Distributed Energy Management of the Hybrid AC/DC MicrogridwithHigh Penetration ofDistributed Energy Resources Based on ADMM

%is paper aims to investigate energymanagement of the hybrid AC/DCmicrogrid with the high penetration of distributed energy resources (DERs), such as electrical vehicles, heat pumps, and photovoltaics. In the previous studies, energy management of the hybrid microgrid is usually carried out by the system operator in a centralized manner, which suffers from the compromise of privacy information protection and the risk of single-point failure. %erefore, this paper proposes a distributed energy management scheme of the hybrid microgrid using the projection function-based alternating direction method of multipliers (PADMM), which allows each subgrid, i.e., AC subgrid and DC subgrid, to make day-ahead schedules independently with information exchanges while obtaining the optimal energy management solution. %e energy management problem of the hybrid microgrid is formulated as a mixed-integer quadratic programming (MIQP) model, considering DER and energy storage system operation constraints, system operation constraints, and converter operation constraints. %en, the MIQP model is decomposed and distributed into smaller-scale QP models between subgrids using the P-ADMM algorithm, which can handle binary variables through projection functions. %e numerical results conducted on the hybrid microgrid demonstrate that the proposed distributed scheme can effectively achieve optimal energy management for the hybrid AC/DC microgrid in a distributed manner.


Introduction
e next-generation distribution system involves the massive deployment of distributed energy resources (DERs), such as electrical vehicles (EVs), heat pumps (HPs), and photovoltaics (PVs) [1]. Adopting the microgrid concept (a small-scale autonomous energy system) has become an effective and promising technology for the integration of DERs due to its technical and economic benefits including the lower operation cost, higher reliability, and stronger resiliency [2].
Generally, microgrids can be put into three main categories according to the voltage type: (1) AC microgrids, (2) DC microgrids, and (3) hybrid AC/DC microgrids [3]. e hybrid AC/DC microgrid separates the AC and DC power supplies and loads, with the AC bus and DC bus linked through a bidirectional converter (BC) [4]. Compared with the conventional AC microgrid requiring a number of AC/ DC converters to convert power, the hybrid AC/DC microgrid has the lower equipment investment and can reduce the energy loss during power conversion. erefore, the hybrid AC/DC microgrid has been considered as an important type of microgrids as more DC power sources and loads are connected [5].
Although the utilization of the microgrid brings about technical and economic benefits, there exist challenges on the microgrid operation, such as energy management. Energy management refers to the optimized operation of dispatchable resources within microgrids such as controllable generation units and loads in order to realize the best tech-economic operation of the microgrid based on system information and forecast data [6].
Energy management for AC or DC microgrids has been widely studied. In [6], a double-layer coordinated energy management approach was proposed for microgrids considering grid-connected and islanding modes, which consists of two layers: the schedule layer obtains an economic operation scheme based on forecast data, while the dispatch layer dispatches controllable units based on real-time data. A centralized optimal microgrid scheduling model which considers multiperiod islanding constraints was proposed in [7]. e optimal scheduling model is solved using the bender decomposition technique, and the obtained solution ensures the microgrid has sufficient capacity to switch to the islanded mode if needed by examining the solution with a proposed islanding criterion. In [8], a practical energy management model for microgrids which considers the operational constraints of DERs, active-reactive power balance, unbalanced system configuration, and voltage-dependent loads was formulated. Moreover, a novel linearization approach was used to reduce the computation complexity. In [9], a model predictive control-based microgrid energy management framework was proposed, which considers the demand response in smart loads and models power flow and unit commitment constraints simultaneously. A multiobjective optimal scheduling model for a DC microgrid considering PVs and EVs was formulated in [10] and solved by the NSGA-II algorithm. In [11], energy management of a DC microgrid with energy storage systems (ESSs) and EVs was studied. Moreover, the uncertainty of renewable energy resources was widely studied and modeled in the microgrid energy management problem in [12][13][14].
For hybrid AC/DC microgrids, the energy management problem of the hybrid microgrid was modeled in [14] considering multiple distributed ESSs. In [15], a decentralized power sharing method was proposed to share power demands between the AC and DC sources without requiring communication between power sources. A temporally coordinated energy management strategy for the hybrid microgrid was proposed in [16] to deal with uncertainties of the RESs, loads, and converters while considering the dynamic conversion efficiency model of the converter. A twostage min-max-min robust optimal energy management for the hybrid microgrid was proposed in [17] to deal with the uncertainties of generation sources and loads. An up-down energy management scheme of the hybrid microgrid was proposed in [18], which consists of two levels: the generation and load balance are realized at the system level, while power converter-based resources are used to control the voltage variations at the device level. However, energy management of the hybrid microgrid in the aforementioned studies is carried out by the system operator in a centralized manner, which suffers from the compromise of privacy information protection and single-point failure risk.
To resolve the aforementioned issues, distributed methods, such as analytical target cascading (ATC) and alternating direction method of multipliers (ADMM), have been widely applied to solve optimization models in a distributed/decentralized manner in order to design distributed/decentralized schemes. For instance, in [19][20][21][22], the ATC method was used. In [19], a decentralized data-driven load restoration scheme was proposed for the coupled transmission and distribution system based on ATC. e uncertainty of wind power and load consumption was dealt with by formulating a data-driven robust model. In [20], an ATC-based decentralized generator startup sequence scheme was proposed for bulk system restoration considering the available black-start resources in the distribution system. In [21], a spatiotemporal decomposition and coordination scheme was proposed for load restoration in an AC/DC hybrid based on the ATC algorithm. In [22], the ADMM algorithm was used to solve the local flexibility market clearing problem and to design a distributed market clearing scheme.
For the energy management of the microgrid, in [23], a distributed energy management scheme was proposed for the AC microgrid based on the predictor-corrector proximal multiplier method. In the proposed scheme, the local controller of each unit (DERs and loads) optimizes its cost function and sends an optimal schedule to the centralized controller, and the centralized controller optimizes the total cost function based on the system operational constraints. However, the hybrid AC/DC configuration is not considered. In [24], a distributed robust energy management scheme for networked hybrid AC/DC microgrids was proposed, in which each hybrid AC/DC microgrid operator solves the adjustable robust optimization model to optimize its operational cost considering the uncertainty of the renewable generation and load demand. For the networked microgrids, the coordination of energy sharing is carried out to minimize the power loss based on the alternating direction method of multipliers (ADMM). However, in order to use the ADMM to solve the energy management model in a distributed manner, the model was reformulated as a convex model by removing nonlinear terms based on strict assumptions regarding the structure of the objective function, which makes the proposed distributed energy management only suitable for specific models. In order to resolve the aforementioned issues, we propose a distributed energy management scheme for the hybrid AC/DC microgrid based on the projection function-based alternating direction method of multipliers (P-ADMM) that can deal with binary variables efficiently. e P-ADMM algorithm has been widely used to solve nonconvex instances. In [25][26][27][28], the P-ADMM algorithm was used to solve the service restoration problem of distribution networks in a hierarchical/ distributed manner. In [29], a P-ADMM-based distributed scheme was proposed to coordinate the fast dispatch of PV inverters with the slow dispatch of on-load tap changer and capacitor banks for voltage regulation in unbalanced distribution systems. e practical evidence in these studies denotes that the P-ADMM is an effective method to obtain quality solutions for mixed-integer quadratic programming (MIQP) and mixed-integer linear programming (MILP) models.
In this study, the energy management problem of the hybrid microgrid is firstly formulated as a MIQP model, considering DER and ESS operation constraints, system operation constraints, and converter operation constraints.
en, the MIQP model is decomposed and distributed into 2 Complexity smaller-scale submodels using the P-ADMM-based algorithm that can handle binary variables through projection functions. In the proposed scheme, each subgrid, i.e., AC subgrid and DC subgrid, can make day-ahead schedules independently with information exchanges while achieving optimal energy management. e main contributions of this paper are summarized as follows: (1) proposes a distributed energy management scheme for the hybrid AC/DC microgrid, which allows each subgrid to make energy schedules independently with information exchanges while realizing optimal energy management and (2) proposes a P-ADMM-based algorithm to efficiently solve the MIQP energy management model in a distributed manner. e rest of this paper is organized as follows. e framework of the proposed distributed energy management scheme and energy management model formulations are presented in Section 2. e P-ADMM-based algorithm to solve the energy management model is presented in Section 3. Case studies are presented and discussed in Section 4, followed by conclusions.

Framework of the Distributed Energy Management
Scheme. Figure 1 illustrates the physical and communication architecture of the hybrid AC/DC microgrid, to which various sources and loads are connected. In the DC subgrid, photovoltaics (PVs), ESSs, EVs, and heat pumps (HPs) are connected, while diesel generators (DGs) and conventional AC loads are connected to the AC subgrid. AC subgrid and DC subgrid are tied by a BC.
In the proposed distributed scheme, the AC subgrid operator and DC subgrid operator communicate with each other to make day-ahead energy schedules for respective sources and loads independently.

Optimal Energy Management Model.
In this section, the optimal energy management model of the hybrid microgrid is formulated.

Objective Function.
For the energy management of microgrids, the objective function is to minimize operational costs during the scheduling period. e objective function is formulated as follows: (1) e objective function costs of three terms are as follows: the first two terms are to minimize energy costs of DGs, where N T is the set of scheduling periods, c G 1 and c G 2 are generation cost parameters, and p DG t is the DG generation; the last term is to minimize the cost of purchasing energy from the external grid, where λ t is the energy sport price, b is the price sensitivity coefficient, and p UG t is the purchased power.

Power Balance Equations
Equations (2) and (3) are active power balance equations of the AC subgrid and DC subgrid, respectively, where N h and N e are sets of HPs and EVs, respectively; p dta t is the power transferring from the DC side to the AC side at hour t and p atd t is the power transferring from the AC side to the DC side at hour t; p AC_L t is the conventional AC load consumption; p ev i,t is the EV charging power of the ith EV at hour t; p hp i,t is the power consumption of the ith heat pump at hour t; p ec t and p ed t are the ESS charging and discharging power, respectively; p pv t is the PV generation power; and η dta and η atd are power transfer coefficients.

EV Operation Constraints
Constraint (6) is the EV SOC level constraint, where d ev is the power consumption of the ith EV at hour t due to driving, e i,0 is the initial SOC level of the ith EV, and e min i,t and e max i,t are the minimum and maximum SOC level limits of the ith EV. Constraint (7) is the EV charging power constraint, where p ev,min i,t and p ev,max i,t are the minimum and maximum charging power limits of the ith EV at hour t and a i,t is the charging availability of the ith EV at hour t.

HP Operation Constraints
Constraints (8) and (9) represent thermal equations of the house equipped with the HP [30,31], where c cop i is the coefficient of performance; k 1 , k 2 , k 3 , k 4 , and k 5 are thermal efficiency coefficients; and K h i,t , K u i,t , and K s i,t are the household inside temperature, outside temperature, and structure temperature, respectively. Constraint (10)  are the maximum and minimum limits of the ith heat pump power consumption at hour t.

ESS Operation Constraints
x ed t p ed, min Constraints (12) and (13)

Bidirectional Converter Operation Constraint
Constraint (17) represents the transferring power limit, where p bc, max t is the maximum active power converted on the BC and x atd t and x dta t are binary variables representing the power conversion direction, power is transferred from the AC side to the DC side if x atd t � 1, and power is transferred from the DC side to the AC side if x dta t � 1 and also represents that the BC can operate either in the charging mode or the discharging mode.
Finally, the energy management problem is formulated as a MIQP problem in (1)- (17). It is noted that the AC subproblem and DC subproblem are coupled through the DC conversion power p atd t and p dta t in power balance equations (2) and (3) and constraint (17). In order to realize distributed energy management, the AC subgrid operator and DC subgrid operator should solve their own problems independently. erefore, a P-ADMM-based algorithm is proposed to solve the energy management problem in a distributed manner, which will be detailed in the next section.

P-ADMM-Based Solution Method
In this section, the P-ADMM algorithm is used to solve the optimal energy management problem in a distributed manner. Firstly, the MIQP problem is reformulated for the 4 Complexity implementation of the P-ADMM algorithm.
en, the augmented Lagrangian of the problem is introduced. Finally, the P-ADMM-based algorithm is presented.

Decoupling of Coupled Constraints.
To decouple power balance equations and BC operational constraints, a set of auxiliary variables are introduced as follows: We replace original variables with auxiliary variables, and the model is reformulated.
Objective function of the reformulated model is (1).
Constraints of the reformulated model include original constraints (4)-(11), auxiliary constraint (18), and It is assumed that variables with superscripts of "atd" are affiliated variables of the AC subgrid operator and variables with superscripts of "dta" are affiliated variables of the DC subgrid operator. erefore, constraints (4)-(11) and (19)- (25) are fully decomposable between the AC and DC subgrids, whereas equality constraint (18) is still coupled.

Augmented Lagrangian.
By adding equality constraint (18) into objective function (1) through dual variables Λ � :{λ t , λ t , δ t , δ t , χ t , θ t , κ t , ψ t } and the penalty parameter ρ, the augmented Lagrangian of the MIQP model is formulated as follows: Complexity Subject to (4)- (11) and (19)- (25). e augmented Lagrangian is optimized over two groups of primal variables X � : {p DG 3.3. P-ADMM-Based Algorithm. For the ease of presentation, the objective function of the augmented Lagrangian is represented using a compact form L (X, Y, Λ). e augmented Lagrangian is solved by the P-ADMM-based algorithm with an iterative procedure, as shown in Algorithm 1, in which optimizations of two subproblems and the dualvariable update procedure are carried out iteratively.
s.t. constraints (4)- (11) and (19)- (25). Subproblem I is a QP problem, and its objective function and constraints can be fully decoupled between AC and DC subgrids. erefore, subproblem I can be solved in a distributed manner between AC and DC subgrids.

Subproblem II.
e primal variables Y k are optimized in subproblem II, given by (27) with parameters X * ,k and Λ * k−1 obtained in the kth and (k − 1)th iteration.
It is noted that subproblem II is a nonconstrained MIQP problem. Instead of directly solving the MIQP problem to optimize binary variables, the binary variables are optimized using the projection function in (28), where P x denotes the projection function, which rounds each entry to its nearest binary value. e remaining primal variables are optimized by solving the QP problem after excluding binary variables, which can be fully decoupled between AC and DC subgrids. erefore, subproblem II can be solved in a distributed manner as well.
3.6. Update of Dual Variables. After solving subproblems I and II, dual variables are updated in (29).

Stop
Criteria. e iterative procedure continues till primal and dual residuals are lower than specified thresholds, respectively, as follows:

Case Studies
A case study was conducted on the hybrid AC/DC system shown in Figure 1 to demonstrate the effectiveness of the proposed distributed energy management scheme. In the system, one DG and conventional loads are connected to the AC bus, whereas one ESS, 200 EVs [30,32], 200 HPs, and one PV are connected to the DC bus. Figures 2 and 3 show the forecasted spot price and PV active power generation profile. e key parameters of DGs, EVs, HPs, and ESSs are given in Table 1. e P-ADMM-based algorithm is implemented using GAMS [31,33], and the solution is obtained with the CPLEX solver [34]. In the algorithm, convergence thresholds σ 1 and σ 2 are set as 0.01, and penalty parameter ρ is set as 50.

Computational Performance.
Although the convergence of the P-ADMM-based algorithm cannot be guaranteed for the MIQP model, the algorithm shows good convergence performance. A convergence analysis is conducted by solving the MIQP model with different penalty parameter values between 10 and 150. Table 2 shows that the algorithm can converge within 650 iterations when the penalty parameter value varies within 10 and 150. It can be seen that the convergence performance of the algorithm highly depends on the selection of the penalty parameter value. For example, an improper value ρ � 30 would cause nonconvergence within the preset iteration limit (1000). Within the certain range, the algorithm with ρ � 50 has the best convergence performance. e convergence process of the P-ADMM-based algorithm with ρ � 50 is shown in Figure 4. It can be seen that the primal and dual residuals are lower than thresholds after 91 iterations. e distributed solution S d is compared with the centralized solution S c in Table 3(take hour t 12 as an example). e centralized solution is obtained by solving the MIQP model using the GAMS/CPLEX solver with a zero optimality gap. It is shown in Table 2 that the distributed solution is almost the same as the centralized solution. e maximum relative error (RE) between the centralized and distributed solutions is 1.28% (the RE is calculated by (31)), which is very small and demonstrates that the proposed P-ADMM-based algorithm can reproduce the optimal centralized solution. Figures 5 and 6 show the optimal day-ahead energy schedules of the AC and DC subgrids, respectively. In the DC subgrid, the PV active power, ESS discharging power, and power converted from the AC subgird support the total power consumption of EVs and HPs. For load consumption, it can be seen that EVs and HPs tend to charge more power at hours with relatively low spot prices in order to minimize day-ahead energy costs. For power generations, in most hours, e.g., hours t 1 -t 9 and t 19 -t 24 , AC power is converted into DC power to support EV and HP power consumption. At hours t 10 -t 16 , since there is sufficient PV active power, there is no need to convert AC power into DC power. Especially at hours t 14 and t 16 , the DC subgrid delivers surplus power to the AC grid. In addition, the ESS would store surplus power at hours with relatively low prices and discharge power at those hours with relatively high spot prices to minimize energy costs.

Energy Management Results.
In the AC subgrid, the AC operator chooses to dispatch the DG unit or buy external electricity for AC load consumption according to DG energy costs and spot prices. In this case, since the DG energy cost is smaller than the energy cost of purchasing external electricity in most hours, the DG maximum capacity is almost used. In addition, the external purchased electricity profile has almost opposite variation P-ADMM algorithm (1) input: system and DER parameters (2) output: AC and DC energy schedules and BC operation status (3) while: ‖X ex, * ,k − Y * ,k ‖ 2 ≤ σ 1 , ‖X ex, * ,k − X ex, * ,k− 1 ‖ 2 ≤ σ 2 do Solve subproblem I to update primal variables X k Solve subproblem I to update primal variables Y k Update dual variables Λ k+1 using (30) end ALGORITHM 1: Pseudo-code of the P-ADMM algorithm.    Complexity 7 tendency as compared with the forecasted spot price profile in order to minimize energy costs.

Conclusions
To resolve the privacy information protection compromise and single-point failure issues when the centralized energy management scheme is applied, this paper proposes a distributed energy management scheme for the hybrid AC/DC microgrid based on the proposed P-ADMM-based algorithm. e energy management problem of the hybrid microgrid is formulated as a MIQP model, considering DER operation constraints, system operation constraints, and converter operation constraints. en, the MIQP model is decomposed and distributed into smaller-scale submodels between subgrids using the P-ADMM-based algorithm. e numerical results demonstrate that the proposed algorithm can efficiently solve the MIQP model in a distributed manner and can deal with binary variables. e proposed distributed scheme allows each subgrid operator to make day-ahead schedules independently with information exchanges while acquiring the optimal energy management solution.
Data Availability e data used to support the findings of this study are available from the corresponding author upon request. Table 3: Comparison between centralized and distributed solutions (take hour t 12 as an example).   Complexity