Adaptive Consensus Algorithm for Distributed Heat-Electricity Energy

energies Article

Adaptive Consensus Algorithm for Distributed Heat-Electricity Energy Management of an Islanded Microgrid Xiaofeng Dong 1,3 , Xiaoshun Zhang 2, * and Tong Jiang 1 1 2 3

*

Department of Electrical and Electronic Engineering, North China Electric Power University, Beijing 100000, China; [email protected] (X.D.); [email protected] (T.J.) College of Engineering, Shantou University, Shantou 515000, China State Grid Suzhou Power Supply Company, Suzhou 215004, China Correspondence: [email protected]; Tel.: +86-150-1752-7246

Received: 10 July 2018; Accepted: 19 August 2018; Published: 26 August 2018







Abstract: This paper proposes a novel adaptive consensus algorithm (ACA) for distributed heat-electricity energy management (HEEM) of an islanded microgrid. In order to simultaneously satisfy the heat-electricity energy balance constraints, ACA is implemented with a switch between unified consensus and independent consensus according to the dynamic energy mismatches. The feasible operation region of a combined heat and power (CHP) unit is decomposed into eight searching sub-regions, thus its electricity and heat energy outputs can simultaneously match the incremental cost consensus requirement and the heat-electricity energy balance constraints. Case studies are thoroughly carried out to verify the performance of ACA for distributed HEEM of an islanded microgrid. Keywords: adaptive consensus algorithm; distributed heat-electricity energy management; eight searching sub-regions; islanded microgrid

1. Introduction Over the past decades, microgrids have attracted extensive attention and study as they provide an efficient and flexible way to integrate various distributed energy resources (DERs), local loads, and energy storage devices [1]. In general, a microgrid is a local energy grid which can be operated in either grid-connected or islanded modes [2]. When a microgrid is islanded, it needs to achieve an energy balance between the energy supply and the demand without the adequate power supply from the main grid [3]. In order to handle this issue, the economic dispatch (ED) is usually employed to minimize the total operating cost while satisfying various operating constraints (e.g., energy balance constraints) [4]. So far, ED of an islanded microgrid can be implemented with two frameworks, including the centralized and distributed frameworks. Under the first framework, the energy management system (EMS) needs to collect the operating parameters of all the energy suppliers and consumers [5], then an optimal dispatch scheme can be determined by a centralized optimization method. As a result, it will inevitably result in three critical problems:

• • •

Communication bottleneck [6] due to the great increasing amount of data from the large integration of DERs; Expensive computation [7] for the growing controllable variables and operating constraints from the large integration of DERs; Low individual privacy and security [8].

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Compared EDcan canautomatically automaticallyaddress address above Comparedwith withthe thecentralized centralized ED, ED, a distributed distributed ED allall of of thethe above problems Owing to advantage, this advantage, distributed optimization techniques haveproposed been problems [6].[6]. Owing to this manymany distributed optimization techniques have been for a ED distributed ED in asuch microgrid, such as distributed [9], population game forproposed a distributed in a microgrid, as distributed λ-iteration λ-iteration [9], population game method [10], method [10], and dual based decomposition based(DDO) optimization (DDO)these [11].approaches, Among these approaches, and dual decomposition optimization [11]. Among consensus-based consensus-based algorithms were due to their remarkable self-organizing ability, algorithms were widely studied duewidely to theirstudied remarkable self-organizing ability, significant robustness, significant robustness, and easy scalability [12–16]. In [12], a simple consensus-based optimization and easy scalability [12–16]. In [12], a simple consensus-based optimization was designed for optimal was designed for optimal management anconsidering islanded microgrid. Byrate considering the ramp resource management in anresource islanded microgrid.inBy the ramp limitations, a novel rate limitations, a novel consensus and innovations [13] were presented for a multistep ED a consensus and innovations [13] were presented for a multistep ED in a microgrid. Moreover, ainnovel microgrid. Moreover, a novel consensus algorithm based ED was proposed by taking the impacts of consensus algorithm based ED was proposed by taking the impacts of communication time delays communication time delays into account [15]. However, all of these consensus algorithms did not into account [15]. However, all of these consensus algorithms did not address two important issues: address two important issues: considersthe theelectricity electricityenergy energy dispatch, and •  Multi-energy Multi-energydispatch: dispatch:The Theabove above ED ED only only considers dispatch, and diddid notnot consider the optimal dispatch of other energies, e.g., the heat energy dispatch; consider the optimal dispatch of other energies, e.g., the heat energy dispatch; •  Tight coupling Asthe theparticipation participationofofa acombined combined heat and Tight couplingfeatures featuresamong among various various energies: energies: As heat and power energy outputs outputsare aretightly tightlycoupled coupled because power(CHP) (CHP)unit, unit,the theelectricity electricity and and heat heat energy because of of thethe feasible operation needs to tobe becarefully carefullydesigned designed the distributed feasible operationregion regionconstraint, constraint, which needs inin the distributed ED.ED. Therefore,this thispaper paperproposes proposes aa novel novel adaptive adaptive consensus forfor distributed Therefore, consensusalgorithm algorithm(ACA) (ACA) distributed heat-electricity energy management (HEEM) of an islanded microgrid, which can not only realize heat-electricity energy management (HEEM) of an islanded microgrid, which can not only realize the optimal multi-energy dispatch but also consider the tight coupling features between heat andand the optimal multi-energy dispatch but also consider the tight coupling features between heat electricity energies. electricity energies. The remainder of this paper is organized as follows: Section 2 introduces the mathematical The remainder of this paper is organized as follows: Section 2 introduces the mathematical model model of distributed HEEM, including the objective function, the operation constraints, and a of distributed HEEM, including the objective function, the operation constraints, and a detailed feature detailed feature analysis of the incremental cost. Section 3 presents the optimization principle of analysis of the incremental cost. Section 3 presents the optimization principle of ACA for distributed ACA for distributed HEEM, while the detailed solving process is provided. Case studies on a HEEM, while the detailed solving process is provided. Case studies on a microgrid with ten energy microgrid with ten energy suppliers and seven energy consumers are given in Section 4, in which suppliers and sevenmethods energy consumers are given in Section 4, in which four four optimization are introduced for performance comparison withoptimization ACA. Finally,methods Section 5are introduced for performance comparison with ACA. Finally, Section 5 concludes the paper. concludes the paper.

2. 2. Mathematical MathematicalModel Modelof ofDistributed Distributed HEEM HEEM InIn this study, thethe distributed HEEM aimsaims to minimize the total operating cost ofcost theof entire this study, distributed HEEM to minimize the total operating the islanded entire microgrid while satisfying the heat and electricity energy balance constraints and other operating islanded microgrid while satisfying the heat and electricity energy balance constraints and other constraints, as illustrated Figure 1. that eachthat controllable unit only communicates with the operating constraints, as in illustrated in Note Figure 1. Note each controllable unit only communicates adjacent units during theduring computation of distributed HEEM, which is which the main difference compared with the adjacent units the computation of distributed HEEM, is the main difference with the centralized ED [17]. compared with the centralized ED [17].

Figure 1. Framework of distributed heat-electricity energy management (HEEM) in an islanded Figure 1. Framework of distributed heat-electricity energy management (HEEM) in an islanded microgrid. microgrid.

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2.1. Objective Function The total operating cost f total is equal to the sum of all the energy suppliers and consumers, which can be written as: f total = ∑ f i ( PGi , HGi ) + ∑ f i (∆PDi ), (1) i ∈ ΩG

i ∈ ΩD

where PGi and HGi are the electricity and heat energy outputs of the ith energy supplier, respectively; ∆PDi is electricity energy curtailment of the ith energy consumer which participates in demand response (DR); ΩG and ΩD are the sets of the energy suppliers and consumers, respectively; and fi denotes the operating cost of the ith energy supplier or consumer, which can be calculated as follows [18]:  0, for WT or PV unit     2 ,  α + β P + γ P for diesel generator  i i Gi i Gi 2 f i ( PGi , HGi ) = αi + β i HGi + γi HGi , for heat − only unit ,   2 +  α + β P + γ P  i i Gi i Gi   2 + ξ H P , for CHP unit δi HGi + θi HGi i Gi Gi f i (∆PDi ) =

P0 − a i −1 2 ∆PDi + Di ∆PDi , i ∈ ΩD , bi bi

(2)

(3)

where αi , βi , γi , δi , θ i , and ξ i are the operating cost coefficients of the ith energy supplier; ai and bi are the operating cost coefficients of the ith energy consumer; WT and PV represent the the wind turbine and photovoltaic unit, respectively; and PDi 0 is the current initial electricity energy demand of the ith energy consumer. 2.2. Constraints 2.2.1. Energy Balance Constraints The total energy outputs of all the energy supplier needs to match the total energy demands of all the energy consumers, is as follows: ∆E =

∑

i ∈ ΩG

PGi −

∆H =

∑

i ∈ ΩG

∑



i ∈ ΩD

HGi −

 0 PDi − ∆PDi = 0,

(4)

∑

(5)

i ∈ ΩD

HDi = 0,

where HDi is the heat energy demand of the ith energy consumer; ∆E and ∆H are the electricity energy mismatch and heat energy mismatch, respectively, which will be combined into ACA in the latter section. 2.2.2. Lower and Upper Capability Limits The energy outputs of each energy supplier, and the electricity energy curtailment of each energy consumer should be limited within their lower and upper bounds, as [18,19]:  min ≤ P ≤ Pmax ,  PGi for diesel generator Gi  Gi   H min ≤ H ≤ H max , for heat − only unit Gi Gi Gi , min max  PGi ( HGi ) ≤ PGi ≤ PGi ( HGi ), for CHP unit    min max ( P ), for CHP unit HGi ( PGi ) ≤ HGi ≤ HGi Gi

(6)

0 0 ≤ ∆PDi ≤ ηi PDi , i ∈ ΩD ,

(7)

where PDi min and PDi max are the minimum and maximum electricity energy outputs of the ith energy supplier, respectively; HDi min and HDi max are the minimum and maximum heat energy outputs of the

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Energies 2018,supplier, 11, x FOR respectively; PEER REVIEW ith energy

4 of 17 and η i is the maximum allowable electricity energy curtailment factor of the ith energy consumer. Note WT and and PV PV units units are are operated operated at at their their maximum maximum power power points points under under the the Note that that both both the the WT current weather conditions [18], thus they do not require a consensus interaction with other current weather conditions [18], thus they do not require a consensus interaction with other controllable controllable devices. Besides, it can from be found from Equation (6) that theand lower and limits upper of limits devices. Besides, it can be found Equation (6) that both theboth lower upper the of the electrical energy output of CHP units are determined by different heat energy outputs and electrical energy output of CHP units are determined by different heat energy outputs and vice versa, vice versa, which the of energy outputs of be CHP units by should be enclosed the which indicates that indicates the energythat outputs CHP units should enclosed the boundary curveby ABCD boundary curve operating ABCD (i.e., the feasible (i.e., the feasible region) [19], asoperating shown inregion) Figure [19], 2. as shown in Figure 2.

Figure 2. Feasible operating region of a combined heat and power (CHP) unit.

2.3. Feature Analysis Since only region is considered for for eacheach CHPCHP unit,unit, the only the the strictly strictlyconvex convexfeasible feasibleoperating operating region is considered proposed distributed HEEM is aisstrictly convex optimization with a unique optimum according to the proposed distributed HEEM a strictly convex optimization with a unique optimum according the quadratic objective functions Equations (1)–(3) and the linear constraints Equations (4)–(7). to the quadratic objective functions Equations (1)–(3) and the linear constraints Equations (4)–(7). Hence, a feasible solution which simultaneously simultaneously satisfies satisfies all all the constraints constraints can can be regarded as the global optimum optimumofofdistributed distributed HEEM allenergy the energy suppliers and consumers reach a HEEM if allifthe suppliers and consumers can reachcan a consensus consensus on the incremental cost, as [20]: on the incremental cost, as [20]: ∂f1 ( PG1 ) ∂fi ( PGi ) ∂fi ( H Gi ) ∂fn ( ΔPDn ) , Dn ) =) = Gi )= (8) ∂ f i ( PGi ∂ fi ( H ∂ f= ∂ f 1 ( PG1 ) =  = n (λ∆P = = ·∂Δ · ·P= = λ, (8) ∂PG1 = · · · =∂PGi ∂H Gi Dn ∂PG1 ∂PGi ∂HGi ∂∆PDn where n is the number of controllable devices; and λ is the incremental cost. where n is that the number of controllable devices; and λ is incremental cost. Note such a consensus condition Equation (8)the will not hold for a constrained optimization Note that such a consensus condition Equation (8) will not hold for a optimization problem, as well as for distributed HEEM. In order to approximate the constrained global optimum, all the problem, well as for distributed orderthe to consensus approximate the global optimum, all the constraintsasEquations (4)–(7) must beHEEM. satisfiedInwhile condition should be satisfied as constraints Equations much as possible [20]. (4)–(7) must be satisfied while the consensus condition should be satisfied as muchAccording as possible to[20]. Equation (8), the incremental cost of each agent can be calculated as follows: According to Equation cost of each agent can be calculated as follows: λiE (8), P incremental generator = 2γthe + β i , for diesel i Gi , (9)  H( λi = 2Eγ i H Gi + β i , for heat-only unit λi = 2γi PGi + β i , for diesel generator , (9) λiE = 2λγHi PG=i +2γ +β , ξ iiH HGGi for heat − only unit i + iβ i , i , for CHP unit (10)  H λi (= 2θ i H Gi + ξ i PGi + δ i , λEi = + βi , P 0ξ i−HaGi −22γi PGi + for CHP unit, (10) E i λi λH = = 2θ ΔPiDHi Gi + +Di ξ i PGi ，+i ∈ (11) δi ,Ω D , i b bi i 0 −a incremental costs of the ith energy supplier or where λiE and λiH denote the electricity −2and heatPDi i λEi = ∆PDi + , i ∈ ΩD , (11) consumer, respectively. bi bi Since all the operating cost coefficients except bi are the positive constants during each dispatch where λi E and λi H denote the electricity and heat incremental costs of the ith energy supplier or time interval, the incremental costs of the diesel generator, heat-only unit, and energy consumer, consumer, respectively. increase linearly with the electricity energy output, heat energy output, and electricity energy curtailment, respectively, as shown in Figure 3. In contrast, the electricity and heat incremental costs of the CHP unit are determined by both the electrical and heat energy outputs, as illustrated in Figure 4. For example, both the electrical and heat incremental costs will increase when the CHP unit is operated from the current operating point to the green region.

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Since all the operating cost coefficients except bi are the positive constants during each dispatch time interval, the incremental costs of the diesel generator, heat-only unit, and energy consumer, increase linearly with the electricity energy output, heat energy output, and electricity energy curtailment, respectively, as shown in Figure 3. In contrast, the electricity and heat incremental costs of the CHP unit are determined by both the electrical and heat energy outputs, as illustrated in Figure 4. For example, both the electrical and heat incremental costs will increase when the CHP unit Energies 2018, 11, x FOR REVIEW 5 of 17 is operated thePEER current operating point to the green region. Energies 2018, from 11, x FOR PEER REVIEW 5 of 17

((aa))

((bb))

((cc))

Figure 3. cost of various units. cost of diesel generator; (b) heat Figure3. 3.Incremental Incrementalcost costof ofvarious variousunits. units.(a) (a)Electrical Electricalincremental incrementalcost costof ofdiesel dieselgenerator; generator;(b) (b)heat heat Figure Incremental (a) Electrical incremental incremental cost of heat-only unit; (c) electrical incremental cost of energy consumer. incremental incrementalcost costof ofheat-only heat-onlyunit; unit;(c) (c)electrical electricalincremental incrementalcost costofofenergy energyconsumer. consumer.

λiEE − β i λi − β i 2γ i 2γ i H λi H − δ i λi − δ i ξi ξi

λiEE↑ λi λ↑iH ↓

λiH ↓

λiEE = 2γ i PGi + ξi H Gi + β i λi = 2γ i PGi + ξi H Gi + β i

λiEE ↑ λiHH ↑ λi ↑ λi ↑

λiEE ↓ λiHH ↓ λi ↓ λi ↓

λiEE↓ λi λ↓H ↑ i H

λi ↑

λ = 2θi H Gi + ξi PGi + δ i λ = 2θi H Gi + ξi PGi + δ i H i H i

Figure Figure 4. Illustration of incremental costs of CHP. Figure4. 4.Illustration Illustrationof ofincremental incrementalcosts costsof ofaaaCHP. CHP.

3. Distributed HEEM 3.Design Designof ofACA ACAfor forDistributed DistributedHEEM HEEM 3. Design of ACA for

3.1. Network 3.1. Graph Theory ofofInteraction Interaction 3.1.Graph GraphTheory Theory of InteractionNetwork Network The be typically built with aadirected graph GG== The interaction amongdifferent differentagents agentscan can typically built with a directed graph Theinteraction interactionnetwork networkamong among different agents can bebe typically built with directed graph ⊆ (V, where VV =V , v2, …, the set of edges G = E, (V, E, A), where = , . .v.vnn,}}visis nodes(agents); (agents);EEE ⊆ ⊆V V denotes the the (V, E, A), A), where = {v {v1{v 1, 1v,2,v2…, thethe setset ofofnodes nodes (agents); V ×× × VV denotes the edges edges n } is n ×× n is ij] ∈ Rn a weighted adjacency matrix [21]. Based on these basic elements, (interactions); and A = [a n (interactions); weightedadjacency adjacencymatrix matrix [21]. [21]. Based Based on on these basic elements, elements, (interactions);and andAA==[a[aijij]]∈∈RRn × n isisaaweighted n and row stochastic matrix D = [dij] ∈ Rn ×nn × ∈∈R RRnnn×× graph GG can be the Laplacian matrix × nn the androw rowstochastic stochasticmatrix matrixD D==[d [dijij] ]∈∈RRn × n of of the graph G can and ofn the the graph can be theLaplacian Laplacianmatrix matrixLLL===[l[l[lijijij]]]∈ determined as be determined as follows: determined asfollows: follows:  llij == i ≠−j aij , ∀i 6= j −aijl,ij∀=  ij −naij , ∀i ≠ j n , (12) , a (12)   nlii = ∑ (12) aaij i=1,i, 6= j ij lliiii ==  i ≠ j ij i = 1,  i = 1, i ≠ j n n lij , i = 1, 2, . . . , n =n llij, / i∑ (13) ddij [[kk]]=d=ijllij[k]  ==1, (13) ij , l i 1,2,..., 2,...,nn (13)  ij ij ij j = 1 j =1 j =1

where isisthe the discrete time index. where wherekkkis thediscrete discretetime timeindex. index. In this paper, the weighted In this paper, the weighted adjacency adjacency matrix matrix isis set set to to be be aa simple simple (0, (0, 1)-matrix, 1)-matrix, thereby thereby aaijij == 11 ifif ij = 0. the ith agent and the jth agent communicate with each other, otherwise a the ith agent and the jth agent communicate with each other, otherwise aij = 0. 3.2. 3.2.Adaptive AdaptiveConsensus ConsensusAlgorithm Algorithm The basic principle The basic principle of of ACA ACA isis that that each each agent agent aims aims to to reach reach aa consensus consensus on on aa specific specific state state with with the adjacent agents by regulating its own state based on the current states from the adjacent agents. the adjacent agents by regulating its own state based on the current states from the adjacent agents.

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In this paper, the weighted adjacency matrix is set to be a simple (0, 1)-matrix, thereby aij = 1 if the ith agent and the jth agent communicate with each other, otherwise aij = 0. 3.2. Adaptive Consensus Algorithm The basic principle of ACA is that each agent aims to reach a consensus on a specific state with the adjacent agents by regulating its own state based on the current states from the adjacent agents. This process can be described by the first-order consensus, as [20]: n

x i [ k + 1] =

∑ dij [k]x j [k],

(14)

j =1

where xi is the state of the ith agent, which refers to the incremental cost of each agent for distributed HEEM on the basis of Equation (8). In this study, each agent will transmit its own energy output or demand to the microgrid EMS at each iteration, then EMS will update ∆E and ∆H, and send them to each agent. In order to satisfy the energy balance constraints Equations (4) and (5), these two mismatches need to be fully considered in the consensus interaction among the agents, which can be achieved as follows:

•

Unified consensus: If the signs of ∆E and ∆H are consistent, i.e., ∆E∆H ≥ 0, then all the agents can update their incremental cost state in a unified interaction network, as

λ i [ k + 1] =

 n    ∑ dij [k]λ j [k] − µ∆E,

i ∈ ΩE

   ∑ dij [k]λ j [k] − µ∆H,

i ∈ ΩH

j =1 n j =1

•

,

(15)

Independent consensus: If the signs of ∆E and ∆H are inconsistent, i.e., ∆E·∆H < 0, then the electricity agents and heat agents need to be separated to update their incremental cost state in two independent interaction networks, as:  E E E   λi [k + 1] = ∑ dij [k]λ j [k ] − µ∆E,

i ∈ ΩE

H H H   λi [k + 1] = ∑ dij [k]λ j [k ] − µ∆H,

i ∈ ΩH

j ∈ ΩE

j ∈ ΩH

,

(16)

where ΩE and ΩH represent the sets of electricity agents and heat agents, respectively; dij E is the (i, j) entry of the row stochastic matrix of the interaction network among the electricity agents; dij H is the (i, j) entry of the row stochastic matrix of the interaction network among the heat agents; and µ denotes the adjustment factor of energy mismatch, µ > 0. Therefore, each agent will regulate its incremental cost between these two consensus modes according to the sign of (∆E·∆H), as illustrated in Figure 5. After a series of consensus interactions by Equations (15) and (16), the energy balance constraints Equations (4) and (5) can be satisfied since both the electricity energy mismatch ∆E and heat energy mismatch ∆H will be sufficiently small. It is important that each interaction network should be strongly connected, i.e., any vertex can be realized from any other vertex by a directed path, thereby the consensus convergence can be guaranteed.

according to the sign of (ΔE·ΔH), as illustrated in Figure 5. After a series of consensus interactions by Equations (15) and (16), the energy balance constraints Equations (4) and (5) can be satisfied since both the electricity energy mismatch ΔE and heat energy mismatch ΔH will be sufficiently small. It is important that each interaction network should be strongly connected, i.e., any vertex can be realized from any other vertex by a directed path, thereby the consensus convergence can Energies 2018, 11, 2236 7 ofbe 17 guaranteed.

Figure 5. Principle of adaptive consensus algorithm. Figure 5. Principle of adaptive consensus algorithm.

3.3. Constraints Handling 3.3. Constraints Handling Owing to the lower and upper capability limits Equations (6) and (7), all the agents may not Owing to the lower and upper capability limits Equations (6) and (7), all the agents may not reach a consensus on the incremental cost. Hence, a virtual incremental cost [17] is designed in ACA, reach a consensus on the incremental cost. Hence, a virtual incremental cost [17] is designed in ACA, which corresponds to the actual incremental cost of each agent. Note that each agent is responsible which corresponds to the actual incremental cost of each agent. Note that each agent is responsible for computing its own incremental cost. More specifically, each agent can update its virtual incremental cost via a consensus interaction with adjacent agents by Equations (15) and (16), which is not limited by the constraints Equations (6) and (7). After updating the virtual incremental cost at each iteration, each agent can calculate its controllable variable by fully considering the constraints, while the actual incremental cost can be determined by Equations (9)–(11). Hence, all the constraints of distributed HEEM can be satisfied, while all the agents can reach a consensus on the incremental cost as much as possible. 1.

Diesel generator: The electrical energy output can be modified as follows:   c PGi = λEi − β i /2γi ,

PGi

2.

 min c min   PGi , if PGi < PGi c , if Pmin ≤ Pc ≤ Pmax , = PGi Gi Gi Gi   Pmax , if Pc > Pmax Gi Gi Gi

(18)

where PGi c is the consensus value of the electrical energy output of the ith energy supplier. Heat-only unit: The heat energy output can be modified as follows:   c HGi = λH i − β i /2γi

HGi

3.

(17)

 min c min   HGi , if HGi < HGi c min c ≤ H max , = HGi , if HGi ≤ HGi Gi   H max , if H c > H max Gi Gi Gi

(19)

(20)

where HGi c is the consensus value of the heat energy output of the ith energy supplier. Energy consumer: The electricity energy curtailment can be modified as follows:   c 0 ∆PDi = PDi − ai − bi λEi /2,

(21)

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∆PDi

 c η P0 i i Di Di Di

(22)

where ∆PDi c is the consensus value of the electrical energy curtailment of the ith energy consumer. 4. CHP unit: Since the electrical and heat energy outputs are highly coupled, the incremental cost should be controlled to meet the energy balance constraints and the feasible operating region constraint. Hence, the feasible operating region is decomposed into eight searching sub-regions, See Figure 6, allowing the CHP unit to adjust its energy outputs based on the current energy Energies 2018, 11, x FOR PEER REVIEW 8 of 17 mismatches and the consensus value of incremental costs, as given in Table 1. PG (MW) A

8

7

B

1

6 2 5 D

4

3

λiE = 2γ i PGi + ξi H Gi + β i

H C λi = 2θi H Gi + ξi PGi + δ i

HG (MW) Eight searching searching sub-regions of the CHP unit. Figure 6. Eight

Table Table 1. Adjusting rules of energy outputs of CHP unit. ΔE > 0 ∆E > 0 True True True True True True True True True True True True True True True True False False False False False False False False False False False False False False False False

∆H > 0 True True True True False False False False True True True True False False False False

ΔH >E0 λiE[k] > λiAE[k −H1] λiH[k] > λiAH[k−1] λi [k] > λi AH [k−1] λi [k] > λi AE [k − 1] True True True True True True True False True False True False False True True True False False False False False True True True True True False False True False False True True False False False False False False False True True True True True False True True True False True False False False True True False True True False True False True False False True True False True False True False False False False False True False False False

(PGi, HGi) (PGi , H Gi ) No adjustment No adjustment No adjustment No adjustment No adjustment No adjustment Sub-region Sub-region #5 #5 Sub-region Sub-region #2 #2 NoNo adjustment adjustment Sub-region #3 #3 Sub-region Sub-region #4 Sub-region #4 Sub-region #8 Sub-region Sub-region #7 #8 No Sub-region adjustment#7 Sub-region #6 No adjustment Sub-region #1 #6 Sub-region No adjustment Sub-region #1 No adjustment adjustment NoNo adjustment No adjustment No adjustment

Note that the CHP unit does not need to adjust its electrical and heat energy outputs if the consensus and energy constraints be satisfied For instance, Note requirement that the CHP unit doesbalance not need to adjustcannot its electrical andsimultaneously. heat energy outputs if the E [k], λ H [k]) are larger than the last actual incremental when both the current virtual incremental costs (λ i i consensus requirement and energy balance constraints cannot be satisfied simultaneously. For costs (λi AEwhen [k − 1], λi AH − 1]), while both the energy costs mismatches positive (∆E >than 0, ∆Hthe > 0), then the instance, both the[kcurrent virtual incremental (λiE[k], are λiH[k]) are larger last actual CHP unit will readjust the energy balances by increasing its incremental costs, thus its electrical and AE AH incremental costs (λi [k − 1], λi [k − 1]), while both the energy mismatches are positive (ΔE > 0, ΔH > heatthen energy will remain unchanged. In addition, the CHP its unitincremental needs to adjust itsthus energy 0), the outputs CHP unit will readjust the energy balances when by increasing costs, its outputs, the and outputs heat energy outputsunchanged. can be updated according to the mismatches, electrical andelectrical heat energy will remain In addition, when theenergy CHP unit needs to as follows: adjust its energy outputs, the electrical and heat energy outputs can be updated according to the

energy mismatches, as follows:  PGi [ k + 1] = PGi [ k ] − μE ΔE ,   H Gi [ k + 1] = H Gi [ k ] − μH ΔH

(23)

where μE and μH denote the adjustment factors of electrical and heat energy outputs, respectively.

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(

PGi [k + 1] = PGi [k ] − µE ∆E , HGi [k + 1] = HGi [k] − µH ∆H

(23)

where µE and µH denote the adjustment factors of electrical and heat energy outputs, respectively. If the operating point of the CHP unit is beyond the corresponding sub-region, then the electrical and heat energy outputs should be modified by the closest point (the point with the shortest Euclidean distance to the updated operating point) within the sub-region. 3.4. Execution Procedure To sum up, the detailed execution procedure of ACA for distributed HEEM of an islanded microgrid is given in Algorithm 1, where τ is the energy mismatch tolerance, which is set to be 0.001 in this paper. Algorithm 1. ACA for distributed HEEM. 1: 2: 3: 4: 5: 6: 7:

Initial the algorithm parameters; Design the interaction network among different agents; Input the operating data of the current optimization task; Calculate the electricity and heat energy mismatches by Equations (4) and (5); While |∆E| > τ or |∆H| > τ If ∆E·∆H ≥ 0 then Update the virtual incremental cost of each agent by unified consensus Equation (15); 8: Else 9: Update the virtual incremental cost of each agent by independent consensus Equation (16); 10: End If 11: Calculate the electricity energy output of each diesel generator by Equations (17) and (18); 12: Calculate the heat energy output of each heat-only unit by Equations (19) and (20); 13: Calculate the electricity energy curtailment of each energy consumer by Equations (21) and (22); 14: Modify the energy outputs of each CHP unit based on the adjusting rule in Table 1 and the eight searching sub-regions in Figure 6; 15: Calculate the electricity and heat energy mismatches by Equations (4) and (5); 16: Set k: = k + 1; 17: End While Output the optimal energy dispatch strategy of each agent.

4. Case Studies 4.1. Simulation Model In order to test the multi-energy dispatch, the islanded microgrid [18] with three PV units, two WTs, two diesel generators, one heat-only unit, two CHP units, and seven controllable energy consumers, is used for the simulation. Hence, both the electrical and heat parts are simultaneously considered in simulation, where the detailed mathematical model of distributed HEEM can be constructed by acquiring the operating constraints and operating cost function of energy for each supplier or consumer. Furthermore, the operating cost coefficients are given in Table 2; the physical topology is provided in Figure 7, and the interaction network among them is illustrated in Figure 8. In addition, three operating scenarios (i.e., scenarios #1 to #3) with different energy outputs of renewables (i.e., 0.8, 0.6, and 1 MW), instead of a single operating scenario, are designed for evaluating the optimization performance of different algorithms more scientifically. The adjustment factors µ, µE , and µH are set to be 10, 0.1, and 0.1, respectively. The following simulations will be carried out in

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Energies 2018, 11, x FOR PEER REVIEW Matlab R2016a a PEER personal computer Energies 2018, 11, xby FOR REVIEW

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with Intel(R) Xeon (R) E5-2670 v3 CPU at 2.3 GHz with 10 64ofGB 17

of RAM. Table 2. Operating cost coefficients of controllable controllable units. Table Table 2. 2. Operating Operating cost cost coefficients coefficients of of controllableunits. units.

Type No. Type Type No. G1 G1 Diesel generator Diesel Diesel generator 2 generatorG G2 Heat-only unit G3 Heat-only unit unit G3 Heat-only G4 G4 CHP unit CHP unit G 5 CHP unit G5 L1 L1 L2 L2 L3 L3 Energy L4 Energy consumer L4 Energy consumer consumer L5 L5 L6 L6 L7 L7

βi βii β 210.36 210.36 210.36 301.4 301.4 301.4 12.3 12.3 12.3 185.7 185.7 185.7 288 288 288 −0.002 −0.002 − 0.002 −0.002 −0.002 − 0.002 −0.001 − 0.001 −0.001 −0.001 − 0.001 −0.001 −0.001 −0.001 −0.001 −0.0035 −0.0035 −0.0035 −0.0035 −0.0035 −0.0035

αi αi 10.193αi 10.193 10.193 2.305 2.305 2.305 33 33 33 339.5 339.5 100339.5 100 100 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

No. G1 G2 G3 G4 G5 L1 L2 L3 L4 L5 L6 L7

G3 G3 H P

WT2 WT2

Disconnecting switch Disconnecting switch

γi δi δi 250.2 δi - θi 250.2 250.2 1100 - 1100 - 1100 6.9 6.9 - 6.944.2 - 53.8 44.2 53.8 44.2 34.5 53.8 21.638.4 34.534.5 21.6 21.621.6 - - - - - - - - -- - - - - - γ i γi

ζi ζi 40 40 8.8 8.8 -

PV3 PV3

H P

Main Main grid grid

Circuit breaker Circuit breaker

θi θζi i --38.438.4 40 21.6 21.6 8.8 --------

L7 L7

L6 L6

PV2 PV2

G4 G4 H P

H P

PV1 PV1

G1 G1 L4 L4

WT1 WT1 G5 G5 L2 L2

G2 G2

L1 L1

L3 L3

Islanded microgrid Islanded microgrid

L5 L5

Figure 7. Physical topology of the testing islanded microgrid. Figure 7. 7. Physical topology of the testing islanded islanded microgrid. microgrid. E H G3 Unified Unified E H G3 consensus 5 consensus G G5

L6 L6 L7 L7 G1 G1

L2 L2 L1 L1

L5 L5 L3 L3

(a) (a)

Independent Independent consensus consensus L6 L6

E H G3 Independent Independent E H G3 consensus 5 consensus G G5 L7 L7

E H G4 E H G4

G1 G1

G2 G2 L4 L4

L2 L2 L1 L1

L5 L5 L3 L3

E H G4 E H G4 G2 G2 L4 L4

(b) (b)

Figure 8. Interaction network of different agents. (a) Unified consensus; (b) independent consensus. Figure 8. 8. Interaction Interaction network network of of different different agents. agents. (a) (a) Unified Unified consensus; consensus; (b) (b) independent independent consensus. consensus. Figure

4.2. Study of Convergence 4.2. Study of Convergence This case study is executed to reveal the convergence of ACA. Figure 9 shows the convergence This case study is executed to reveal the convergence of ACA. Figure 9 shows the convergence process of ACA for distributed HEEM under scenario #1. It can be found from Figure 9a that the process of ACA for distributed HEEM under scenario #1. It can be found from Figure 9a that the virtual incremental cost of each agent will update between unified consensus mode and virtual incremental cost of each agent will update between unified consensus mode and independent consensus mode according to the dynamic energy mismatches, in which the independent consensus mode according to the dynamic energy mismatches, in which the

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4.2. Study of Convergence This case study is executed to reveal the convergence of ACA. Figure 9 shows the convergence process of ACA for distributed HEEM under scenario #1. It can be found from Figure 9a that the virtual incremental cost of each agent will update between unified consensus mode and independent consensus mode according to the dynamic energy mismatches, in which the incremental11heat costs Energies 2018, 11, x FOR PEER REVIEW of 17 cannot reach a consensus with other incremental electrical costs due to the energy balance constraints incremental heat cannot reachenergy a consensus incremental electrical costs due to theafter a Equations (4) and (5).costs Besides, some agentswith haveother reached their energy capability limits energy balance Equations and (5). Besides, some energy agents haveadjust reached their few interactions, asconstraints shown in Figure 9b. (4) Moreover, two CHP units can adaptively their energy energy capability after arule few in interactions, as shown Figure 9b.the Moreover, two CHPsub-region units outputs based on thelimits adjusting Table 1, see Figurein9c, where zero searching can adaptively adjust their energy outputs based on the adjusting rule in Table 1, see Figure 9c, indicates that the energy outputs of the CHP unit remain unchanged. Finally, both the electrical where the zero searching sub-region indicates that the energy outputs of the CHP unit remain and heat energy mismatches (∆E and ∆H) can simultaneously satisfy the energy mismatch tolerance unchanged. Finally, both the electrical and heat energy mismatches (ΔE and ΔH) can simultaneously aftersatisfy approximately 150 iterations, see Figure 9d, i.e., |∆E| < τ and |∆H|