.13	212.95	635.59	1550.37

present the impact of varying sample sizes on the cost improvement, voltage reliability, and computation time within the proposed DRCCO framework. As the sample size increases from 30 to 300, the voltage reliability improves steadily from 96.83% to 98.82%, while the cost improvement declines from 44.7% to 38.9%. Notably, the computation time increases sharply, from 53.05 [s] at 30 samples to 1550.37 [s] at 300 samples.

Increasing the sample size improves voltage reliability but also leads to a substantial rise in computation time. For example, the reliability increases from 96.83% (30 samples) to 98.07% (100 samples), while the computation time grows from 53 to 213 s. Beyond 100 samples, the reliability gain becomes marginal, whereas the computation time increases sharply. Therefore, a sample size between 30 and 100 is recommended to balance the reliability and computational efficiency for day-ahead scheduling applications. These results highlight the trade-off between computational burden and performance accuracy, underscoring the importance of selecting an appropriate sample size that balances reliability, cost efficiency, and tractability.

Additionally, to evaluate the scalability of the proposed DRCCO framework for more complex DNs, we extended the case study to three DNs interconnected with three independent MVDC links. The detailed configuration of the system is illustrated in Figure 11. The data of DN1 (IEEE-33 bus system) are from [40], DN2 (IEEE-69 bus system) and DN3 (IEEE-85 bus system) are from [41] and [42], respectively. This setup enables enhanced power exchange flexibility among multiple DNs, each equipped with its own BESS.

The detailed specifications of each test system are summarized in

Table 6. Detailed specification of each test system in the multiple DNs.

DN	PV Location	MVDC Link Location			BESS Location
		MVDC1	MVDC2	MVDC3
1 (IEEE-33)	8, 10, 24, 28, 33	18	-	33	25
2 (IEEE-69)	16, 20, 24, 56, 62	65	27	36	69
3 (IEEE-85)	30, 34, 52, 54, 63, 69, 82	-	51	-	71
DN	PV Capacity	MVDC Link Capacity			BESS Capacity
1 and 2	700 kVA (Each)	1000 kVA			500 kVA/1500 kWh
3	500 kVA (Each)	1000 kVA			500 kVA/1500 kWh

. As shown, the system includes three MVDC links (MVDC1, MVDC2, and MVDC3) connecting the DNs, with each DN equipped with its own BESS to enhance operational flexibility.

Figure 12 and Figure 13 illustrate the optimal power dispatch of MVDC links and BESS in the three DNs. These dispatch profiles reflect the hourly net load variations of each DN in Figure 14. During low net load periods in DN3 (e.g., 1:00–7:00), active power is transferred from DN3 to DN1 and DN2 through MVDC2 and MVDC3. Similarly, during midday hours (12:00–17:00), surplus power from DN1 and DN3 is delivered to DN2, while in the evening (18:00–24:00), DN2 and DN3 supply power to DN1 to meet its higher net load.

In parallel, the BESS units perform charging during low-price periods and discharging during high-price hours. The coordinated operation of MVDC links and BESSs effectively balances the net load differences among the DNs while minimizing the operational costs under the proposed DRCCO framework.

Under the simulation using a three-DN interconnected test system,

Table 7. Detailed specification of each test system.

Scenarios	Bus Voltage Confidence Level $(1-{\mathit{\alpha }}_{\mathit{V}})$	Optimal Cost [USD/Day]	Cost Improvement [%]	Bus Voltage Reliability [%]
II	-	325.19	35.17	25.4
III	100%	501.64	-	100
IV	95%	403.12	19.64	98.86
	90%	381.11	24.03	94.60
	85%	371.40	25.96	90.30
	80%	365.52	27.13	84.50

summarizes the performance comparison for Scenarios II, III, and IV under different voltage confidence levels. Similar to the previous case study, Scenario II (deterministic approach) achieved the lowest cost but showed poor voltage reliability (25.4%). Scenario III (robust approach) ensured 100% reliability but with the highest cost. In contrast, Scenario IV (proposed DRCCO method) achieved a well-balanced trade-off, maintaining high voltage reliability (above 90%) while significantly reducing the operational costs compared to the robust approach. In addition, the calculation time for determining the day-ahead MVDC link set-points under Scenario IV was 146.24 s, demonstrating the computational efficiency and practical applicability of the proposed method from the perspective of computational burden.

Furthermore, the current study focuses solely on operational cost minimization for day-ahead scheduling, assuming that the MVDC link and BESS infrastructure are already deployed. A more holistic techno-economic analysis that incorporates capital investment and maintenance costs remains an important direction for future research.

4. Conclusions

In this paper, we proposed a distributionally robust chance-constrained optimal dispatch strategy for MVDC-linked distribution networks with BESS integration. The strategy effectively handles uncertainty in PV and load forecasts through a tractable DRCCO framework.

The simulation results demonstrated that the proposed method achieves up to 44.7% cost improvement and maintains voltage reliability above 96.8% under various uncertainty scenarios. The impact of key DRCC parameters—such as the confidence level and ambiguity set radius—was analyzed, revealing a clear trade-off between operational efficiency and robustness.

However, some limitations remain. The computational time increases significantly with larger sample sizes, and the solution quality depends on the accuracy of historical forecast data. Additionally, the current model assumes that forecast errors have a zero mean, which may not always hold in real-world scenarios. If forecast bias exists, it could affect both the cost estimation and system reliability. Future research may focus on improving the computational efficiency, extending the method to multi-time-step or real-time applications and incorporating non-zero mean error models or bias-correction techniques to address potential forecast bias. Furthermore, more detailed modeling of converter dynamics could further enhance the practical applicability.

The proposed DRCCO framework can be also extended to accommodate other controllable assets such as smart inverters, demand response resources, and EV charging stations. This would involve adding new decision variables and constraints specific to each asset type, updating sensitivity coefficients, and incorporating additional uncertainty models where necessary.