# Distributed Energy Management of the Hybrid AC/DC Microgrid with High Penetration of Distributed Energy Resources Based on ADMM

Authors: Yaling Chen; Luxi Hao; Gaowen Yin
Journal: Complexity
 (2021)

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## Abstract

This paper aims to investigate energy management of the hybrid AC/DC microgrid 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. Therefore, this paper proposes a distributed energy management scheme of the hybrid microgrid using the projection function-based alternating direction method of multipliers (P-ADMM), 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. The 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. Then, 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. The 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.


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## Body

## 1. Introduction

The 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]. The 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. Therefore, 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]. The 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 [12–14].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 two-stage 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 [19–22], 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. The 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. The P-ADMM algorithm has been widely used to solve nonconvex instances. In [25–28], 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. The 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. Then, the MIQP model is decomposed and distributed into 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.The 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.The rest of this paper is organized as follows. The framework of the proposed distributed energy management scheme and energy management model formulations are presented in Section2. The 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.


## 2. Framework and Model Formulations of the Distributed Energy Management Scheme

### 2.1. Framework of the Distributed Energy Management Scheme

Figure1 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.Figure 1
Physical and communication architecture of the hybrid AC/DC microgrid.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.


### 2.2. Optimal Energy Management Model

In this section, the optimal energy management model of the hybrid microgrid is formulated.


#### 2.2.1. Objective Function

For the energy management of microgrids, the objective function is to minimize operational costs during the scheduling period. The objective function is formulated as follows:(1)min∑t∈NTc1GptDG2+c2GptDG+λtptUG+bptUG2.The objective function costs of three terms are as follows: the first two terms are to minimize energy costs of DGs, whereNT is the set of scheduling periods, c1G and c2G are generation cost parameters, and ptDG is the DG generation; the last term is to minimize the cost of purchasing energy from the external grid, where λt is the energy sport price, b is the price sensitivity coefficient, and ptUG is the purchased power.


#### 2.2.2. Power Balance Equations

(2)ptUG+ptDG+ηdtaptdta=ptAC_L+ptatd,∀t∈NT,(3)pted+ptpv+ηatdptatd=ptec+∑i∈Nepi,tev+∑i∈Nhpi,thp+ptdta,∀t∈NT.Equations (2) and (3) are active power balance equations of the AC subgrid and DC subgrid, respectively, where Nh and Ne are sets of HPs and EVs, respectively; ptdta is the power transferring from the DC side to the AC side at hour t and ptatd is the power transferring from the AC side to the DC side at hour t; ptAC_L is the conventional AC load consumption; pi,tev is the EV charging power of the ith EV at hour t; pi,thp is the power consumption of the ith heat pump at hour t; ptec and pted are the ESS charging and discharging power, respectively; ptpv is the PV generation power; and ηdta and ηatd are power transfer coefficients.


#### 2.2.3. DG Operation Constraints

(4)ptDG,min≤ptDG≤ptDG,max,∀t∈NT,(5)ptDG,dr≤ptDG−pt−1DG≤ptDG,urpt1DG,dr≤pt1DG−p0DG≤pt1DG,ur,∀t∈NT.Constraint (4) is the DG capacity constraint, where ptDG,max and ptDG,min are the maximum and minimum limits of the DG active power generation. Constraint (5) is the DG ramping constraint, where ptDG,ur and ptDG,dr are the upramping and downramping limits.


#### 2.2.4. EV Operation Constraints

(6)ei,tmin≤ei,0+∑t−≤tpi,t−ev−di,t−ev≤ei,tmax,∀i∈Ne,t∈Nt,(7)pi,tev,min≤pi,tev≤ai,tpi,tev,max,∀i∈Ne,t∈Nt.Constraint (6) is the EV SOC level constraint, where di,tev is the power consumption of the ith EV at hour t due to driving, ei,0 is the initial SOC level of the ith EV, and ei,tmin and ei,tmax are the minimum and maximum SOC level limits of the ith EV. Constraint (7) is the EV charging power constraint, where pi,tev,min and pi,tev,max are the minimum and maximum charging power limits of the ith EV at hour t and ai,t is the charging availability of the ith EV at hour t.


#### 2.2.5. HP Operation Constraints

(8)cicoppi,thp−k1Ki,th−Ki,tu−k2Ki,th−Ki,ts=k3Ki,th−Ki,t−1h,∀t∈NT,i∈Nh,(9)−k4Ki,ts−Ki,tu+k2Ki,th−Ki,ts=k5Ki,ts−Ki,t−1s,∀t∈NT,i∈Nh,(10)Ki,th,min≤Kt,ih≤Ki,th,max,∀t∈NT,i∈Nh,(11)pi,thp,min≤pi,thp≤pi,thp,max,∀t∈NT,i∈Nh.Constraints (8) and (9) represent thermal equations of the house equipped with the HP [30, 31], where cicop is the coefficient of performance; k1, k2, k3, k4, and k5 are thermal efficiency coefficients; and Ki,th, Ki,tu, and Ki,ts are the household inside temperature, outside temperature, and structure temperature, respectively. Constraint (10) represents the household temperature constraint, where Ki,th,max and Ki,th,min are the maximum and minimum limits of the household inside temperature. Constraint (11) represents the HP power consumption constraint, where pi,thp,max and pi,thp,min are the maximum and minimum limits of the ith heat pump power consumption at hour t.


#### 2.2.6. ESS Operation Constraints

(12)xtecpiec,min≤ptec≤xtecpiec,max,∀t∈NT,(13)xtedpied,min≤pted≤xtedpied,max,∀i∈Ne,t∈NT,(14)xi,tec+xi,ted=1,t∈NT,(15)Etes=Et−1es+ηcptecΔt−1ηdptedΔt,∀t∈NT,(16)Etes,min≤Etes≤Etes,max.Constraints (12) and (13) are ESS charging and discharging power constraints, respectively, where piec,min and piec,max are the minimum and maximum limits of ESS charging power, respectively; pied,min and pied,max are the minimum and maximum limits of ESS discharging power, respectively; and xtec and xtedare the binary variables representing ESS charging/discharging status, ESS is in the charging mode if xtec  = 1, and ESS is in the discharging mode if xted  = 1. Constraint (15) is the ESS energy balance constraint, where Etes is the ESS SOC level, Δt is the time interval of an hour, and ηc and ηd are the charging and discharging coefficients. Constraint (16) is the ESS SOC level constraint, where Etes,min and Etes,max are the minimum and maximum limits of the ESS SOC level.


#### 2.2.7. Bidirectional Converter Operation Constraint

(17)0≤ptatd≤xtatdptbc,max,∀t∈NT,0≤ptdta≤xtdtaptbc,max,∀t∈NT,xtatd+xtdta=1,∀t∈NT.Constraint (17) represents the transferring power limit, where ptbc,max is the maximum active power converted on the BC and xtatd and xtdta are binary variables representing the power conversion direction, power is transferred from the AC side to the DC side if xtatd  = 1, and power is transferred from the DC side to the AC side if xtdta  = 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 ptatd and ptdta 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. Therefore, 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.


## 2.1. Framework of the Distributed Energy Management Scheme

Figure1 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.Figure 1
Physical and communication architecture of the hybrid AC/DC microgrid.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.


## 2.2. Optimal Energy Management Model

In this section, the optimal energy management model of the hybrid microgrid is formulated.


### 2.2.1. Objective Function

For the energy management of microgrids, the objective function is to minimize operational costs during the scheduling period. The objective function is formulated as follows:(1)min∑t∈NTc1GptDG2+c2GptDG+λtptUG+bptUG2.The objective function costs of three terms are as follows: the first two terms are to minimize energy costs of DGs, whereNT is the set of scheduling periods, c1G and c2G are generation cost parameters, and ptDG is the DG generation; the last term is to minimize the cost of purchasing energy from the external grid, where λt is the energy sport price, b is the price sensitivity coefficient, and ptUG is the purchased power.


### 2.2.2. Power Balance Equations

(2)ptUG+ptDG+ηdtaptdta=ptAC_L+ptatd,∀t∈NT,(3)pted+ptpv+ηatdptatd=ptec+∑i∈Nepi,tev+∑i∈Nhpi,thp+ptdta,∀t∈NT.Equations (2) and (3) are active power balance equations of the AC subgrid and DC subgrid, respectively, where Nh and Ne are sets of HPs and EVs, respectively; ptdta is the power transferring from the DC side to the AC side at hour t and ptatd is the power transferring from the AC side to the DC side at hour t; ptAC_L is the conventional AC load consumption; pi,tev is the EV charging power of the ith EV at hour t; pi,thp is the power consumption of the ith heat pump at hour t; ptec and pted are the ESS charging and discharging power, respectively; ptpv is the PV generation power; and ηdta and ηatd are power transfer coefficients.


### 2.2.3. DG Operation Constraints

(4)ptDG,min≤ptDG≤ptDG,max,∀t∈NT,(5)ptDG,dr≤ptDG−pt−1DG≤ptDG,urpt1DG,dr≤pt1DG−p0DG≤pt1DG,ur,∀t∈NT.Constraint (4) is the DG capacity constraint, where ptDG,max and ptDG,min are the maximum and minimum limits of the DG active power generation. Constraint (5) is the DG ramping constraint, where ptDG,ur and ptDG,dr are the upramping and downramping limits.


### 2.2.4. EV Operation Constraints

(6)ei,tmin≤ei,0+∑t−≤tpi,t−ev−di,t−ev≤ei,tmax,∀i∈Ne,t∈Nt,(7)pi,tev,min≤pi,tev≤ai,tpi,tev,max,∀i∈Ne,t∈Nt.Constraint (6) is the EV SOC level constraint, where di,tev is the power consumption of the ith EV at hour t due to driving, ei,0 is the initial SOC level of the ith EV, and ei,tmin and ei,tmax are the minimum and maximum SOC level limits of the ith EV. Constraint (7) is the EV charging power constraint, where pi,tev,min and pi,tev,max are the minimum and maximum charging power limits of the ith EV at hour t and ai,t is the charging availability of the ith EV at hour t.


### 2.2.5. HP Operation Constraints

(8)cicoppi,thp−k1Ki,th−Ki,tu−k2Ki,th−Ki,ts=k3Ki,th−Ki,t−1h,∀t∈NT,i∈Nh,(9)−k4Ki,ts−Ki,tu+k2Ki,th−Ki,ts=k5Ki,ts−Ki,t−1s,∀t∈NT,i∈Nh,(10)Ki,th,min≤Kt,ih≤Ki,th,max,∀t∈NT,i∈Nh,(11)pi,thp,min≤pi,thp≤pi,thp,max,∀t∈NT,i∈Nh.Constraints (8) and (9) represent thermal equations of the house equipped with the HP [30, 31], where cicop is the coefficient of performance; k1, k2, k3, k4, and k5 are thermal efficiency coefficients; and Ki,th, Ki,tu, and Ki,ts are the household inside temperature, outside temperature, and structure temperature, respectively. Constraint (10) represents the household temperature constraint, where Ki,th,max and Ki,th,min are the maximum and minimum limits of the household inside temperature. Constraint (11) represents the HP power consumption constraint, where pi,thp,max and pi,thp,min are the maximum and minimum limits of the ith heat pump power consumption at hour t.


### 2.2.6. ESS Operation Constraints

(12)xtecpiec,min≤ptec≤xtecpiec,max,∀t∈NT,(13)xtedpied,min≤pted≤xtedpied,max,∀i∈Ne,t∈NT,(14)xi,tec+xi,ted=1,t∈NT,(15)Etes=Et−1es+ηcptecΔt−1ηdptedΔt,∀t∈NT,(16)Etes,min≤Etes≤Etes,max.Constraints (12) and (13) are ESS charging and discharging power constraints, respectively, where piec,min and piec,max are the minimum and maximum limits of ESS charging power, respectively; pied,min and pied,max are the minimum and maximum limits of ESS discharging power, respectively; and xtec and xtedare the binary variables representing ESS charging/discharging status, ESS is in the charging mode if xtec  = 1, and ESS is in the discharging mode if xted  = 1. Constraint (15) is the ESS energy balance constraint, where Etes is the ESS SOC level, Δt is the time interval of an hour, and ηc and ηd are the charging and discharging coefficients. Constraint (16) is the ESS SOC level constraint, where Etes,min and Etes,max are the minimum and maximum limits of the ESS SOC level.


### 2.2.7. Bidirectional Converter Operation Constraint

(17)0≤ptatd≤xtatdptbc,max,∀t∈NT,0≤ptdta≤xtdtaptbc,max,∀t∈NT,xtatd+xtdta=1,∀t∈NT.Constraint (17) represents the transferring power limit, where ptbc,max is the maximum active power converted on the BC and xtatd and xtdta are binary variables representing the power conversion direction, power is transferred from the AC side to the DC side if xtatd  = 1, and power is transferred from the DC side to the AC side if xtdta  = 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 ptatd and ptdta 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. Therefore, 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.


## 2.2.1. Objective Function

For the energy management of microgrids, the objective function is to minimize operational costs during the scheduling period. The objective function is formulated as follows:(1)min∑t∈NTc1GptDG2+c2GptDG+λtptUG+bptUG2.The objective function costs of three terms are as follows: the first two terms are to minimize energy costs of DGs, whereNT is the set of scheduling periods, c1G and c2G are generation cost parameters, and ptDG is the DG generation; the last term is to minimize the cost of purchasing energy from the external grid, where λt is the energy sport price, b is the price sensitivity coefficient, and ptUG is the purchased power.


## 2.2.2. Power Balance Equations

(2)ptUG+ptDG+ηdtaptdta=ptAC_L+ptatd,∀t∈NT,(3)pted+ptpv+ηatdptatd=ptec+∑i∈Nepi,tev+∑i∈Nhpi,thp+ptdta,∀t∈NT.Equations (2) and (3) are active power balance equations of the AC subgrid and DC subgrid, respectively, where Nh and Ne are sets of HPs and EVs, respectively; ptdta is the power transferring from the DC side to the AC side at hour t and ptatd is the power transferring from the AC side to the DC side at hour t; ptAC_L is the conventional AC load consumption; pi,tev is the EV charging power of the ith EV at hour t; pi,thp is the power consumption of the ith heat pump at hour t; ptec and pted are the ESS charging and discharging power, respectively; ptpv is the PV generation power; and ηdta and ηatd are power transfer coefficients.


## 2.2.3. DG Operation Constraints

(4)ptDG,min≤ptDG≤ptDG,max,∀t∈NT,(5)ptDG,dr≤ptDG−pt−1DG≤ptDG,urpt1DG,dr≤pt1DG−p0DG≤pt1DG,ur,∀t∈NT.Constraint (4) is the DG capacity constraint, where ptDG,max and ptDG,min are the maximum and minimum limits of the DG active power generation. Constraint (5) is the DG ramping constraint, where ptDG,ur and ptDG,dr are the upramping and downramping limits.


## 2.2.4. EV Operation Constraints

(6)ei,tmin≤ei,0+∑t−≤tpi,t−ev−di,t−ev≤ei,tmax,∀i∈Ne,t∈Nt,(7)pi,tev,min≤pi,tev≤ai,tpi,tev,max,∀i∈Ne,t∈Nt.Constraint (6) is the EV SOC level constraint, where di,tev is the power consumption of the ith EV at hour t due to driving, ei,0 is the initial SOC level of the ith EV, and ei,tmin and ei,tmax are the minimum and maximum SOC level limits of the ith EV. Constraint (7) is the EV charging power constraint, where pi,tev,min and pi,tev,max are the minimum and maximum charging power limits of the ith EV at hour t and ai,t is the charging availability of the ith EV at hour t.


## 2.2.5. HP Operation Constraints

(8)cicoppi,thp−k1Ki,th−Ki,tu−k2Ki,th−Ki,ts=k3Ki,th−Ki,t−1h,∀t∈NT,i∈Nh,(9)−k4Ki,ts−Ki,tu+k2Ki,th−Ki,ts=k5Ki,ts−Ki,t−1s,∀t∈NT,i∈Nh,(10)Ki,th,min≤Kt,ih≤Ki,th,max,∀t∈NT,i∈Nh,(11)pi,thp,min≤pi,thp≤pi,thp,max,∀t∈NT,i∈Nh.Constraints (8) and (9) represent thermal equations of the house equipped with the HP [30, 31], where cicop is the coefficient of performance; k1, k2, k3, k4, and k5 are thermal efficiency coefficients; and Ki,th, Ki,tu, and Ki,ts are the household inside temperature, outside temperature, and structure temperature, respectively. Constraint (10) represents the household temperature constraint, where Ki,th,max and Ki,th,min are the maximum and minimum limits of the household inside temperature. Constraint (11) represents the HP power consumption constraint, where pi,thp,max and pi,thp,min are the maximum and minimum limits of the ith heat pump power consumption at hour t.


## 2.2.6. ESS Operation Constraints

(12)xtecpiec,min≤ptec≤xtecpiec,max,∀t∈NT,(13)xtedpied,min≤pted≤xtedpied,max,∀i∈Ne,t∈NT,(14)xi,tec+xi,ted=1,t∈NT,(15)Etes=Et−1es+ηcptecΔt−1ηdptedΔt,∀t∈NT,(16)Etes,min≤Etes≤Etes,max.Constraints (12) and (13) are ESS charging and discharging power constraints, respectively, where piec,min and piec,max are the minimum and maximum limits of ESS charging power, respectively; pied,min and pied,max are the minimum and maximum limits of ESS discharging power, respectively; and xtec and xtedare the binary variables representing ESS charging/discharging status, ESS is in the charging mode if xtec  = 1, and ESS is in the discharging mode if xted  = 1. Constraint (15) is the ESS energy balance constraint, where Etes is the ESS SOC level, Δt is the time interval of an hour, and ηc and ηd are the charging and discharging coefficients. Constraint (16) is the ESS SOC level constraint, where Etes,min and Etes,max are the minimum and maximum limits of the ESS SOC level.


## 2.2.7. Bidirectional Converter Operation Constraint

(17)0≤ptatd≤xtatdptbc,max,∀t∈NT,0≤ptdta≤xtdtaptbc,max,∀t∈NT,xtatd+xtdta=1,∀t∈NT.Constraint (17) represents the transferring power limit, where ptbc,max is the maximum active power converted on the BC and xtatd and xtdta are binary variables representing the power conversion direction, power is transferred from the AC side to the DC side if xtatd  = 1, and power is transferred from the DC side to the AC side if xtdta  = 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 ptatd and ptdta 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. Therefore, 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.


## 3. 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 implementation of the P-ADMM algorithm. Then, the augmented Lagrangian of the problem is introduced. Finally, the P-ADMM-based algorithm is presented.


### 3.1. Decoupling of Coupled Constraints

To decouple power balance equations and BC operational constraints, a set of auxiliary variables are introduced as follows:(18)ptdta=p^tdta,ptdta=p˜tatd,ptatd=p^tatd,ptatd=p˜tdta,xtatd=x^tatd,xtdta=x^tdta,xtdta=x˜tatd,xtec=x^tec,xtdc=x^tdc.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(19)x^tecpiec,min≤ptec≤x^tecpiec,max,∀t∈NT,(20)x^tedpied,min≤pted≤x^tedpied,max,∀i∈Ne,t∈NT,(21)x^i,tec+x^i,ted=1,t∈NT,(22)ptUG+ptDG+ηdtap˜tatd=ptAC_L+p^tatd,∀t∈NT,(23)pted+ptpv+ηatdp˜tdta=ptec+∑i∈Nepi,tev+∑i∈Nhpi,thp+p^tdta,∀t∈NT,(24)0≤p^tatd≤x^tatdptbc,max,∀t∈NT,0≤p^tdta≤x^tdtaptbc,max,∀t∈NT,(25)x^tatd+x˜tatd=1,∀t∈NT.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. Therefore, constraints (4)–(11) and (19)–(25) are fully decomposable between the AC and DC subgrids, whereas equality constraint (18) is still coupled.


### 3.2. Augmented Lagrangian

By adding equality constraint (18) into objective function (1) through dual variables Λ = :{λt, λ^t, δt, δ^t, χt, θt, κt, ψt} and the penalty parameter ρ, the augmented Lagrangian of the MIQP model is formulated as follows:(26)min∑t∈NTc1GptDG2+c2GptDG+λtptUG+ρ2ptdta−p˜tatd+λt2−λt2+ptdta−p^tdta+λ^t2−λ^t2+ptatd−p˜tdta+δt2−δt2+ptatd−p^tatd+δ^t2−δ^t2++xtatd−x^tatd+χt2−χt2+xtdta−x^tdta+θt2−θt2+xtdta−x˜tatd+ξt2−ξt2+xtec−x^tec+κt2−κt2+xted−x^ted+ψt2−ψt2.Subject to (4)–(11) and (19)–(25).The augmented Lagrangian is optimized over two groups of primal variablesX = : {ptDG, ptUG, pi,tev, pi,thp, ptec, pted, p˜tatd, p^tdta, p˜tdta, p^tatd, x^tatd, x˜tatd, x^tdta, x^tec, x^ted} and Y = : {ptdta, ptatd, xtatd, xtdta, xtec, xted} and a group of dual variables Λ.


### 3.3. P-ADMM-Based Algorithm

For the ease of presentation, the objective function of the augmented Lagrangian is represented using a compact formL (X, Y, Λ). The 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 dual-variable update procedure are carried out iteratively.Algorithm 1: Pseudo-code of the P-ADMM algorithm.
P-ADMM algorithm(1) input: system and DER parameters(2) output: AC and DC energy schedules and BC operation status(3) while:Xex,∗,k−Y∗,k2≤σ1,Xex,∗,k−Xex,∗,k−12≤σ2doSolve subproblem I to update primal variablesXkSolve subproblem I to update primal variablesYkUpdate dual variablesΛk+1using (30)end


### 3.4. Subproblem I

In thekth iteration, primal variables Xk are optimized in subproblem I, given by (26), subject to (4)–(11) and (19)–(25), with parameters Y∗,k−1 and Λ∗k−1 obtained in the (k− 1)th iteration.(27)Subproblem⋅I:⋅LXk,Y∗,k−1,Λ∗,k−1,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. Therefore, subproblem I can be solved in a distributed manner between AC and DC subgrids.


### 3.5. Subproblem II

The primal variablesYk 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.(28)Subproblem⋅II:⋅LX∗,k,Yk,Λ∗,k−1.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 Px denotes the projection function, which rounds each entry to its nearest binary value. The remaining primal variables are optimized by solving the QP problem after excluding binary variables, which can be fully decoupled between AC and DC subgrids. Therefore, subproblem II can be solved in a distributed manner as well.(29)xtatd=Pxx^tatd−χt,xtdta=Pxx^tdta−θt+x˜tatd−ξt2.


### 3.6. Update of Dual Variables

After solving subproblems I and II, dual variables are updated in (29).(30)Λk=Λ∗,k−1+Xex,∗,k−1−Y∗,k,Xex=:p˜tatd,p^tdta,p˜tdta,p^tatd,x^tatd,x˜tatd,x^tdta.


### 3.7. Stop Criteria

The iterative procedure continues till primal and dual residuals are lower than specified thresholds, respectively, as follows:(31)Xex,∗,k−Y∗,k2≤σ1,Xex,∗,k−Xex,∗,k−12≤σ2.


## 3.1. Decoupling of Coupled Constraints

To decouple power balance equations and BC operational constraints, a set of auxiliary variables are introduced as follows:(18)ptdta=p^tdta,ptdta=p˜tatd,ptatd=p^tatd,ptatd=p˜tdta,xtatd=x^tatd,xtdta=x^tdta,xtdta=x˜tatd,xtec=x^tec,xtdc=x^tdc.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(19)x^tecpiec,min≤ptec≤x^tecpiec,max,∀t∈NT,(20)x^tedpied,min≤pted≤x^tedpied,max,∀i∈Ne,t∈NT,(21)x^i,tec+x^i,ted=1,t∈NT,(22)ptUG+ptDG+ηdtap˜tatd=ptAC_L+p^tatd,∀t∈NT,(23)pted+ptpv+ηatdp˜tdta=ptec+∑i∈Nepi,tev+∑i∈Nhpi,thp+p^tdta,∀t∈NT,(24)0≤p^tatd≤x^tatdptbc,max,∀t∈NT,0≤p^tdta≤x^tdtaptbc,max,∀t∈NT,(25)x^tatd+x˜tatd=1,∀t∈NT.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. Therefore, constraints (4)–(11) and (19)–(25) are fully decomposable between the AC and DC subgrids, whereas equality constraint (18) is still coupled.


## 3.2. Augmented Lagrangian

By adding equality constraint (18) into objective function (1) through dual variables Λ = :{λt, λ^t, δt, δ^t, χt, θt, κt, ψt} and the penalty parameter ρ, the augmented Lagrangian of the MIQP model is formulated as follows:(26)min∑t∈NTc1GptDG2+c2GptDG+λtptUG+ρ2ptdta−p˜tatd+λt2−λt2+ptdta−p^tdta+λ^t2−λ^t2+ptatd−p˜tdta+δt2−δt2+ptatd−p^tatd+δ^t2−δ^t2++xtatd−x^tatd+χt2−χt2+xtdta−x^tdta+θt2−θt2+xtdta−x˜tatd+ξt2−ξt2+xtec−x^tec+κt2−κt2+xted−x^ted+ψt2−ψt2.Subject to (4)–(11) and (19)–(25).The augmented Lagrangian is optimized over two groups of primal variablesX = : {ptDG, ptUG, pi,tev, pi,thp, ptec, pted, p˜tatd, p^tdta, p˜tdta, p^tatd, x^tatd, x˜tatd, x^tdta, x^tec, x^ted} and Y = : {ptdta, ptatd, xtatd, xtdta, xtec, xted} and a group of dual variables Λ.


## 3.3. P-ADMM-Based Algorithm

For the ease of presentation, the objective function of the augmented Lagrangian is represented using a compact formL (X, Y, Λ). The 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 dual-variable update procedure are carried out iteratively.Algorithm 1: Pseudo-code of the P-ADMM algorithm.
P-ADMM algorithm(1) input: system and DER parameters(2) output: AC and DC energy schedules and BC operation status(3) while:Xex,∗,k−Y∗,k2≤σ1,Xex,∗,k−Xex,∗,k−12≤σ2doSolve subproblem I to update primal variablesXkSolve subproblem I to update primal variablesYkUpdate dual variablesΛk+1using (30)end


## 3.4. Subproblem I

In thekth iteration, primal variables Xk are optimized in subproblem I, given by (26), subject to (4)–(11) and (19)–(25), with parameters Y∗,k−1 and Λ∗k−1 obtained in the (k− 1)th iteration.(27)Subproblem⋅I:⋅LXk,Y∗,k−1,Λ∗,k−1,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. Therefore, subproblem I can be solved in a distributed manner between AC and DC subgrids.


## 3.5. Subproblem II

The primal variablesYk 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.(28)Subproblem⋅II:⋅LX∗,k,Yk,Λ∗,k−1.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 Px denotes the projection function, which rounds each entry to its nearest binary value. The remaining primal variables are optimized by solving the QP problem after excluding binary variables, which can be fully decoupled between AC and DC subgrids. Therefore, subproblem II can be solved in a distributed manner as well.(29)xtatd=Pxx^tatd−χt,xtdta=Pxx^tdta−θt+x˜tatd−ξt2.


## 3.6. Update of Dual Variables

After solving subproblems I and II, dual variables are updated in (29).(30)Λk=Λ∗,k−1+Xex,∗,k−1−Y∗,k,Xex=:p˜tatd,p^tdta,p˜tdta,p^tatd,x^tatd,x˜tatd,x^tdta.


## 3.7. Stop Criteria

The iterative procedure continues till primal and dual residuals are lower than specified thresholds, respectively, as follows:(31)Xex,∗,k−Y∗,k2≤σ1,Xex,∗,k−Xex,∗,k−12≤σ2.


## 4. Case Studies

A case study was conducted on the hybrid AC/DC system shown in Figure1 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. The key parameters of DGs, EVs, HPs, and ESSs are given in Table 1. The 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 σ1 and σ2 are set as 0.01, and penalty parameter ρ is set as 50.Figure 2
Forecasted spot prices.Figure 3
Forecasted PV active power generation.Table 1
Key parameters used in simulations.
ESS parametersESS battery sizeEtes,max200 kWhMin/max charging power limitpiec,min/piec,max0/50 kWMin/max discharging power limitpied,min/pied,max0/50 kWCharing/discharging coefficientsηc/ηd0.90/0.90HP parametersCOP of HPcicop2.3Min/max temperature of the houseKi,th,min/Ki,th,min20/24oCMin/max HP power consumptionpi,thp,min/pi,thp,max0/6 kWEV parametersMin/max HP power consumptionpi,tev,min/pi,tev,max0/11 kWEV availabilityai,t1DG parametersMin/max DG active power generationptDG,min/ptDG,max0/200 kWDown- and upramping limitsptDG,dr/ptDG,ur80 kWDG energy costsc1G/c2G0.001/0.15BC parametersMax. limit of active power convertedptbc,max1000 kWPower transfer coefficientsηdta/ηdta0.9/0.9


### 4.1. 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. Table2 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. The 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.Table 2
Numbers of iterations with different penalty parameter values.
ρ1020304050100150Num. of iterations633434—17591245404Figure 4
Convergence process of the P-ADMM-based algorithm (ρ = 50).The distributed solutionSd is compared with the centralized solution Sc in Table 3(take hour t12 as an example). The 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. The 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.(32)RE=Sc−SdSc.Table 3
Comparison between centralized and distributed solutions (take hourt12 as an example).
kWptDG∑pi,tev∑pi,thpptecptedptatdptdtaCentralized139.9219.2559.546.2000Distributed139.9218.4559.646.8000


### 4.2. Energy Management Results

Figures5 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 t1–t9 and t19–t24, AC power is converted into DC power to support EV and HP power consumption. At hours t10–t16, since there is sufficient PV active power, there is no need to convert AC power into DC power. Especially at hours t14 and t16, 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.Figure 5
Day-ahead scheduling in the DC grid.Figure 6
Day-ahead scheduling in the AC grid.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 tendency as compared with the forecasted spot price profile in order to minimize energy costs.


## 4.1. 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. Table2 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. The 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.Table 2
Numbers of iterations with different penalty parameter values.
ρ1020304050100150Num. of iterations633434—17591245404Figure 4
Convergence process of the P-ADMM-based algorithm (ρ = 50).The distributed solutionSd is compared with the centralized solution Sc in Table 3(take hour t12 as an example). The 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. The 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.(32)RE=Sc−SdSc.Table 3
Comparison between centralized and distributed solutions (take hourt12 as an example).
kWptDG∑pi,tev∑pi,thpptecptedptatdptdtaCentralized139.9219.2559.546.2000Distributed139.9218.4559.646.8000


## 4.2. Energy Management Results

Figures5 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 t1–t9 and t19–t24, AC power is converted into DC power to support EV and HP power consumption. At hours t10–t16, since there is sufficient PV active power, there is no need to convert AC power into DC power. Especially at hours t14 and t16, 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.Figure 5
Day-ahead scheduling in the DC grid.Figure 6
Day-ahead scheduling in the AC grid.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 tendency as compared with the forecasted spot price profile in order to minimize energy costs.


## 5. 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. The energy management problem of the hybrid microgrid is formulated as a MIQP model, considering DER operation constraints, system operation constraints, and converter operation constraints. Then, the MIQP model is decomposed and distributed into smaller-scale submodels between subgrids using the P-ADMM-based algorithm. The numerical results demonstrate that the proposed algorithm can efficiently solve the MIQP model in a distributed manner and can deal with binary variables. The proposed distributed scheme allows each subgrid operator to make day-ahead schedules independently with information exchanges while acquiring the optimal energy management solution.



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