Data-Driven Distributionally Robust Optimization for Day-Ahead Operation Planning of a Smart Transformer-Based Meshed Hybrid AC/DC Microgrid Considering the Optimal Reactive Power Dispatch

Abstract

Smart Transformer (ST)-based Meshed Hybrid AC/DC Microgrids (MHMs) present a promising solution to enhance the efficiency of conventional microgrids (MGs) and facilitate higher integration of Distributed Energy Resources (DERs), simultaneously managing active and reactive power dispatch. However, MHMs face challenges in resource management under uncertainty and control of electronic converters linked to the ST and DERs, complicating the pursuit of optimal system performance. This paper introduces a Data-Driven Distributionally Robust Optimization (DDDRO) approach for day-ahead operation planning in ST-based MHMs, focusing on minimizing network losses, voltage deviations, and operational costs by optimizing the reactive power dispatch of DERs. The approach accounts for uncertainties in photovoltaic generator (PVG) output and demand. The Column-and-Constraint Generation (C&CG) algorithm and the Duality-Free Decomposition (DFD) method are employed. The initial mixed-integer non-linear planning problem is also reformulated into a mixed-integer (MI) Second-Order Cone Programming (SOCP) problem using second-order cone relaxation and a positive octagonal constraint method. Simulation results on a connected MHM system validate the model’s efficacy and performance. The study also highlights the advantages of the meshed MG structure and the positive impact of integrating the ST into MHMs, leveraging the multi-stage converter’s flexibility for optimal energy management under uncertain conditions.

1. Introduction

To address the need to reduce greenhouse gas emissions, it is essential to promote technological advances that make the electricity system more flexible by incorporating Battery Energy Storage Systems (BESSs) integrated with Distributed Generation (DG) systems [1]. Microgrids (MGs) present a practical approach to overcoming the barriers associated with integrating Distributed Energy Resources (DERs) into the Distribution System (DS). They provide enhanced flexibility and significantly improve the reliability and energy efficiency of the electrical grid [2]. Hybrid AC/DC Microgrids (HMGs) combine the advantages of both AC and DC microgrids, providing high reliability, flexibility, and cost-effectiveness [3]. HMGs integrate AC and DC networks within the same DS through Interlinking Converters (ICs) and have traditionally been implemented using radial distribution schemes with results within the proposed objective [4]. However, high DER penetration can lead to voltage increases [5]. In contrast, high load demand can result in considerable voltage drops at the far end of the line and excessive loading of distribution lines and transformers [6]; consequently, voltage regulation becomes particularly difficult, especially for long distribution feeders [7]. Recently, ST-based Meshed Hybrid AC/DC Microgrids (MHMs) [8] have emerged as a promising solution for enhancing the integration of DERs [9] and optimizing reactive power control [10]. STs enable the formation of AC/DC loops within the Energy Router (ER) framework [11]. With the flexibility provided by STs, MHMs can be implemented by connecting the Power Electronic Interface (PEI) of each DER to the ST stages at either the DC or AC ports. Additionally, STs eliminate the need for ICs, allowing for various energy flow paths and management strategies [8].

Despite the advantages of MHMs, the uncertainties of high penetration of DERs and demand have made day-ahead optimal scheduling of MGs difficult and complex [12]. Thus, it is necessary to introduce Energy Management Systems (EMSs) to maintain the optimal operation of the MG. The EMS based on deterministic optimization algorithms loses accuracy due to the uncertainty of DERs and consumer behavior [13]. In this context, optimization techniques that account for uncertainties, particularly those related to DERs, are incorporated into solving the power-flow optimization problem. According to the optimization taxonomy outlined in [14], when dealing with uncertainty in decision variables, one can employ Stochastic Programming (SP) or Robust Optimization (RO) methods [15]. The probability distribution functions for demand profiles and DER often do not fully capture the variability of random variables, making SP complex and challenging to implement [16]. In contrast, RO is generally more conservative, prioritizing the worst-case scenarios over a broader range of possibilities [17].

A Data-Driven Distributionally Robust Optimization (DDDRO) method has been created to determine the worst-case probability distribution within a defined ambiguity set, integrating features of both SP and RO [18,19]. This approach leverages historical data to generate various scenarios, considering the worst-case probabilities through a moment-based ambiguity set to establish a probability distribution [20,21]. DDDRO allows the formulation of a two-stage Robust Optimization problem defined by the maximum cost within the uncertainty set [22,23]. To solve such complex issues, a decomposition framework based on the Column-and-Constraint Generation (C&CG) method [24] and Bender’s Decomposition (BD) [25] has been proposed [26]. The complexity escalates when the number of scenarios increases, and the problem becomes non-linear [27]. To address this, a Duality-Free Decomposition (DFD) method has been introduced, transforming the bi-level (max–min) problem into separate subproblems [19,27,28].

Table 1 compares the primary papers reviewed in this study with the model introduced in our research. Initially, we aimed to determine the publication date and the type of system discussed. Likewise, we aimed to reach problem types related to generation, transmission, DSs, expansion, or operation planning and that consider if the integration of a ST is considered. On the other hand, we sought to identify the works that study active and reactive power dispatch both in a deterministic and under uncertainties approaches, considering the reactive power regulation of DG and ST. Finally, a scenario under uncertainty was introduced, in which our goal was to identify the reported studies that have explored approaches to managing uncertainty, whether in demand or DERs, as well as the formulation of the optimization problem, a data-driven approach, and the solution approach adopted, mainly in the subproblem.

According to the information compiled in Table 1, it can be observed that typically the subproblem is approached from DT to convert the max-min subproblem into a single maximization problem; despite this, recent reports have started to introduce DFD techniques as an alternative to reduce the computational cost and convergence times [27], with favorable results in problems related to UC [27,29], PL [12], and OP [19,28,30].

Table 1. Comparison of key elements in hybrid microgrids and distribution systems under uncertainty.

Ref.	Year	Type System	Type Problem	ST	PQ Power Dispatch	Uncertainties	DDDRO Approach	Solution Algorithm	Subproblem
[23]	2018	DS	OP	-	Y	PVG—DM	Y	-	-
[31]	2018	HMG	OP	-	-	PVG—DM	Y	C&CG	nested-C&CG
[32]	2018	DS	OP	Y	Y	-	-	PSO	-
[33]	2018	DS	OP	Y	Y	-	-	-	-
[27]	2019	TS	UC	-	-	WT—DM	Y	C&CG	DFD
[34]	2019	TS	UC	-	-	WT—DM	-	BD	DT
[29]	2019	MG	UC	-	-	WT	Y	C&CG	DFD
[35]	2019	DS	UC	-	-	WT—DM	-	BD	DT
[36]	2019	HMG	OP	-	-	PVG—DM	-	C&CG	DT
[37]	2019	TS	UC	-	-	WT	-	C&CG	DT
[26]	2019	TS	PL	-	-	WT—PVG	Y	C&CG	DT
[6]	2020	MG	OP	Y	Y	-	-	-	-
[38]	2020	MG	PL	-	Y	WT—PVG—DM	Y	-	DT
[28]	2020	MG	OP	-	Y	WT—PVG—DM	Y	C&CG	DFD
[12]	2020	DS	PL	-	Y	PVG—DM	Y	C&CG	DF
[39]	2020	MG	OP	Y	Y	-	-	GA	-
[40]	2021	TS	UC	-	-	WT—DM	Y	C&CG	DT
[41]	2021	TS	PL	-	Y	PVG	Y	C&CG	DT
[42]	2021	TS	UC	-	-	WT	Y	C&CG	DT
[8]	2021	DS	OP	Y	Y	-	-	-	-
[43]	2021	HMG	OP	Y	Y	-	-	-	-
[44]	2021	HMG	OP	Y	Y	-	-	GA	-
[45]	2021	HMG	OP	Y	Y	-	-	-	-
[46]	2021	HMG	OP	Y	-	-	-	GA	-
[47]	2021	HMG	OP	Y	Y	-	-	GA	-
[48]	2021	DS	PL	-	-	PVG—DM	-	C&CG	DT
[19]	2021	TS	OP	-	-	WT	Y	C&CG	DFD
[49]	2022	MG	OP	-	Y	EV—PVG	Y	C&CG	DT
[50]	2022	HMG	OP	Y	Y	WT—PVG—DM—EV	-	PSO	PSO
[51]	2022	DS	OP	-	Y	WT	Y	C&CG	DT
[52]	2022	HMG	OP	Y	-	-	-	-	-
[10]	2023	DS	OP	Y	Y	WT—PVG—DM—EV	-	PSO	-
[53]	2023	DS	OP	Y	Y	WT—PVG—DM—EV	-	PSO	-
[30]	2023	MG	OP	-	-	PVG—DM	Y	C&CG	DFD
Proposed	2023	HMG	OP	Y	Y	PVG—DM	Y	C&CG	DFD

TS: Transmission System; HMG: Hybrid Microgrid; DS: Distribution System; UC: unit commitment; OP: operation; PL: planning; WT: wind turbine; DM: demand; PVG: photovoltaic generator; EV: electric vehicle; BD: Bender’s Decomposition; C&CG: Column-and-Constraint Generation; PSO: Particle Swarm Optimization; GA: Genetic Algorithm; DT: Duality Theory; DFD: Duality-Free Decomposition; Y: reported; -: no reported.

Table 1. Comparison of key elements in hybrid microgrids and distribution systems under uncertainty.

Ref.	Year	Type System	Type Problem	ST	PQ Power Dispatch	Uncertainties	DDDRO Approach	Solution Algorithm	Subproblem
[23]	2018	DS	OP	-	Y	PVG—DM	Y	-	-
[31]	2018	HMG	OP	-	-	PVG—DM	Y	C&CG	nested-C&CG
[32]	2018	DS	OP	Y	Y	-	-	PSO	-
[33]	2018	DS	OP	Y	Y	-	-	-	-
[27]	2019	TS	UC	-	-	WT—DM	Y	C&CG	DFD
[34]	2019	TS	UC	-	-	WT—DM	-	BD	DT
[29]	2019	MG	UC	-	-	WT	Y	C&CG	DFD
[35]	2019	DS	UC	-	-	WT—DM	-	BD	DT
[36]	2019	HMG	OP	-	-	PVG—DM	-	C&CG	DT
[37]	2019	TS	UC	-	-	WT	-	C&CG	DT
[26]	2019	TS	PL	-	-	WT—PVG	Y	C&CG	DT
[6]	2020	MG	OP	Y	Y	-	-	-	-
[38]	2020	MG	PL	-	Y	WT—PVG—DM	Y	-	DT
[28]	2020	MG	OP	-	Y	WT—PVG—DM	Y	C&CG	DFD
[12]	2020	DS	PL	-	Y	PVG—DM	Y	C&CG	DF
[39]	2020	MG	OP	Y	Y	-	-	GA	-
[40]	2021	TS	UC	-	-	WT—DM	Y	C&CG	DT
[41]	2021	TS	PL	-	Y	PVG	Y	C&CG	DT
[42]	2021	TS	UC	-	-	WT	Y	C&CG	DT
[8]	2021	DS	OP	Y	Y	-	-	-	-
[43]	2021	HMG	OP	Y	Y	-	-	-	-
[44]	2021	HMG	OP	Y	Y	-	-	GA	-
[45]	2021	HMG	OP	Y	Y	-	-	-	-
[46]	2021	HMG	OP	Y	-	-	-	GA	-
[47]	2021	HMG	OP	Y	Y	-	-	GA	-
[48]	2021	DS	PL	-	-	PVG—DM	-	C&CG	DT
[19]	2021	TS	OP	-	-	WT	Y	C&CG	DFD
[49]	2022	MG	OP	-	Y	EV—PVG	Y	C&CG	DT
[50]	2022	HMG	OP	Y	Y	WT—PVG—DM—EV	-	PSO	PSO
[51]	2022	DS	OP	-	Y	WT	Y	C&CG	DT
[52]	2022	HMG	OP	Y	-	-	-	-	-
[10]	2023	DS	OP	Y	Y	WT—PVG—DM—EV	-	PSO	-
[53]	2023	DS	OP	Y	Y	WT—PVG—DM—EV	-	PSO	-
[30]	2023	MG	OP	-	-	PVG—DM	Y	C&CG	DFD
Proposed	2023	HMG	OP	Y	Y	PVG—DM	Y	C&CG	DFD

TS: Transmission System; HMG: Hybrid Microgrid; DS: Distribution System; UC: unit commitment; OP: operation; PL: planning; WT: wind turbine; DM: demand; PVG: photovoltaic generator; EV: electric vehicle; BD: Bender’s Decomposition; C&CG: Column-and-Constraint Generation; PSO: Particle Swarm Optimization; GA: Genetic Algorithm; DT: Duality Theory; DFD: Duality-Free Decomposition; Y: reported; -: no reported.

Algorithms are implemented from decomposition methods such as C&CG and BD to address two-stage optimization problems. However, some work reports a heuristic approach using PSO-based techniques in both the master problem [10,32,53] and the subproblem [50]. The sources of uncertainty have been mainly contemplated in WTs, PVGs, and DM. However, EVs have started to play a more relevant role given the particular conditions of this type of element in power systems, whose management complexity implies a heuristic approach, typically PSO-based techniques [10,49,50,53]. Data-driven approaches have been widely used for both TSs and DSs; however, representative contributions have been identified for MGs and HMGs, typically for OP problems. In consideration of MHMs, although relevant works have focused on the management and control of this type of MG that has been identified, no contributions to the study of optimal management of MGs under uncertainty conditions involving a data-driven approach have been identified. This is evidenced in Table 1, in which only the work reported in [50] contemplates uncertainty conditions whose solution alternative for the master problem and the subproblem is approached using heuristic techniques such as PSO, considering the ST as a back-to-back converter or two-stage ST with a DC link port. On the other hand, EMS results have been identified from heuristic methods based on GA [39,44,46,47], but none of them contemplates uncertainty in the formulation of the optimization problem. On the other hand, it is observed that the works that consider reactive power dispatch and STs are typically formulated under a deterministic approach [6,32,33,39,44], taking advantage of the benefits of ST as an ER. However, works have recently been identified that consider STs for reactive power dispatch under uncertain conditions but propose a heuristic approach to solve the optimization problem [10,53].

Despite the degree of progress and relevant contributions of the DDDRO approach to deal with optimization problems under uncertainty both in UC, PL, and OP, applied to TSs, DSs, and MGs, no studies have been found that fully exploit the advantages of MHMs, considering that integrating the ST increases the degrees of freedom for a better performance of the EMS, as well as auxiliary services such as active and reactive power dispatch. In this context, a gap exists regarding EMS developments in MHMs integrating STs under uncertain operating conditions and reactive power dispatch; thus, the main contributions of this paper are the following:

✓

An equivalent ST and Voltage-Sourced Converter (VSC) model is formulated to determine the impact of reactive power regulation from variable power factors on voltage deviation and loss minimization in a ST-based MHM in an optimal power-flow problem under uncertainty;

✓

For day-ahead operation planning of the ST-based MHM, a DDDRO is proposed to consider the DG’s active and reactive power dispatch, the uncertainty of photovoltaic generators (PVGs), and demand;

✓

A tri-level master–subproblem framework based on the DFD method and the C&CG algorithm is developed to solve the day-ahead operation planning of a ST-based MHM.

The paper is organized as follows. Section 2 proposes a Deterministic Optimal Model, organized according to constraint (Section 2.1), objective function (Section 2.2), second-order cone relaxation, and positive octagonal constraint method (Section 2.3), followed by modeling and solution under uncertainty by DDDRO, in Section 3, where an ambiguity is set for uncertainty (Section 3.1), a two-stage Robust Optimization Model (Section 3.2) is used, and Duality-Free Decomposition (Section 3.3) is proposed. Section 4 presents case studies and results, in which the system’s performance, both on the AC and DC sides, is analyzed. The discussion of the results is presented in Section 5. Finally, the conclusions are given in Section 6. An abbreviation table provides the list of acronyms and symbols.

2. Deterministic Optimization Model

This study focuses on optimizing the ST, VSC, and BESS to enhance the control of active and reactive power dispatch. It aims to increase efficiency and lower operational costs in the HMG. This section outlines the objective function and constraints within the deterministic optimization model.

2.1. Constraints

The optimization model presented in this paper must adhere to the constraints associated with the VSC, ST, BESS, and power flow. The primary constraints are outlined as follows.

2.1.1. Voltage-Sourced Converter

In this study, the DER operates in connected mode via a VSC and a bidirectional DC–DC converter. The VSC facilitates independent regulation of active and reactive power on the AC side, while the bidirectional DC–DC converter manages active power control on the DC side. This operation is based on the resource availability of the PVG, the state of charge (SoC) of the BESS, and the equilibrium of active power between the AC and DC sides.

Figure 1 shows the simplified equivalent power-flow model for the VSC, acting as the PEI for the PVG, BESS, and the third stage of the ST. The AC-side variables encompass voltage and phase at bus $B_{A{C}_k}$ ($V_k^{AC},{\delta }_k$), whereas the DC-side variable is the voltage at bus $B_{D{C}_i}$ ($V_i^{DC}$). The interaction between AC and DC power flow is characterized by the transferring between active power on the AC side ($P_k^{{VSC}_{AC}}$) and the active power on the DC side ($P_i^{{VSC}_{DC}}$), including bidirectional power transfer losses ($P_{Los{s}_i}^{VSC}$) due to the switching of the VSC’s Insolate Gate Bipolar Transistor (IGBT) [54]. Based on the equivalent model, the power-flow equations for the VSCs in the microgrid are formulated from Equations (1) and (2). These equations relate $P_{Los{s}_i}^{VSC}$ as a non-linear function of the loss coefficient ($A_i^{VSC}$) and the inverter’s capacity limit, which depend on $P_k^{VS{C}_{AC}}$ and reactive power ($Q_k^{{VSC}_{AC}}$) transferred by the VSC on the AC side. The coefficient $A_i^{VSC}$ is influenced by the power and characteristics of the inverter [13,55].

$$P_k^{VS{C}_{AC}}+P_i^{VS{C}_{DC}}+P_{Los{s}_i }^{VSC}=0$$

(1)

$$P_{Los{s}_i}^{VSC}=A_i^{VSC}\sqrt{{P_k^{{VSC}_{AC}}}^2+{Q_k^{{VSC}_{AC}}}^2}$$

(2)

According to the proposed methodology, the DER’s VSCs are intended to offer auxiliary services related to active and reactive power control in the MHM, limited by the VSC’s apparent power rating ($S_k^{{VSC}_{A{C}_{max}}}$) in Equation (3). The operational range of the inverter is depicted on a PQ plane as a semicircle centered around the axis of positive active power, illustrating the functioning limits of the VSC [57].

$$\sqrt{{P_k^{{VSC}_{AC}}}^2+{Q_k^{{VSC}_{AC}}}^2}\le S_k^{{VSC}_{A{C}_{max}}}$$

(3)

Figure 2a shows the operating region of the VSC (highlighted in yellow) in conjunction with the PVG on the PQ plane. This region is considered a minimum power factor ($P{F}_{min}$) value that sets the operational limit of the VSC under $S_k^{{VSC}_{A{C}_{max}}}$ (green dashed line). Therefore, the reactive power ($Q_k^{{VSC}_{AC}}=Q_k^{{PV}_{AC}}$) available for dispatch is limited by the active power ($P_k^{{VSC}_{AC}}=P_k^{{PV}_{AC}}$) generated by the PVG according to a Maximum Power Point Tracking (MPPT) algorithm.

Figure 2b illustrates the operational zones (blue zone) of the BESS’s VSC across all four quadrants. The BESS functions in either charge ($P_k^{{ESS}_{AC}}$) or discharge ($-P_{k,t}^{{ESS}_{AC}}$) mode, constrained by the maximum active power capacity of the VSC and the physical characteristics of the BESS ($P_k^{{ESS}_{{AC}_{max}}}$). In either mode of operation, the BESS’s VSC can regulate the injection of reactive power ($Q^{{VSC}_{AC}}=Q_k^{{ESS}_{AC}}$). A similar operation is proposed for VSCs coupled between the AC and DC sides, whose operating point is set by the active power balance given by Equation (1). The PVG and BESS’s operational restrictions of the reactive power injection are defined as follows, in Equations (4) and (5), respectively, where ${\theta }_{P{F}_{min}}^{{ESS,PVG}_{AC}}={\mathit{cos}}^{-1}\left(P{F}_{min}\right)$, and it is equivalent to the phase at $P{F}_{min}$.

$$-tan\left({\theta }_{P{F}_{min}}^{{PVG}_{AC}}\right)P_k^{{PVG}_{AC} }\le Q_k^{{PVG}_{AC} }\le tan\left({\theta }_{P{F}_{min}}^{{PVG}_{AC}}\right)P_k^{{PVG}_{AC} }$$

(4)

$$-tan\left({\theta }_{P{F}_{min}}^{{ESS}_{AC}}\right)P_k^{{ESS}_{{AC}_{max}}}\le Q_k^{{ESS}_{AC} }\le tan\left({\theta }_{P{F}_{min}}^{{ESS}_{AC}}\right)P_k^{{ESS}_{{AC}_{max}}}$$

(5)

2.1.2. ST Equivalent Power-Flow Model

The ST is a multi-stage power electronic converter that supports AC and DC connections with multiple low voltage (LV) and medium voltage (MV) ports. Beyond simply replacing conventional Low-Frequency Transformers (LFTs) operating at 60 Hz or 50 Hz, the ST provides numerous services and facilitates the incorporation of advanced hybrid AC and DC microgrid architectures. This system can inject or absorb reactive power, reduce harmonics [32], address voltage sags [58], and restrict current during short-circuit faults [59]. In a grid managed by a ST, DGs are connected either to the $L{V}_{DC}$ side or the $L{V}_{AC}$ asynchronously on the grid side, allowing the low-voltage DC or AC frequency to function as a feedback signal for regulating power flow. This configuration enables effective power routing.

Figure 3 presents the structure of a ST, detailing its control and communication layers. Each stage utilizes a decoupled control architecture, enabling independent and bidirectional power-flow management. This design offers greater flexibility in controlling power flow on both the AC and DC sides. The ST concept, as proposed in [60], offers DC connectivity [61] and lessens the necessity for reinforcement in the $L{V}_{AC}$ distribution network owing to the rising adoption of DERs and BESSs. The ST enables the formation of a hybrid meshed DS that allows simultaneous voltage and power-flow regulators through a centralized controller.

In examining the power-flow equations of MHMs, the secondary output of the ST, known as the regulated DC feeder ($G_{D{C}_i}$), operates as a slack node, maintaining the voltage level on the DC side.

In the ST’s third stage, the active and reactive power flows are optimized by replacing the traditional VSC with an active VSC, which manages $P_i^{{VSC}_{DC}}$ on the DC side, and $P_k^{{VSC}_{AC}}$ and $Q_k^{VS{C}_{AC}}$ on the AC side. Consequently, a DC generator ($P_i^{G_{DC}}$) is incorporated into the DC-side load-flow equations. Figure 4 shows the equivalent model for analyzing power flow at the ST’s AC and DC coupling points. The power-flow equations are detailed in Equations (13) and (14) for the AC side and in Equation (16) for the DC side. Reactive power control follows the descriptions in Equations (4) and (5).

2.1.3. Battery Energy Storage System

The power-flow model for the BESS defines the constraints for energy storage components on both AC and DC interfaces. The power-flow equations applicable to the BESS, outlined in Equations (6)–(10), originate from the framework suggested in reference [13].

$$U_{k,t}^{{ESS}_{{AC}_C}}+U_{k,t}^{{ESS}_{{AC}_D}}\le 1$$

(6)

$$P_{k,t}^{{ESS}_{AC}}=P_{k,t}^{{{ESS}_{AC}}_C}{\eta }_C-{\displaystyle \frac{P_{k,t}^{{{ESS}_{AC}}_D}}{{\eta }_D}}$$

(7)

$$0\le P_{k,t}^{{{ESS}_{AC}}_C}\le U_{k,t}^{{ESS}_{{AC}_C}}{P}_k^{{ESS}_{{AC}_{max}}}$$

(8)

$$0\le P_{k,t}^{{{ESS}_{AC}}_D}\le U_{k,t}^{{ESS}_{{AC}_D}}{P}_k^{{ESS}_{{AC}_{max}}}$$

(9)

$$E_{k,t}^{A{C}_{ESS}}=E_{k,t-1}^{A{C}_{ESS}}+P_{k,t}^{A{C}_{ESS}}∆t$$

(10)

For the $k_{th}$ BESS, $U_{k,t}^{{ESS}_{{AC}_C}}$ denotes the binary variable indicating charging, and $U_{k,t}^{{ESS}_{{AC}_D}}$ denotes the binary variable indicating discharging. The variables $P_{k,t}^{{{ESS}_{AC}}_C}$ and $P_{k,t}^{{{ESS}_{AC}}_D}$ denote the power for charging and discharging, respectively. The efficiencies for these processes are indicated by ${\eta }_C$ and ${\eta }_D$. Finally, the battery state of charge is denoted by $E_{k,t}^{{ESS}_{AC}}$. The model for the $i_{th}$battery on the DC side follows a similar structure to Equations (6)–(10), with appropriate adjustments to indexes and superscripts. The reactive power dispatch for the BESS on the AC side is described by Equation (5).

2.1.4. Power-Flow Constraints

Power-flow constraints are defined for both the AC and DC sides of the MHM. For the DC side, the equations are formulated using the conductance matrix and the voltage at the DC bus points, with conductance represented by $g_{ij}$. On the AC side, the equations are expressed in polar form, derived from the network admittance matrix according to the pi-line model. The series admittance of the line is denoted by the set $g_{kl}-b_{kl}$, and its shunt susceptance is denoted by $b_{k{l}_{sh}}$. The active power $P_{kl}^{AC}$ and reactive power $Q_{kl}^{AC}$ transmitted through the line on the AC side, in per unit (p.u.), are determined using Equations (11) and (12).

$$P_{kl}^{AC}=g_{kl}{V_k^{AC}}^2-V_k^{AC}{V}_l^{AC}\left(g_{kl}Cos {\delta }_{kl}+b_{kl}Sin {\delta }_{kl}\right)$$

(11)

$$Q_{kl}^{AC}=-\left(b_{kl}+b_{k{l}_{sh}}\right){V_k^{AC}}^2-V_k^{AC}{V}_l^{AC}\left({-b}_{kl}Cos {\delta }_{kl}+g_{kl}Sin {\delta }_{kl}\right)$$

(12)

where ${\delta }_{kl}={\delta }_k-{\delta }_l$. The balance of active and reactive power is delineated in Equations (13) and (14).

$$\sum _{l\in \mathcal{N}\left(k\right)}{P}_{kl}^{AC}=P_k^{G_{AC}}+P_k^{{PV}_{AC}}+P_k^{{VSC}_{AC}}-P_k^{{ESS}_{AC}}-P_k^{L_{AC}}$$

(13)

$$\sum _{l\in \mathcal{N}\left(k\right)}{Q}_{kl}^{AC}=Q_k^{G_{AC}}+Q_k^{{PV}_{AC}}+Q_k^{{VSC}_{AC}}-Q_k^{{ESS}_{AC}}-Q_k^{L_{AC}}$$

(14)

Nodes $l\in \mathcal{N}\left(k\right)$ represent those directly connected to node $k$. The calculation of active power $P_{ij}^{DC}$ flowing through the line on the DC side, in per unit (p.u.), is addressed in Equation (15). For node $i$, the active power balance is described by Equation (16), where $j\in \mathcal{N}\left(i\right)$ are directly connected to node $i$.

$$P_{ij}^{DC}=g_{ij}{V_i^{DC}}^2-V_{i,t}^{DC}{V}_j^{DC}{g}_{ij}$$

(15)

$$\sum _{j\in \mathcal{N}\left(i\right)}{P}_{ij}^{DC}=P_i^{G_{DC}}+P_i^{{PV}_{DC}}+P_i^{{VSC}_{DC}}-P_i^{{ESS}_{DC}}-P_i^{L_{DC}}$$

(16)

2.2. Objective Functions

The optimization problem focuses on minimizing expected operational costs, losses, and voltage deviations within a 24 h period, while adhering to the physical constraints of the MHM. At any given time $t$, the objective function ($f)$ comprises (i) the operation cost ($f_{cost}$); (ii) network losses ($f_{loss}$); and (iii) voltage deviation ($f_{d{V}_{AC}}; f_{d{V}_{DC}}$), as described in Equation (17). The weight factors $w_c$, $w_l$, and $w_d$ represent the respective weights of operational costs, network losses, and voltage deviations, with the condition $w_c+w_l+w_d=1$. The terms $c_k^{G_{AC}}$, $c_i^{G_{DC}}$, $c_k^{ES{S}_{AC}}$, and $c_i^{ES{S}_{DC}}$ represent the costs associated with energy transactions with the main medium voltage grid and the operational expenses of the BESS on both AC and DC sides, respectively. Furthermore, $N^{G_{AC}}$, $N^{{ESS}_{AC}}$, $N^{{PVG}_{AC}}$, $N^{L_{AC}}$, $N^{G_{DC}}$, $N^{{ESS}_{DC}}$, $N^{{PVG}_{DC}}$, and $N^{L_{DC}}$ denote the sets of generators, BESS, photovoltaic generators, and loads on the AC and DC sides, respectively.

$$\begin{gathered} f=min \sum _{t=0}^T\left(w_c{f}_{c,t }+w_l{f}_{l,t}+w_d\left(f_{d{V}_{AC,t}}+f_{d{V}_{DC,t}}\right)\right) \\ {f}_{c,t}=c_k^{G_{AC}}\sum _{k=0}^{N^{G_{AC}}}{P}_{k,t}^{G_{AC}}+c_i^{G_{DC}}\sum _{i=0}^{N^{G_{DC}}}{P}_{i,t}^{G_{DC}}+c_k^{ES{S}_{AC}}\sum _{k=0}^{N^{{ESS}_{AC}}}\left(P_{k,t}^{{ESS}_{{AC}_C}}+P_{k,t}^{{ESS}_{{AC}_D}}\right)+c_i^{ES{S}_{DC}}\sum _{i=0}^{N^{{ESS}_{DC}}}\left(P_{i,t}^{{ESS}_{{DC}_C}}+P_{i,t}^{{ESS}_{{DC}_D}}\right) \\ {f}_{l,t}=\left(\sum _{k=0}^{N^{G_{AC}}}{P}_{k,t}^{G_{AC}}+\sum _{i=0}^{N^{G_{DC}}}{P}_{i,t}^{G_{DC}}+\sum _{k=0}^{N^{{PVG}_{AC}}}{P}_{k,t}^{{PV}_{AC}}+\sum _{i=0}^{N^{{PVG}_{DC}}}{P}_{i,t}^{{PV}_{DC}}\right)-\left(\sum _{k=0}^{N^{{ESS}_{AC}}}{P}_{k,t}^{{ESS}_{AC}}+\sum _{i=0}^{N^{{ESS}_{DC}}}{P}_{i,t}^{{ESS}_{DC}}+\sum _{k=0}^{N^{L_{AC}}}{P}_{k,t}^{L_{AC}}+\sum _{i=0}^{N^{L_{DC}}}{P}_{i,t}^{L_{DC}}\right) \\ {f}_{d{V}_{AC,t}}=\sum _{k=0}^{N^{AC}}{\left(V_{k,t}^{AC}-V_{ref}^{AC}\right)}^2; f_{d{V}_{DC,t}}=\sum _{i=0}^{N^{DC}}{\left(V_{i,t}^{DC}-V_{ref}^{DC}\right)}^2 \end{gathered}$$

(17)

2.3. Second-Order Cone Relaxation and Positive Octagonal Constraint Method

This work transfers the power-flow model to a second-order conic model based on the relaxation technique for its fast solution. Thus, second-order cone relaxation [62] and the positive octagonal constraint method [13] are used. Firstly, the power flow on a distribution network can be precisely reformulated by introducing the variables $C_{kl}=V_k{V}_{l}Cos\left({\delta }_k-{\delta }_l\right)$, $S_{kl}=V_k{V}_{l}Sin\left({\delta }_k-{\delta }_l\right)$, $C_{kk}=V_k^2$, and $C_{ll}=V_l^2$ [62] in the rectangular formulation provided in Equations (11) and (12). Thus, we obtain Equations (18) and (19).

$$P_{kl}=g_{kl}{C}_{kk}-g_{kl}{C}_{kl}-b_{kl}{S}_{kl}$$

(18)

$$Q_{kl}=-\left(b_{kl}+b_{k{l}_{sh}}\right)C_{kk}+b_{kl}{C}_{kl}-g_{kl}{S}_{kl}$$

(19)

On the DC side, the variables $C_{ij}=V_i{V}_j$, $C_{ii}=V_i^2$, and $C_{jj}=V_j^2$ are induced in the Equation (15). Thus, we obtain Equation (20).

$$P_{ij}^{DC}=g_{ij}{C}_{ii}-g_{ij}{C}_{ij}$$

(20)

Equations (18)–(20) linearize the expressions for the power flow through the lines on both the AC and DC sides. Still, additional constraints must be imposed to represent the underlying trigonometric nature of the AC power flow. The trigonometric relationships in Equations (21)–(23) ensure the symmetry property of the sine and cosine functions.

$$C_{kl}^2+S_{kl}^2+\left({\displaystyle \frac{C_{kk}-C_{ll}}{2}}\right)\le \left({\displaystyle \frac{C_{kk}+C_{ll}}{2}}\right)$$

(21)

$$C_{kl}=C_{lk}$$

(22)

$$S_{kl}=-S_{lk}$$

(23)

VSC loss constraint in Equation (2) is a non-linear quadratic constraint, which can be further relaxed to the following second-order cone constraints given by Equation (24) [13].

$${‖\begin{array}{c}{P}_k^{{VSC}_{AC}}\\ Q_k^{{VSC}_{AC}}\end{array}‖}_2\le {\displaystyle \frac{P_{Los{s}_i}^{VSC}}{A_i^{VSC}}}$$

(24)

$$\begin{gathered} -S_k^{A{C}_{VS{C}_{max}}}\le P_k^{A{C}_{VSC}}\le S_k^{A{C}_{VS{C}_{max}}} \\ -S_k^{A{C}_{VS{C}_{max}}}\le Q_k^{A{C}_{VSC}}\le S_k^{A{C}_{VS{C}_{max}}} \\ -\sqrt{2}{S}_k^{A{C}_{VS{C}_{max}}}\le P_k^{A{C}_{VSC}}+Q_k^{A{C}_{VSC}}\le \sqrt{2}{S}_k^{A{C}_{VS{C}_{max}}} \\ -\sqrt{2}{S}_k^{A{C}_{VS{C}_{max}}}\le P_k^{A{C}_{VSC}}-Q_k^{A{C}_{VSC}}\le \sqrt{2}{S}_k^{A{C}_{VS{C}_{max}}} \end{gathered}$$

(25)

On the other hand, the circle constraints from Equation (3) can be transferred into Equation (25) based on the convex relaxation technique and positive octagonal constraint [13], respectively. By employing convex relaxation and positive octagonal constraints, the initial optimization model is transformed into the Second-Order Cone Programming (SOCP) model presented in Equation (26) for all time periods $t$.

$$\begin{gathered} f=min \sum _{t=0}^T\left(w_c{f}_{c,t }+w_l{f}_{l,t}+w_d\left(f_{d{V}_{AC,t}}+f_{d{V}_{DC,t}}\right)\right) \\ \mathrm{subject} \mathrm{to} \mathrm{Equations} (1), (4)–(10), (13), (14), (16), \mathrm{and} (18)–(25). \end{gathered}$$

(26)

3. Modeling and Solution under Uncertainty by DDDRO

This section develops a DDDRO approach to solve the day-ahead operation planning of a MHM under uncertainties based on a combination of the C&CG and the DFD methods.

3.1. Ambiguity Set for Uncertainties

In real applications, probability distribution functions of DERs may not be available. In this way, historical data are a more suitable for obtaining an approximation of the probabilities of a scenario of interest [28]. Thus, historical data can be converted into data bins, where an estimated probability distribution function (E-PDF) is established from the data bins, which allows the definition of the true probability distribution function (T-PDF) within a tolerance range. In [28,41], a confidence uncertainty set is proposed to cover all possible probability realizations by leveraging historical data and subsequently estimating the distribution of worst-case uncertainties across all scenarios ($\mathcal{S}$) according to the number of data bins ($M_D$). Two norms $L_1$ and $L_{\infty }$ are used to construct the confidence uncertainty set based on T-PDF (${\mathcal{p}}_{\mathcal{s}}^{\mathcal{S}}$) and E-PDF (${\mathcal{p}}_{\mathcal{s}}^{{\mathcal{S}}^0}$) [63].

The tolerance coefficients ${\delta }_1$ and ${\delta }_{\infty }$ are used to define the confidence levels and account for historical data variability. As the sample size ($N$) of historical data increases, the uncertainty set narrows, and the empirical probability distribution function (E-PDF) approaches the theoretical probability distribution function (T-PDF). The confidence levels for the two norms, ${\phi }_1$ and ${\phi }_{\infty }$, establish the tolerance ranges, as outlined in Equation (27). The ambiguity set is crucial for reconstructing the T-PDF within the specified confidence level. This paper utilizes the idealization approach from [41], detailed in Equations (28)–(33). It is important to note that within the ambiguity set, the probability ${\mathcal{p}}_{\mathcal{s}}^{\mathcal{S}}$ can deviate from the initial ${\mathcal{p}}_{\mathcal{s}}^{{\mathcal{S}}^0}$ based on historical data, with a maximum error ${\mathcal{p}}_{\mathcal{s}}^{{\mathcal{S}}^{\pm }}$, determined by ${\delta }_1$ and ${\delta }_{\infty }$.

$${\delta }_1={\displaystyle \frac{M_D}{2N}}\mathit{ln}\left({\displaystyle \frac{2M_D}{1-{\phi }_1}}\right); {\delta }_{\infty }={\displaystyle \frac{1}{2N}}\mathit{ln}\left({\displaystyle \frac{2M_D}{1-{\phi }_{\infty }}}\right)$$

(27)

$$\sum _{\mathcal{s}}{\mathcal{p}}_{\mathcal{s}}^{\mathcal{S}}=1$$

(28)

$${\mathcal{p}}_{\mathcal{s}}^{\mathcal{S}}\ge 0$$

(29)

$${\mathcal{p}}_{\mathcal{s}}^{\mathcal{S}}={\mathcal{p}}_{\mathcal{s}}^{{\mathcal{S}}^0}+{\mathcal{p}}_{\mathcal{s}}^{{\mathcal{S}}^{\pm }}$$

(30)

$$-{\delta }_1\le \sum _{\mathcal{s}}{\mathcal{p}}_{\mathcal{s}}^{{\mathcal{S}}^{\pm }}\le {\delta }_1$$

(31)

$$-{\delta }_1\le {\mathcal{p}}_{\mathcal{s}}^{{\mathcal{S}}^{\pm }}\le {\delta }_1$$

(32)

$$-{\delta }_{\infty }\le {\mathcal{p}}_{\mathcal{s}}^{{\mathcal{S}}^{\pm }}\le {\delta }_{\infty }$$

(33)

3.2. The Two-Stage Robust Optimization Model

For the DDDRO problem statement, the objective function in Equation (17) was reorganized; thus, the variables associated with the energy storage battery system, such as the active power charging $\left(P_{k,t}^{{{ESS}_{AC}}_C}, P_{i,t}^{{{ESS}_{DC}}_C}\right)$, active power discharge $\left(P_{k,t}^{{{ESS}_{AC}}_D}, P_{i,t}^{{{ESS}_{DC}}_D}\right)$, and active power available in the battery $\left(P_{k,t}^{{ESS}_{AC}}, P_{i,t}^{{ESS}_{DC}}\right)$, both on the AC and DC sides, are considered decision variables in the first stage of the DDDRO problem statement. In this stage, the decision variables that minimize the operating costs of the BESS and the worst-case operational scenario of the second stage are optimized.

The exogenous variables subject to uncertainty are the PV generator profile $\left(P_{k,t,\mathcal{s}}^{{PV}_{AC}},P_{i,t,\mathcal{s}}^{{PV}_{DC}}\right)$ and the demand profile $\left(P_{k,t,\mathcal{s}}^{L_{AC}},P_{i,t,\mathcal{s}}^{L_{DC}}\right)$; given these variables’ stochastic nature, the other variables are minimized after the worst-case scenario has occurred. A DDDRO model seeks to identify the optimal solution under the worst-case probability distribution ${\mathcal{p}}_{\mathcal{s}}^{\mathcal{S}}$ under each scenario $\mathcal{s}\in \mathcal{S}$. Thus, a two-stage, three-level problem is defined in Equation (34), where $\mathbf{x}$ represents the decision variables of the first stage; $\mathbf{y}$ represents the decision variables of the third level in the second stage; and finally, ${\mathcal{s}}_{\mathbf{p}}$ represents the decision variables of the second level in the second stage.

$$\begin{gathered} f=\underset{\mathbf{x}}{\mathit{min}}\sum _{t=0}^T\left(\begin{array}{c}{w}_c\left(c_k^{ES{S}_{AC}}{\displaystyle \sum _{k=0}^{N^{{ESS}_{AC}}}}\left(P_{k,t}^{A{C}_{ES{S}_C}}+P_{k,t}^{A{C}_{ES{S}_D}}\right)+c_i^{ES{S}_{DC}}{\displaystyle \sum _{i=0}^{N^{{ESS}_{DC}}}}\left(P_{i,t}^{D{C}_{ES{S}_C}}+P_{i,t}^{D{C}_{ES{S}_D}}\right)\right)\\ -w_l\left({\displaystyle \sum _{k=0}^{N^{{ESS}_{AC}}}}{P}_{k,t}^{{ESS}_{AC}}+{\displaystyle \sum _{i=0}^{N^{{ESS}_{DC}}}}{P}_{i,t}^{{ESS}_{DC}}\right)\end{array}\right)+\psi \\ \psi =\underset{{\mathcal{s}}_{\mathbf{p}}}{\mathit{max}}\underset{\mathbf{y}}{\mathit{min}}\sum _{\mathcal{s}}{\mathcal{p}}_{\mathcal{s}}^{\mathcal{S}}\sum _{t=0}^T\left(\begin{array}{c}{w}_c\left(c_k^{AC}{\displaystyle \sum _{k=0}^{N^{G_{AC}}}}{P}_{k,t,\mathcal{s}}^{G_{AC}}+c_i^{DC}{\displaystyle \sum _{i=0}^{N^{G_{DC}}}}{P}_{i,t,\mathcal{s}}^{G_{DC}}\right)\\ +w_l\left(\left({\displaystyle \sum _{k=0}^{N^{G_{AC}}}}{P}_{k,t,\mathcal{s}}^{G_{AC}}+{\displaystyle \sum _{i=0}^{N^{G_{DC}}}}{P}_{i,t,\mathcal{s}}^{G_{DC}}+{\displaystyle \sum _{k=0}^{N^{{PVG}_{AC}}}}{P}_{k,t,\mathcal{s}}^{{PV}_{AC}}+{\displaystyle \sum _{i=0}^{N^{{PVG}_{DC}}}}{P}_{i,t,\mathcal{s}}^{{PV}_{DC}}\right)-\left({\displaystyle \sum _{k=0}^{N^{L_{AC}}}}{P}_{k,t,\mathcal{s}}^{L_{AC}}+{\displaystyle \sum _{i=0}^{N^{L_{DC}}}}{P}_{i,t,\mathcal{s}}^{L_{DC}}\right)\right)\\ +w_d\left({\displaystyle \sum _{k=0}^{N^{AC}}}{\left(0.5\left(C_{kk,t,\mathcal{s}}+1\right)-V_{ref}^{AC}\right)}^2+{\displaystyle \sum _{i=0}^{N^{DC}}}{\left(0.5\left(C_{ii,t,\mathcal{s}}+1\right)-V_{ref}^{DC}\right)}^2\right)\end{array}\right) \\ \mathrm{Subject} \mathrm{to} \mathrm{Equations} \left(1\right), \left(4\right)–\left(10\right), \left(13\right), \left(14\right), \left(16\right), \left(18\right)–\left(25\right), \mathrm{and} \left(28\right)–\left(33\right), \mathrm{with} \forall t=1\dots T;\forall \mathcal{s}=1\dots \mathcal{S}. \end{gathered}$$

(34)

3.3. Duality-Free Decomposition Method

The C&CG algorithm employs a master problem (MP), as described in Equation (38) and a subproblem (SP), detailed in Equation (35). The subproblem’s objective is to identify the critical scenario within the uncertainty set that establishes an upper bound (UB). Subsequently, new variables and constraints are incorporated into the master problem to determine a lower bound (LB). This iterative process continues, with the master problem and subproblem being solved repeatedly (denoted as the ${\mathcal{l}}_{th}$ iteration), until the relative difference between the upper and lower bounds is less than a specified convergence tolerance $\mathcal{E}$.

3.3.1. Subproblem

In the ${\mathcal{l}}_{th}$ iteration, given a specific set of first-stage variables, denoted as $\left({P_{k,t}^{{ESS}_{AC}}}^{\ast },{P_{i,t}^{{ESS}_{DC}}}^{\ast }\right)$, a second-stage bi-level “max–min” model can be formulated to identify the worst-case scenario $\left({{\mathcal{p}}_{\mathcal{s}}^{\mathcal{S}}}^{\ast }\right)$. Two parameters are set for this stage: input parameters $\mathcal{i}\left({P_{k,t}^{{ESS}_{AC}}}^{\ast },{P_{i,t}^{{ESS}_{DC}}}^{\ast }\right)$ and output parameters, after completion of each iteration $\mathcal{l}$, $\mathcal{o}\left({{\mathcal{p}}_{\mathcal{s}}^{\mathcal{S}}}^{\ast },{P_{k,t,\mathcal{s}}^{G_{AC}}}^{\ast }, {P_{i,t,\mathcal{s}}^{G_{DC}}}^{\ast },C_{kk,t,\mathcal{s}}^{\ast },C_{ii,t,\mathcal{s}}^{\ast }\right)$. Each output parameter corresponds to a feasible solution of the SP at each iteration $\mathcal{l}$, for all scenarios $\mathcal{s}=1\dots \mathcal{S}$. The complete SP formulation is defined by Equation (35).

$$\begin{gathered} \underset{{\mathcal{s}}_{\mathbf{p}}}{\mathit{max}}\underset{\mathbf{y}}{\mathit{min}}\sum _{\mathcal{s}}{\mathcal{p}}_{\mathcal{s}}^{\mathcal{S}}\sum _{t=0}^T\left(\begin{array}{c}{w}_c\left(c_k^{AC}{\displaystyle \sum _{k=0}^{N^{G_{AC}}}}{P}_{k,t,\mathcal{s}}^{G_{AC}}+c_i^{DC}{\displaystyle \sum _{i=0}^{N^{G_{DC}}}}{P}_{i,t,\mathcal{s}}^{G_{DC}}\right)\\ +w_l\left(\left({\displaystyle \sum _{k=0}^{N^{G_{AC}}}}{P}_{k,t,\mathcal{s}}^{G_{AC}}+{\displaystyle \sum _{i=0}^{N^{G_{DC}}}}{P}_{i,t,\mathcal{s}}^{G_{DC}}+{\displaystyle \sum _{k=0}^{N^{{PVG}_{AC}}}}{P}_{k,t,\mathcal{s}}^{{PV}_{AC}}+{\displaystyle \sum _{i=0}^{N^{{PVG}_{DC}}}}{P}_{i,t,\mathcal{s}}^{{PV}_{DC}}\right)-\left({\displaystyle \sum _{k=0}^{N^{L_{AC}}}}{P}_{k,t,\mathcal{s}}^{L_{AC}}+{\displaystyle \sum _{i=0}^{N^{L_{DC}}}}{P}_{i,t,\mathcal{s}}^{L_{DC}}\right)\right)\\ +w_d\left({\displaystyle \sum _{k=0}^{N^{AC}}}{\left(0.5\left(C_{kk,t,\mathcal{s}}+1\right)-V_{ref}^{AC}\right)}^2+{\displaystyle \sum _{i=0}^{N^{DC}}}{\left(0.5\left(C_{ii,t,\mathcal{s}}+1\right)-V_{ref}^{DC}\right)}^2\right)\end{array}\right) \\ \mathrm{Subject} \mathrm{to} \mathrm{Equations} \left(1\right), \left(4\right), \left(13\right), \left(13\right), \left(16\right), \left(18\right)–\left(25\right), \mathrm{and} \left(28\right)–\left(33\right), \mathrm{with} \forall t=1\dots T;\forall \mathcal{s}=1\dots \mathcal{S} \end{gathered}$$

(35)

The subproblem is a max–min bilevel problem with a structure that can be decomposed into several small subproblems without the duality information [27]. Given that between constraints associated with $\mathbf{y}$ and associated with ${\mathcal{s}}_{\mathbf{p}}$, there are no variables in common, and the feasible region bounded by the variable $\mathbf{y}$ is disjointed with the confidence set ${\mathcal{s}}_{\mathbf{p}}$; therefore, the summation (∑) operator and the min( ) operator can be interchanged [19,27,28,29,30]. Thus, the max–min problem of the second stage can be formulated as follows in Equations (36) and (37).

$$\begin{gathered} p_{\mathcal{S}}=\underset{{\mathcal{s}}_{\mathbf{p}}}{\mathit{max}}\sum _{\mathcal{s}}{\mathcal{p}}_{\mathcal{s}}^{\mathcal{S}}{h}_{\mathcal{S}}^{\ast } \\ \mathrm{Subject} \mathrm{to} \mathrm{Equations} \left(28\right)–\left(33\right), \mathrm{with} \forall \mathcal{s}=1\dots \mathcal{S}. \end{gathered}$$

(36)

For each scenario $\mathcal{s}$, an optimal solution ($h_{\mathcal{S}}^{\ast }$) is obtained from Equation (37), which is fixed in Equation (36), to find the probability of the worst-case scenario ${\mathcal{p}}_{\mathcal{s}}^{\mathcal{S}}$.

$$\begin{gathered} h_{\mathcal{S}}=\underset{\mathbf{y}}{\mathit{min}}\sum _{t=0}^T\left(\begin{array}{c}{w}_c\left(c_k^{AC}{\displaystyle \sum _{k=0}^{N^{G_{AC}}}}{P}_{k,t,\mathcal{s}}^{G_{AC}}+c_i^{DC}{\displaystyle \sum _{i=0}^{N^{G_{DC}}}}{P}_{i,t,\mathcal{s}}^{G_{DC}}\right)\\ +w_l\left(\left({\displaystyle \sum _{k=0}^{N^{G_{AC}}}}{P}_{k,t,\mathcal{s}}^{G_{AC}}+{\displaystyle \sum _{i=0}^{N^{G_{DC}}}}{P}_{i,t,\mathcal{s}}^{G_{DC}}+{\displaystyle \sum _{k=0}^{N^{{PVG}_{AC}}}}{P}_{k,t,\mathcal{s}}^{{PV}_{AC}}+{\displaystyle \sum _{i=0}^{N^{{PVG}_{DC}}}}{P}_{i,t,\mathcal{s}}^{{PV}_{DC}}\right)-\left({\displaystyle \sum _{k=0}^{N^{L_{AC}}}}{P}_{k,t,\mathcal{s}}^{L_{AC}}+{\displaystyle \sum _{i=0}^{N^{L_{DC}}}}{P}_{i,t,\mathcal{s}}^{L_{DC}}\right)\right)\\ +w_d\left({\displaystyle \sum _{k=0}^{N^{AC}}}{\left(0.5\left(C_{kk,t,\mathcal{s}}+1\right)-V_{ref}^{AC}\right)}^2+{\displaystyle \sum _{i=0}^{N^{DC}}}{\left(0.5\left(C_{ii,t,\mathcal{s}}+1\right)-V_{ref}^{DC}\right)}^2\right)\end{array}\right) \\ \mathrm{Subject} \mathrm{to} \mathrm{Equations} \left(1\right), \left(4\right), \left(13\right), \left(14\right), \left(16\right), \mathrm{and} \left(18\right)–\left(25\right), \mathrm{with} \forall t=1\dots T;\forall \mathcal{s}=1\dots \mathcal{S}. \end{gathered}$$

(37)

3.3.2. Master Problem

After solving the subproblem, we derive the optimal values $\left({P_{k,t,\mathcal{s}}^{G_{AC}}}^{\mathcal{l}\mathcal{*}}, {P_{i,t,\mathcal{s}}^{G_{DC}}}^{\mathcal{l}\mathcal{*}},C_{kk,t,\mathcal{s}}^{\mathcal{l}\mathcal{*}},C_{ii,t,\mathcal{s}}^{\mathcal{l}\mathcal{*}}\right)$ along with the worst-case probability $\left({{{\mathcal{p}}_{\mathcal{s}}^{\mathcal{S}}}^{\mathcal{l}}}^{\ast }\right)$. These results provide an upper bound for the initial model. Following this, a new set of variables $\left({P_{k,t,\mathcal{s}}^{G_{AC}}}^{\mathcal{l}+1}, {P_{i,t,\mathcal{s}}^{G_{DC}}}^{\mathcal{l}+1},C_{kk,t,\mathcal{s}}^{\mathcal{l}+1},C_{ii,t,\mathcal{s}}^{\mathcal{l}+1}\right)$ and associated constraints are formulated and incorporated into the master problem, setting the optimal probability $\left({{{\mathcal{p}}_{\mathcal{s}}^{\mathcal{S}}}^{\mathcal{l}}}^{\ast }\right)$ from the subproblem. If the subproblem remains feasible, these newly generated variables and constraints, known as “optimality cuts” are added to the master problem during the ${\mathcal{l}}_{th}$ iteration, with the dummy continuous variable $\eta$ introduced as a constraint in Equation (39). Two parameters are set for this stage: input parameters $\mathcal{i}\left({{{\mathcal{p}}_{\mathcal{s}}^{\mathcal{S}}}^{\mathcal{l}}}^{\ast },{P_{k,t,\mathcal{s}}^{G_{AC}}}^{\mathcal{l}\mathcal{*}}, {P_{i,t,\mathcal{s}}^{G_{DC}}}^{\mathcal{l}\mathcal{*}},C_{kk,t,\mathcal{s}}^{\mathcal{l}\mathcal{*}},C_{ii,t,\mathcal{s}}^{\mathcal{l}\mathcal{*}}\right)$ and output parameters, after completion of each iteration $\mathcal{l}$, $\mathcal{o}\left({P_{k,t}^{A{C}_{ES{S}_C}}}^{\ast },{P_{k,t}^{A{C}_{ES{S}_D}}}^{\ast },{P_{i,t}^{D{C}_{ES{S}_C}}}^{\ast },{P_{i,t}^{D{C}_{ES{S}_D}}}^{\ast },{P_{k,t}^{{ESS}_{AC}}}^{\ast },{P_{i,t}^{{ESS}_{DC}}}^{\ast }\right)$. Each output parameter corresponds to a feasible solution of the MP at each iteration $\mathcal{l}$. The complete MP formulation is defined by Equations (38) and (39).

$$\begin{gathered} \underset{\mathbf{x}}{\mathit{min}}\sum _{t=0}^T\left(\begin{array}{c}{w}_c\left(c_k^{ES{S}_{AC}}{\displaystyle \sum _{k=0}^{N^{{ESS}_{AC}}}}\left(P_{k,t}^{A{C}_{ES{S}_C}}+P_{k,t}^{A{C}_{ES{S}_D}}\right)+c_i^{ES{S}_{DC}}{\displaystyle \sum _{i=0}^{N^{{ESS}_{DC}}}}\left(P_{i,t}^{D{C}_{ES{S}_C}}+P_{i,t}^{D{C}_{ES{S}_D}}\right)\right)\\ -w_l\left({\displaystyle \sum _{k=0}^{N^{{ESS}_{AC}}}}{P}_{k,t}^{{ESS}_{AC}}+{\displaystyle \sum _{i=0}^{N^{{ESS}_{DC}}}}{P}_{i,t}^{{ESS}_{DC}}\right)\end{array}\right)+\eta \\ \mathrm{Subject} \mathrm{to} \mathrm{Equations} \left(1\right), \left(4\right)–\left(10\right), \left(13\right), \left(14\right), \left(16\right), \left(18\right)–\left(25\right), \mathrm{with} \forall t=1\dots T;\forall \mathcal{s}=1\dots \mathcal{S}, \forall \mathcal{l}=1,\dots , \mathcal{L} \end{gathered}$$

(38)

$$\eta \ge \sum _{\mathcal{s}}{{{\mathcal{p}}_{\mathcal{s}}^{\mathcal{S}}}^{\mathcal{l}}}^{\ast }\sum _{t=0}^T\begin{array}{c}\left(\begin{array}{c}{w}_c\left(c_k^{AC}{\displaystyle \sum _{k=0}^{N^{G_{AC}}}}{P_{k,t,\mathcal{s}}^{G_{AC}}}^{\mathcal{l}}+c_i^{DC}{\displaystyle \sum _{i=0}^{N^{G_{DC}}}}{P_{i,t,\mathcal{s}}^{G_{DC}}}^{\mathcal{l}}\right)\\ +w_l\left(\left({\displaystyle \sum _{k=0}^{N^{G_{AC}}}}{P_{k,t,\mathcal{s}}^{G_{AC}}}^{\mathcal{l}}+{\displaystyle \sum _{i=0}^{N^{G_{DC}}}}{P_{i,t,\mathcal{s}}^{G_{DC}}}^{\mathcal{l}}+{\displaystyle \sum _{k=0}^{N^{{PVG}_{AC}}}}{P}_{k,t,\mathcal{s}}^{{PV}_{AC}}+{\displaystyle \sum _{i=0}^{N^{{PVG}_{DC}}}}{P}_{i,t,\mathcal{s}}^{{PV}_{DC}}\right)-\left({\displaystyle \sum _{k=0}^{N^{L_{AC}}}}{P}_{k,t,\mathcal{s}}^{L_{AC}}+{\displaystyle \sum _{i=0}^{N^{L_{DC}}}}{P}_{i,t,\mathcal{s}}^{L_{DC}}\right)\right)\\ +w_d\left({\displaystyle \sum _{k=0}^{N^{AC}}}{\left(0.5\left(C_{kk,t,\mathcal{s}}^{\mathcal{l}}+1\right)-V_{ref}^{AC}\right)}^2+{\displaystyle \sum _{i=0}^{N^{DC}}}{\left(0.5\left(C_{ii,t,\mathcal{s}}^{\mathcal{l}}+1\right)-V_{ref}^{DC}\right)}^2\right)\end{array}\right)\\ \forall t=1\dots T;\forall s=1\dots S; \forall l=1,\dots , L \end{array}$$

(39)

In summary, Figure 5 illustrates the methodology applied in this study. To tackle the data-driven Robust Optimization problem outlined in Equation (34), a tri-level decomposition strategy is employed. The process can be outlined as follows:

The solution methodology has three key elements: it is data-driven, has a master–subproblem framework, and combines C&CG with DFD.

4. Case Studies and Results

This section uses case studies to validate the DDDRO framework for day-ahead operation planning for both VSC-based MHMs and ST-based MHMs. First, the parameters and schemes of the system under study are presented, followed by the simulation results using an uncertainties approach. In addition, comparison results and sensitivity analysis as a function of the ambiguity set are also shown.

The DDDRO framework was programmed in Python 3.9.5 in the Spyder 5.4.2 integrated development environment under the Pyomo 6.5.0 framework. To solve the MP, we use the solver Mixed-Integer Non-linear Decomposition Toolbox in Pyomo (MindtPy) [64], and for SP, we use the Interior Point Optimizer—Ipopt. All algorithms ran on a personal computer with Intel(R) Core (TM) i7-10710U CPU and 16 GB of RAM. Historical sample data for PVGs and demand, from 2012 to 2022, for the city of Bucaramanga are considered and available in [65,66].

4.1. Benchmark Test System

Analysis cases were proposed, referencing the benchmark reported in [67]. It takes the parameters of the lines, loads, and busbar connections of the North American configuration low-voltage DS with radial structure. A one-line equivalent modified MHM schematic of the interconnection of the LV commercial sub-networks is shown in Figure 6.

The MG in Figure 6 consists of 15 AC busbars and 11 DC busbars. It has a feeder connected to $B_{AC0}$, which corresponds to the low-voltage output of the transformer secondary at the common coupling point. The coupling of the 15 AC busbars with the 11 DC busbars is made from two VSCs coupled between $B_{AC14}$ and bus $B_{DC0}$, as well as bus $B_{AC13}$ and bus $B_{DC9}$, and finally, the ST is connected to the bus $B_{AC2}$ and bus $B_{DC10}$. A PVG of 28.6 kWp on the AC side and 28.38 kWp on the DC side is integrated; this equates to approximately 90% of the demand, which is considered a system with high PVG penetration. Likewise, the BESS is introduced on both the AC and DC sides coupled to their respective PEIs with a capacity of 120 kWh. The connection of the PVG and BESS is distributed in a manner according to Figure 6. It should be noted that the MHM features a second 12.47 kV MV feeder connected at the first stage of the ST.

4.2. Tests Cases and Results

Two case studies are proposed to determine the performance of the DDDRO framework for both VSC-based MHMs and ST-based MHMs, in which one level of PVG penetration (high—90%) is presented, considering the uncertainty approach in both the demand and the PVG. The following cases are considered: Case I—data-driven distributionally robust operation of the VSC-based MHM and Case II—data-driven distributionally robust operation of the ST-based MHM.

The data-driven method generates typical power scenarios and associated confidence uncertainty sets. This study uses ten years of historical data (87,600 h) on irradiance and power demand from northwestern Colombia. Drawing on previous research [40,68,69], the sample space is divided into six bins to represent random power output scenarios, leading to discrete probability distributions. Clustering techniques, such as K-means, are employed to aggregate data points into representative scenarios for each bin. Using the average values of irradiance and demand, these scenarios are determined. The probability distribution is then estimated by counting the frequency of data samples within each bin. Figure 7 displays the six clusters (scenarios) with their corresponding probabilities derived from the historical data.

Case I and Case II are compared to determine the performance of the MG by including the ST under a worst-case uncertainty scenario. To determine the performance indicators in Case I and Case II, the optimal solution ${\mathbf{x}}^{\ast }$ of the MP regarding the decision variables of the BESS and the worst-case generation profiles with the respective associated probability were taken as a reference to determine the feasible solution of the operational costs ${\mathbf{y}}^{\ast }$ of SP.

4.2.1. Case I

A day-ahead operation plan for the VSC-based MHM under uncertainty operation is investigated in this case. Operating the BESS without ST interconnection costs USD 14.62918/kW. Figure 8a and Figure 9a show the active and reactive AC power dispatch of the AC feeder, the VSCs, the PVGs, the charge and discharge power of the BESS, and the total demand. The total energy supplied by the feeder and PVGs on the AC side is 812.58 kWh. On the DC side, the total energy provided by the feeder and PVGs is 149.57 kWh. Figure 10a shows the active power dispatch of the DC PVGs, the charge and discharge power of the BESS, and the total demand.

Figure 11a,b show a heat map of the voltage profiles on the AC- and DC-side buses in p.u., respectively, for 24 h. The slack node on the AC side is set by the main feeder connected to the bus $B_{AC0}$, and on the DC side, the low-voltage DC port of the ST connected to the bus $B_{DC10}$. The buses with peak demand present moderate voltage sag and overvoltage in the buses where the PVGs are connected during the generation peak. The voltage deviation on the AC side is close to 0.07031 p.u., and on the DC side, it is 0.02327 p.u.

4.2.2. Case II

In this case, day-ahead operation planning of the ST-based MHM under uncertainty is investigated. Operating the BESS with ST interconnection costs USD 8.9043/kW. Figure 8b and Figure 9b show the active and reactive AC power dispatch of the AC feeder, the VSCs, the PVGs, the charge and discharge power of the BESS, and the total demand. The total energy supplied by the feeder and PVGs on the AC side is 256.32 kWh. On the DC side, the total energy provided by the feeder and PVGs is 701.94 kWh. Figure 10b shows the active power dispatch of the DC feeder, DC PVGs, the charge and discharge power of the BESS, and the total demand.

Figure 12a,b show a heat map of the voltage profiles on the AC- and DC-side buses in p.u., respectively, for 24 h. Likewise, as in Case I, the slack node on the AC side is set by the main feeder connected to the bus $B_{AC0}$, and on the DC side, the low-voltage DC port of the ST is connected to the bus $B_{DC10}$. The effect of the reference buses is shown in Figure 12 since they are the zones that remain with values close to the reference during the operation period. However, the buses with peak demand present moderate voltage sag and overvoltage in the buses where the PVGs are connected during the generation peak. The voltage deviation on the AC side is close to 0.03279 p.u., and on the DC side, it is 0.02527 p.u.

5. Discussion

According to the results of Case I and Case II, the total energy of the AC-side feeder has an 80% reduction in Case II compared to Case I. On the DC side, a variation from 0 kWh in Case I to 552.36 kWh in Case II is observed, given the functionality of the ST. As for the cumulative reactive power of the AC feeder, it presents a reduction closer to 23% in Case II compared to Case I. Similarly, despite testing under the worst case within the uncertainty scenarios, it is observed that the total charging and discharging power of the BESS is reduced by 44% and 34%, respectively, in Case II compared to Case I. Regarding reactive power in the BESS, a slight reduction is observed in Case II concerning Case I, given that, with the higher availability of active power in the PVG, the availability of reactive power increases, and this increase is reflected in the decrease in reactive power available in the BESS, as well as the power available at the ST’s VSC.

Finally, on the DC side, the impact of the ST and the effect of the compensation of the worst-case scenario is observed since the utilization of the BESS is reduced according to the optimization objectives, mainly in the MP; additionally, the effect of the ST as a controlled DC feeder is reflected.

Table 2. Total active and reactive power at the main MHM’s elements.

Cases	${\mathit{P}}_{\mathit{k}}^{{\mathit{G}}_{\mathit{A}\mathit{C}}}$	${\mathit{Q}}_{\mathit{k}}^{{\mathit{G}}_{\mathit{A}\mathit{C}}}$	${\mathit{P}}_{\mathit{i}}^{{\mathit{G}}_{\mathit{D}\mathit{C}}}$	${\mathit{P}}_{\mathit{k}}^{\mathit{A}{\mathit{C}}_{\mathit{E}\mathit{S}{\mathit{S}}_{\mathit{C}}}}$	${\mathit{P}}_{\mathit{k}}^{\mathit{A}{\mathit{C}}_{\mathit{E}\mathit{S}{\mathit{S}}_{\mathit{D}}}}$	${\mathit{P}}_{\mathit{i}}^{\mathit{D}{\mathit{C}}_{\mathit{E}\mathit{S}{\mathit{S}}_{\mathit{C}}}}$	${\mathit{P}}_{\mathit{i}}^{\mathit{D}{\mathit{C}}_{\mathit{E}\mathit{S}{\mathit{S}}_{\mathit{D}}}}$	${\mathit{Q}}_{\mathit{k}}^{\mathit{A}{\mathit{C}}_{\mathit{V}\mathit{S}\mathit{C}}}$	${\mathit{Q}}_{\mathit{k}}^{{\mathit{E}\mathit{S}\mathit{S}}_{\mathit{A}\mathit{C}}}$
Case I	661.84	−11.35	0	34.21	27.64	46.69	37.82	11.45	−79.50
Case II	105.58	−8.75	552.36	18.89	15.30	30.30	24.54	10.60	−74.12

presents comparative data of the system’s total active and reactive power under an uncertainty approach with management and demand profiles according to the system operation case.

Table 3. Objective function.

Case	Operation Cost [USD/kW]	Losses [kW]	Deviation Voltage AC [p.u.]	Deviation Voltage DC [p.u.]	BESS’s Cost [USD/kW]
Case I	66,184.54	14.26	0.07031	0.02327	15.49334
Case II	65,795.18	10.37	0.03279	0.02527	8.76482

compiles the values for each of the optimization problem objectives, which demonstrates the positive impact of ST coupling on the MHM.

According to the system performance indicators, the ST has a strong influence and positive impact on system performance under uncertain conditions. The ST reduces the system’s operational costs, losses, voltage deviation, and operational costs of the BESS. That is, despite the high probability of the worst-case scenario, the Energy Management System for a ST-based MHM under uncertainties can reduce the operational cost of the BESS by 50%, 30% losses, and 50% voltage deviation on the AC side. However, it is observed that, although it is a reduced value, the voltage deviation on the DC side remains stable; this is related to the high penetration of PVG and the power balance of the system. On the other hand, the BESS is the main element for reactive power dispatch in the MG, supported by the PVGs, the bidirectional transfer of reactive power in the VSCs, and the third stage of the ST to meet the demand and voltage regulation. For this analysis, the results of Case II were taken as a reference for the results of this operating case. In this way, we seek to establish the incidence of the size of historical data and the confidence level with the objective function.

Table 4. Effects of the size of historical data on BESS cost.

	Test 1	Test 2	Test 3	Test 4	Test 5	Test 6
Size sample (days)	100	500	1000	2000	3000	3650
Objective function	4062.856	4061.05	4059.85	4058.98	4058.8	4058.77

displays the total cost under various quantities of historical data. As shown in Figure 13, the total planned cost (objective function) tends to decrease and stabilizes at 4058.77 with an increase in historical data. This is because more abundant information reduces the confidence uncertainty set, making the empirical probability distribution closer to the actual distribution. This implies that with sufficient data, the probability distribution uncertainty does not need to be considered in operational planning problems.

Table 5. Total cost in different confidence levels.

${\mathit{\phi }}_1$	${\mathit{\phi }}_{\mathit{\infty }}$
	0.5	0.8	0.99
0.5	4057.22	4058.40	4058.35
0.8	4057.21	4058.03	4058.55
0.99	4057.22	4058.40	4058.76

analyzes the influence of the confidence level on the objective function results. It can be seen that the total cost of the objective function increases with the rise in the confidence level because the model becomes more conservative at higher confidence levels. Thus, a confidence level $\phi$ directly impacts ${\delta }_1$ and ${\delta }_{\infty }$, which, in turn, conditions the maximum error ${\mathcal{p}}_{\mathcal{s}}^{{\mathcal{S}}^{\pm }}$ between the ${\mathcal{p}}_{\mathcal{s}}^{{\mathcal{S}}^0}$ and ${\mathcal{p}}_{\mathcal{s}}^{\mathcal{S}}$, leading to more conservative operational planning schemes in an uncertain environment. Therefore, in the data-driven approach, the level of acceptance according to the worst-case operational scenario can be adjusted by setting the value of $\phi$ or the size of the historical data.

6. Conclusions

This paper presents a Data-Driven Distributionally Robust Optimization (DDDRO) model for a Solid-State Transformer (ST)-based Hybrid Microgrid (HMG). By constructing a confidence set for the probability distribution of uncertainties from historical data, the model ensures a robust yet non-conservative solution. The ST’s low-voltage DC port functions as a controllable feeder, supplying active power to the DC side and coupling the Voltage-Sourced Converter’s (VSC) input port with the HMG’s AC side. Additionally, the ST facilitates power exchange with the medium voltage grid and addresses the day-ahead operation planning problem, while enabling decentralized reactive power dispatch through integration with other Distributed Energy Resources (DERs) and Power Electronics Interfaces (PEIs).

To mitigate computational complexity, the proposed DDDRO model employs a Duality-Free Decomposition method, second-order cone relaxation, and positive octagonal constraint method. The modified HMG studied comprises 14 AC buses and 11 DC buses. Integration with the Column-and-Constraint Generation (C&CG) algorithm allows the ST and PEIs to reach agreements within a few iterations. The model successfully maintains the utilization of Battery Energy Storage Systems (BESSs) without compromising the economic efficiency of the entire system, even under worst-case operational scenarios.