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Applied Energy

journal homepage: www.elsevier.com/locate/apenergy

A cooperative game approach for coordinating multi-microgrid operationwithin distribution systems

Yan Dua, Zhiwei Wangb, Guangyi Liub, Xi Chenb, Haoyu Yuanc, Yanli Weid, Fangxing Lia,⁎

a Dept. of Electrical Engineering and Computer Science, University of Tennessee, Knoxville, USAbGEIRI North America, San Jose, CA, USAc Peak Reliability, Loveland, CO, USAd Southern California Edison, Rosemead, CA, USA

H I G H L I G H T S

• A coalitional operation model for multiple microgrids to achieve global optimum.

• A cost allocation method from cooperative game theory to achieve local optimum.

• A linearized optimal power flow with voltage constraints to realize cooperation.

• The economy benefits of multi-microgrid cooperation are simulated and analyzed.

A R T I C L E I N F O

Keywords:Benders decompositionCoalitional operationCooperative gameCost allocationlinearized optimal power flow for distribution(LOPF-D)Multi-microgrid

A B S T R A C T

This paper focuses on simulating the potential cooperative behaviors of multiple grid-connected microgrids toachieve higher energy efficiency and operation economy. Motivated by the cooperative game theory, a group ofindividual microgrids is treated as one grand coalition with the aim of minimizing the total operation cost. Next,given that each microgrid operator is an independent and autonomous entity with the aim of maximum self-interest, a cost allocation method based on the concept of core in the cooperative game is implemented to ensurea fair cost share among microgrid coalition members, which guarantees the economic stability of the coalition.Considering the combinatorial explosive characteristic of the cost allocation problem, Benders Decomposition(BD) algorithm is applied to locate the core solution with computational efficiency. In addition, since microgridcoalition is formed at the distribution system level, network losses is not negligible. After considering networklosses, the coalition operation model of multi-microgrid becomes an optimal power flow problem. A linearizedoptimal power flow for distribution (LOPF-D) model is applied instead of the conventional ACOPF model toreduce computation burden, meanwhile maintaining adequate accuracy. Case studies on standard IEEE systemsdemonstrate the advantages of multi-microgrid cooperation and the robustness of the formulated grand coali-tion. In addition, comparisons with the conventional ACOPF model verifies the high performance of the pro-posed LOPF-D model.

1. Introduction

The worldwide energy and environmental crisis has led to the large-scale development of renewable energy sources (RES) and distributedenergy resources (DER), which in return has brought microgrid tech-nology under spotlight in the power industry. A microgrid is a small-scale electric power system which contains distributed resources andload, and can operate in either grid-connected mode or islanded mode[1]. Currently, emerging new types of demand-side resources have beenspotted in microgrids, including electrical vehicles, air conditioning

loads, and refrigerators, which add considerable flexibility to microgridoperation. The advantages of integrating microgrids into distributionsystems are multifold: first, the DER units that reside in a microgrid cansupport the local energy demand, hence reduces its reliance on theupper-level utility grid and enhances the reliability of power supply;second, it follows that microgrid facilitates environmentally-friendlyenergy consumption by utilizing renewable energy-fueled generators,i.e., wind turbines, photovoltaic panels, and fuel cells; last but not least,by supplying the energy demand via local distributed generators (DGs),microgrid can reduce long-distance transmission loss, as well as the

https://doi.org/10.1016/j.apenergy.2018.03.086Received 20 December 2017; Received in revised form 27 February 2018; Accepted 25 March 2018

⁎ Corresponding author.E-mail address: fli6@utk.edu (F. Li).

Applied Energy 222 (2018) 383–395

0306-2619/ © 2018 Elsevier Ltd. All rights reserved.

T

investment on large-scale transformers and transmission lines.Most recently, with the increasing penetration of renewable energy

into power systems, the concept of multi-microgrid (MMG) comes up onthe stage, which refers to a cluster of microgrids connected with eachother in close electrical or spatial distance [2,3]. The aim of MMG is toachieve resilience and stability via fast power exchange and to furtherobtain a high and smooth penetration of DERs into the bulk system.Possible architectures for multi-microgrid regarding layout and inter-faces accompanied by cost and reliability analysis are discussed in [4].To achieve a coordinated penetration of multi-microgrid into the bulkpower system, a hierarchical control strategy is proposed in [5,6],which includes the primary droop-control of power electronic devices,the secondary control for voltage/frequency restoration and synchro-nization, and the tertiary control of real and reactive power. The lastone is in association with microgrid energy management system, andcan be formulated as an economic dispatch problem with the aim ofmaximizing economic profit.

The focus of this paper lies in the tertiary control level of a multi-microgrid system. In retrospect, existing works mainly cover two topicsrelated to this field: planning and operation. In terms of the former, Ref.[7] applies the Decision-Tree (DT) method to plan the capacities ofenergy storage devices within microgrids to realize local power bal-ance; Ref. [8] includes the coupling physical and operational con-straints of electrical and heating/cooling networks for multi-energymicrogrids in the design of the capacities of DER units; Ref. [9] com-bines both DER sizing and placement problems into one mixed integerlinear programming, where microgrid is modeled as a multi-nodesystem instead of an aggregated single-node model to better considerpower flow and heat flow balances.

With regard to operation, existing research works mainly adopt twoapproaches to coordinate MMG economic dispatch: the centralizedapproach and the decentralized approach. The main idea behind thecentralized optimization method is to aggregate all the entities into thesystem as one unity with a collective objective. In the case of multi-microgrid coordination, a central controller is selected (i.e. distributionsystem operator, DSO) to organize the operation of all the DGs andloads regardless of their individual interests. In this aspect, Ref. [10]

establishes a centralized control model of a group of microgrids that canexchange power with their neighbors, where the objective is to max-imize the total profit of all microgrid operators. Simulation results in-dicate that local energy exchange improves individual operationeconomy by making full use of the zero-cost renewable energy. In [11],the interactions between the upper-level distribution system and themulti-microgrid system are further considered, and the DSO is includedas an additional independent entity in MMG coordination. To decreasemodel complexity and improve computational efficiency, decentralizeddispatch methods have been applied in [12,13], where the global op-timization model is decomposed into several independent sub-problemsusing Lagrange relaxation method and solved by local entities. Modelpredictive control (MPC) scheme is implemented in [14,15] in a dis-tributed manner to address the uncertainties of load and renewableenergy within the microgrids and to maintain a steady power exchangewith the rest of the distribution system. The authors in [16,17] explorethe optimal risk-constrained bidding strategies of microgrids for pro-viding ancillary service to the utility grids using decentralized andcentralized approaches, respectively.

There exist some challenges with the above conventional models[18]: in the centralized method, since it requires full communicationamong all entities within the entire network, it is not scalable, espe-cially not suitable for plug-and-play DERs like electrical vehicles; in thedecentralized method, since local entities independently work on theirown optimal dispatch schedule without the information from otherentities, this complete isolation from the rest of the system usuallycannot reach global optimum. In summary, the centralized method hasa simple implementation to realize global optimum, while the decen-tralized method focuses on local optimum. Nevertheless, there remainssome gap between the two goals, which may sabotage the coordinatedoperation of the multi-microgrid. The reason is that each microgrid is ahighly independent and profit-driven entity with the goal of max-imizing its self-interest. Thus conflicts of interests between the localmicrogrid (local optimum) and the system operator (global optimum)may drive microgrid away from coordination.

The motivation of this work is to address the above mentionedconcerns between global optimum and local optimum. In this paper, we

Nomenclature

πP(t), πQ(t) active/reactive power exchange price between dis-tribution system and transmission system at interval t,in $/MWh

Pgrid (t) active power exchange (power flow at PCC) at timeinterval t, in MW

Qgrid (t) reactive power exchange (power flow at PCC) at timeinterval t, in MW

PMT,m(t) micro turbine generation of the mth microgrid at timeinterval t, in kW

Cgas fuel cost of micro turbine, in $/kWhηMT efficiency of micro turbineP P,MT

minMTmax lower and upper bound of micro turbine, in kW

P P,bmin

bmax lower and upper bound of boiler, in kW

νh heat to electricity ratio of micro turbineSth,m(t) energy level of thermal energy storage in mth microgrid

at time interval t, in kWhS t( )th m

cap, capacity of thermal energy storage in mth microgrid at

time interval t, in kWhS t( )th m

min, minimum energy level of thermal energy storage in mth

microgrid at time interval t, in kWhPth,m (t) charge/discharge rate of thermal energy storage in mth

microgrid at time interval t, in kWηh charge/discharge efficiency of thermal energy storagePb,m(t) boiler generation of the mth microgrid at time interval t,

in kW

P t( )mD electrical load of the mth microgrid at time interval t, in

kWp t( )Load m

h, thermal load of the mth microgrid at time interval t, in

kWPPV,m(t) PV generation of the mth microgrid at time interval t, in

kWPWT,m(t) wind turbine generation of the mth microgrid at time

interval t, in kWPsolar,m(t) thermal solar energy of the mth microgrid at time in-

terval t, in kWP t Q t( ), ( )i

DiD active/reactive load at bus i in distribution system at

time interval tP t Q t( ), ( )i

GiG active/reactive generation at bus i in distribution

system at time interval trk(rij), xk(xij) the resistance and reactance of the kth line in the dis-

tribution systemP t Q t( ), ( )L Lk k active and reactive power flow on the kth line in the

distribution system at time interval tP t Q t( ), ( )Loss

LLossLk k active and reactive line loss on the kth line in the

distribution system at time interval t∂ ∂ ∂ ∂P P P Q/ , /Loss

LiG

LossL

iGk k active loss factor of the kth line to the ith bus

generation∂ ∂ ∂ ∂Q P Q Q/ , /Loss

LiG

LossL

iGk k rective loss factor of the kth line to the ith bus

loadVi(t) voltage magnitude of bus i at time interval t, in p.u.V V,i

minimax lower and upper voltage level of bus i, in p.u.

Y. Du et al. Applied Energy 222 (2018) 383–395

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propose a cooperative game approach to implement multi-microgridcoordinated operation. Non-cooperative game and cooperative gameare the two fundamental pillars of game theory. Both games intend toreach a globally balanced status where no player can get any furtherimprovement of their interests, which is referred to as a NashEquilibrium (NE) in the former, and a core status in the latter. However,the difference between the two lies in that the non-cooperative gamefocuses on obtaining the maximum individual payoff for each singleplayer without evaluating the global welfare. While in the cooperativegame, a coalitional optimization model is first developed to reach theglobal optimum, then a cost or profit allocation model is established tofairly distribute the collective benefits among all the players to guar-antee local optimum. Hence, it can be safely concluded that the co-operative game approach is a natural fit for multi-microgrid coopera-tion problem to mitigate the potential interest collisions between globaland local stake holders.

In retrospect, considerable efforts have been made in implementingcooperative game into the field of power system, from transmission costallocation [19–21] to revenue distribution among the portfolio ofpower generators [22,23]. The Shapley value from cooperative gametheory is applied in [24] to measure the flexibility of the fast-rampingDERs. However, application of cooperative game in multi-microgridcoordination is still at its initial stage of research, which leaves greatpotentials for further explorations. In this regard, the authors in [25,26]treat each microgrid as an active cooperative player seeking for po-tential coalitions with their neighbors to share power and save trans-mission cost. A merge-and-split algorithm is developed to guide theformation of different coalitions under environmental changes. Theadvantages of direct power exchange among local DERs and consumersare discussed in [27], and the Shapley value is proposed as the optimalcost saving division among the players, which belongs to the set of coresolution. A cooperative generation planning model for interconnectedmicrogrids is proposed in [28,29], in which both the long-term in-vestment cost and short-term operational cost are included in the faircost distribution model. A Nash bargaining solution is implemented asthe optimal cost share scheme. An interactive energy game matrix(IEGM) is developed in [30] to describe the available capacity reservethat each microgrid can provide to their neighbors and the cooperationlevel.

In this paper, we also apply the cooperative game theory to realize acoordinated operation of multiple grid-connected microgrids at thedistribution system level. The present work is an extension from thediscussion in our previous work [31]. The problem under discussionincludes two major parts: the first one is to achieve the group

rationality by conducting a coalitional economic dispatch for multi-microgrid system with the aim of minimizing the total operation cost;the second part is to fairly distribute the cost among group members,under the criterion that each single player, as well as each subset of theplayers, is able to receive some cost savings from the multi-microgridcooperation, which is in essence the local optimal solution. Therefore,they are more willing to cooperate with their neighbors, such that theeconomic stability of multi-microgrid cooperation can be enhanced.

In addition to the application of cooperative game approach to themulti-microgrid coordination, another highlight of this work is that alinearized optimal power flow for distribution (LOPF-D) model is im-plemented to adequately model the network losses, since multi-micro-grid system is connected to the bulk system at the distribution level,where the distribution network losses are considerable as much as5–12%. With a large-scale microgrid penetration into the power system,the conventional ACOPF model becomes highly computationally in-tensive. We validate that LOPF-D can be a substitute of ACOPF withadequate accuracy and much higher computation efficiency, whichpaves the way for potential real-world application of the proposedmulti-microgrid cooperation.

The rest of the paper is organized as follows: Section 2 illustrates themulti-microgrid coalitional operation model that aims at maximizingglobal economy; Section 3 presents the LOPF-D model with multi-mi-crogrid penetration; Section 4 gives the mathematical description of thefair cost allocation among microgrids based on the core concept incooperative game theory and the associated decomposition method toobtain the optimal solution; Section 5 presents the case studies on IEEEstandard test cases; and finally, conclusions are given in Section 6.

2. Coalitional operation model of multi-microgrid system

2.1. A brief introduction of cooperative game theory

Cooperative game, or coalitional game, is the study concerned witha group of rational players who coordinate their actions and pool theirwinning, which consequently leads to the problem of how to divide theextra earnings or total cost among the coalition members [32]. A co-operative game consists of two essential elements: (1) a set of players N= {1,2…,i,…,n} and (2) a characteristic function v that specifies thevalue created by different subsets of the players. A coalition c refers to asubset of the players. The grand coalition includes all the players. Anallocation x is a way to distribute the value created by grand coalition,marked as v(1), among all the players. Several other related definitionsare listed as follows:

81 76543 21110192 13 14 15 16 17 18

19 21 22

23

24 25 26 27 28 29 30 31 32 33

20

MG8 MG2

MG3 MG4

MG5

MG6 MG7

MG1

MG9

MG10

PV panel Wind turbine

Micro turbine

Fuel cell Thermal storage

Distribution system(DNO)

TransmissionSystem(ISO)

PCC G2

Boiler

Fig. 1. IEEE 33-bus distribution system with 10 microgrids.

Y. Du et al. Applied Energy 222 (2018) 383–395

385

(1) an allocation (x1, x2,…, xn) is individually rational if xi≥ v(i) for all i;(2) an allocation (x1, x2,…, xn) is efficient if ∑ =i

n1 xi= v(1);

(3) an allocation (x1, x2,…,xn) is coalitionally rational if ∑i∈c xi≥ v(c) forall the subsets c;

(4) an allocation (x1, x2,…,xn) is said to lie in the core of the game if itsatisfies all three conditions above.

From the above definitions, it can be discovered that a core allo-cation ensures that every player in the grand coalition benefits morethan in the case when they each work alone or form coalitions. Thus, noone would be willing to leave the grand coalition and its stability can beensured.

In the case of multi-microgrid coordination, the players are in-dividual microgrid operators, and all players are assumed to auto-matically form a grand coalition with ex-ante binding contracts. Theassociated characteristic function is the total operation cost. The focusof the problem is to find the core allocation of the total operation costsuch that every microgrid operator can receive some cost saving fromcooperation. This section will specify the objective functions and con-straints of the operation model of multi-microgrid grand coalition, andSection 4 will center on tackling the fair cost allocation problem.

2.2. A coalitional operation model of multi-microgrid system

2.2.1. Objective functionA typical network topology of distribution system with 10 micro-

grids integrated is shown in Fig. 1. In this figure, each microgrid con-tains different types of DGs on the generation side, including microturbines (MT), boilers, thermal energy storage (TES), wind turbines andphotovoltaic solar panels, and electrical load and thermal load on thedemand side. All microgrids form one grand coalition based on bindingcontract, and their objective is to minimize the total operation cost:

∑

∑ ∑ ∑

= ⎛

⎝⎜ +

+ ⎛

⎝⎜ + ⎞

⎠⎟

⎞

⎠⎟

=

∈ = =

v c π t P t π t Q t

CP t

ηC

P tη

min ( ) ( ) ( ) ( ) ( )

( ) ( )

t

N

P grid Q grid

m c MT

N

fuelMT m

MT b

N

fuelb m

b

1

1

,

1

,

T

MT b

(1)

Eq. (1) calculates the total operation cost of microgrid coalition c, inwhich the first two terms are the cost for distribution system to ex-change active and reactive of power exchange with the transmissionsystem. The power exchange is evaluated as the power flow at the pointof common coupling (PCC) shown in Fig. 1. NT is the length of opera-tion cycle, in this study it is set to 24 h. The following two terms are thegeneration cost of micro turbines and boilers that reside in microgrid m.

2.2.2. ConstraintsOperation of microgrid coalition c should obey the following con-

straints to ensure system-wide economy and stability:

(1) The scale of microgrid coalition:

< ⩽c10 MT (2)

In Eq. (2), M is the total number of microgrids, c is an M×1 vectorcomposed of 0–1 binary indices [c1,c2,…,cm,…,cM]T, where cm indicateswhether the mth microgrid belongs to the coalition c or not. It is obviousthat in the grand coalition case, we have 1Tc=M.

(2) Capacity of DGs:

⩽ ⩽c P P t c P( )m MT MT m m MTmin

,max (3)

⩽ ⩽c P P t c P( )m b b m m bmin

,max (4)

Eqs. (3) and (4) are the capacity constraints of DGs, where P MT b{ , }min

and P MT b{ , }max are the lower bound and upper bound, respectively, of the

generator output. If the mth microgrid and its DGs does not belong tocoalition c, cm is 0, and P{MT,b},m(t) equals 0. Otherwise we have

⩽ ⩽P P t P( )MT b MT b m MT b{ , }min

{ , }, { , }max .

(3) Charge/discharge constraints of thermal energy storage:

= − −S t S t tP t η( ) ( 1) Δ ( )/th m th m th m th, , , (5)

⩽ ⩽S S t S( )th th m thmin

,max (6)

⩽ ⩾ −P t c P P t c P( ) , ( )th m m thd

th m m thc

,max

,max (7)

⩾S N S( ) (0)th m T th m, , (8)

Eq. (5) is the inter-temporal constraint of energy level in the thermalstorage. For simplicity, we assume that the charging and dischargingefficiency of the storage are the same. Eq. (6) implies that the energylevel of the thermal storage should be within a certain range. Eq. (7)indicates the maximum charge and discharge rate. Similar to Eqs. (3)and (4), 0–1 binary variable cm indicates whether the mth microgridoperator and its energy storage belongs to the microgrid coalition c ornot. Eq. (8) requires that the energy level at the end of the operationcycle should be no lower than its initial value.

(4) Power balance constraint:

∑ ∑ ∑

∑ ∑

+ + +

− − − =

= = =

= =

P t P t c P t c P t

c P t P t P t

( ) ( ) ( ) ( )

( ) ( ) ( ) 0

gridMT

N

MT mMT

N

m PV mMT

N

m WT m

m

M

m mD

i

n

iD

Loss

1,

1,

1,

1 1

MT PV WT

(9)

∑− − ==

Q t Q t Q t( ) ( ) ( ) 0gridi

n

iD

Loss1 (10)

Eqs. (9) and (10) are the active and reactive power balance con-straint of the distribution system, where PLoss(t) and QLoss(t) are thenetwork losses. Since both terms are nonlinear and nonconvex func-tions of the other control variables, i.e. generator output, which addsgreat model complexity. We further apply a linearized optimal powerflow for distribution (LOPF-D) model to overcome this computationdifficulty. More details of the LOPF-D model will be revealed in Section3. In Eq. (10), for simplicity, we assume that the reactive load is sup-ported by power exchange from the transmission system, and localmicrogrids only generate active power.

(5) Bus voltage constraint:

= − + + +

=

V t V t P t P t r Q t Q t x V t for i

n

( ) ( ) (( ( ) ( )) ( ( ) ( )) )/ ( )

2,3,...,

j i L LossL

k L LossL

k 1kk

kk

(11)

=V t( ) 1.05 p.u.1 (12)

⩽ ⩽V V t V( )i i imin max (13)

Eqs. (11)–(13) is based on the line model illustrated in Fig. 2 [33].The line that has a tail bus k+1 is numbered as Lk. For a line thatconnects a head bus i to a tail bus j (k+1), the relationship betweenbus voltage Vi(t) and Vj(t) are shown in Eq. (11), where +P t P t( ) ( )L Loss

Lk

k

and +Q t Q t( ) ( )L LossL

kk stand for the total line power flow from bus i to

( ( ) ( )) ( ( ) ( ))D G D Gi i i iP t P t j Q t Q t− + −

,( ) ( )k kL LLoss LossP t Q t

( ) ( )k kL LP t jQ t+

Vi(t) Vj(t)(Vk+1(t))rk+jxk

Fig. 2. Line model in distribution system.

Y. Du et al. Applied Energy 222 (2018) 383–395

386

bus j. In addition, the vertical voltage drop is neglected, and the headbus voltage is assumed to be close to the rated value.

(6) Heat balance constraints:

∑ ∑+ + + −

=∈ ∈

ν P t P t P t c P t c p t( ( ) ( ) ( ) ( )) ( )

0m c

h MT m b m th m m solar mm c

m Load mh

, , , , ,

(14)

In Eq. (14), νh is the heat to electricity ratio of micro turbines,Psolar,m(t) is the heat generated by solar energy, and P t( )Load m

h, is the

thermal load of the mth microgrid system.In the above coalitional operation model, when all microgrids form

as one grand coalition, DGs, energy storages and loads owned by dif-ferent microgrid operators are dispatched indiscriminately in a cen-tralized manner. In this way, neighboring microgrids can provide en-ergy support to each other via power exchange, and share excessiveresources like low-cost renewable energy, which consequently lead toreduction of total operation cost. On the other hand, the individualbenefit of each microgrid operator is downplayed under the globaloptimization goal of minimizing the total cost. In order to guarantee theinterest of local microgrid operators, a fair cost allocation method isproposed in this paper, in which case any microgrid operator, and anysubset of microgrid operators can receive some cost saving by partici-pating in the grand coalition. The detailed modeling of the cost allo-cation will be demonstrated in Section 4.

3. Linearized optimal power flow for distribution with multi-microgrid coalition

As has been observed in Section 2, Eqs. (9) and (10), the active andreactive network losses PLoss(t) and QLoss(t) are nonlinear and non-convex terms, which adds to model complexity. To improve computa-tion efficiency, we apply the linearization technique derived from [33]in the model, which are presented as follows:

For the kth line in the distribution system, the active and reactiveline flow can be calculated as follows:

∑ ∑= + − + += + = +

−P t Sub k j P t P t Sub k j P t( ) ( 1, )·( ( ) ( )) ( 1, )· ( )Lj k

n

jD

jG

j k

n

LossL

1 2k

j 1

(15)

∑ ∑= + − + += + = +

−Q t Sub k j Q t Q t Sub k j Q t( ) ( 1, )·( ( ) ( )) ( 1, )· ( )Lj k

n

jD

jG

j k

n

LossL

1 2k

j 1

(16)

In Eqs. (15) and (16), both j and k are bus indices of the distributionsystem, n is the set of buses. Sub is a n× n matrix, where each element(k+1, j) denotes if bus j belongs to the sub-tree of bus (k+1).

The active and reactive line loss in Eqs. (15) and (16) can be ex-pressed as the following linearized function of bus generation:

∑ ∑≈ +∂

∂− +

∂∂

−

∗

=

∗

=

∗

− −− −

P t P tP

PP t P t

PQ

Q t

Q t

( ) ( ) ( ( ) ( )) ( ( )

( ))

LossL

LossL

i

nLossL

iG i

GiG

i

nLossL

iG i

G

iG

1 1

j jj j

1 11 1

(17)

∑ ∑≈ +∂

∂− +

∂∂

−

∗

=

∗

=

∗

− −− −

Q t Q tQ

PP t P t

QQ

Q t

Q t

( ) ( ) ( ( ) ( )) ( ( )

( ))

LossL

LossL

i

nLossL

iG i

GiG

i

nLossL

iG i

G

iG

1 1

j jj j

1 11 1

(18)

In Eqs. (17) and (18), the partial derivative parts, i.e.∂ ∂ ∂ ∂ ∂ ∂− − −P t P t P t Q t Q t P t( )/ ( ), ( )/ ( ), ( )/ ( )Loss

LiG

LossL

iG

LossL

iGj j j1 1 1 , and ∂ −QLoss

Lj 1

∂t Q t( )/ ( )iG , are called loss factors for distribution (LF-D), which de-

scribe the sensitivity of the (j−1)th line loss to the ith bus generation.The LF-D can be obtained as follows:

The losses of the kth line are given by

=+

+P t

P t Q tV t

r( )( ) ( )

( )LossL L L

kk

2 2

12

k k k

(19)

=+

+Q t

P t Q tV t

x( )( ) ( )

( )LossL L L

kk

2 2

12

k k k

(20)

The LF-D is calculated as the partial derivative of line losses to thebus generation:

⎜ ⎟∂∂

= ⎛⎝

∂∂

+∂∂

⎞⎠ +

−P tP t

P tP tP t

Q tQ tP t

rV t

( )( )

2 ( )( )( )

2 ( )( )( ) ( )

LossL

iG L

L

iG L

L

iG

k

k 12

j

kk

kk

1

(21)

⎜ ⎟∂∂

= ⎛⎝

∂∂

+∂∂

⎞⎠ +

−P tQ t

P tP tQ t

Q tQ tQ t

rV t

( )( )

2 ( )( )( )

2 ( )( )( ) ( )

LossL

iG L

L

iG L

L

iG

k

k 12

j

kk

kk

1

(22)

⎜ ⎟∂

∂= ⎛

⎝

∂∂

+∂∂

⎞⎠ +

−Q tP t

P tP tP t

Q tQ tP t

xV t

( )( )

2 ( )( )( )

2 ( )( )( ) ( )

LossL

iG L

L

iG L

L

iG

k

k 12

j

kk

kk

1

(23)

⎜ ⎟∂∂

= ⎛⎝

∂∂

+∂∂

⎞⎠ +

−Q tQ t

P tP tQ t

Q tQ tQ t

xV t

( )( )

2 ( )( )( )

2 ( )( )( ) ( )

LossL

iG L

L

iG L

L

iG

k

k 12

j

kk

kk

1

(24)

In Eqs. (21)–(24), index j is equal to k+1, as is shown in Fig. 2. LF-D is related to the generation shift factors (GSF),∂ ∂ ∂ ∂ ∂ ∂P t P t P t Q t Q t P t( )/ ( ), ( )/ ( ), ( )/ ( )Lk i

GLk i

GLk i

G , and ∂ ∂Q t Q t( )/ ( )Lk iG ,

which is the sensitivity of line power flow to bus power injection, andcan be calculated as follows:

∑∂∂

= − + + +∂∂= +

−P tP t

Sub k i Sub k jP t

P t( )( )

( 1, ) ( 1, )( )

( )L

iG

j k

nLossL

iG

2

kj 1

(25)

∑∂∂

= +∂∂= +

−P tQ t

Sub k jP tQ t

( )( )

( 1, )( )

( )L

iG

j k

nLossL

iG

2

kj 1

(26)

∑∂∂

= +∂

∂= +

−Q tP t

Sub k jQ t

P t( )( )

( 1, )( )

( )L

iG

j k

nLossL

iG

2

kj 1

(27)

∑∂∂

= − + + +∂∂= +

−Q tQ t

Sub k i Sub k jQ t

Q t( )( )

( 1, ) ( 1, )( )

( )L

iG

j k

nLossL

iG

2

kj 1

(28)

It can be observed from Eqs. (21)–(28) that the calculation of gen-eration shift factors and loss factors are nested within each other. Hencea recursive method is applied to obtain their values: to begin with, bothGSF and LF-D are set to 0. GSF are first calculated based on Eqs.(25)–(28). Then LF-D are calculated based on both GSF and the linepower flow results from a linearized power flow model, which can beexpressed as follows:

⎡⎣⎢

⎤⎦⎥

= ⎡⎣−

⎤⎦

⎡⎣

⎤⎦

PQ

B BB B

δV

inj

inj2 1

1 2 (29)

In Eq. (29), Pinj and Qinj are (n−1)× 1 vectors accounting for activeand reactive power injection for all buses except for the slack bus. δ andV are the voltage angle vector and voltage magnitude vector, both withdimension of n×1. The voltage angle and voltage magnitude at slackbus are set to 0 and 1 p.u., respectively. The B matrix contains the re-sistance and reactance information of the system, and is expressed asfollows:

=+

=+

≠B i jr

r xB i j

xr x

i j( , ) , ( , ) ,ij

ij ij

ij

ij ij1 2 2 2 2 2

(30)

∑ ∑=+

=+= ≠ = ≠

B i ir

r xB i i

xr x

( , ) , ( , )j j i

nij

ij ij j j i

nij

ij ij1

1,2 2 2

1,2 2

(31)

Since the power injection equations do not include the slack bus, theB1 and B2 matrices are both (n−1)× n matrices with i traversing from2 to n and j traversing from 1 to n. Eq. (29) holds because originally wehave:

Y. Du et al. Applied Energy 222 (2018) 383–395

387

= ⎧⎨⎩

⎛

⎝⎜

−+

⎞

⎠⎟

⎫⎬⎭

=+

−+

+

∗

P Re VV V

r jxr x

r xV V V δ

xx

r xV V δ

x·

( cos )·

sinij i

i j

ij ij

ij ij

ij ij

i i j ij

ij

ij

ij ij

i j ij

ij2 2

2

2 2

(32)

= ⎧⎨⎩

⎛

⎝⎜

−+

⎞

⎠⎟

⎫⎬⎭

= −+

++

−

∗

Q Im VV V

r jxr x

r xV V δ

x

xr x

V V V δx

·sin

·( cos )

ij ii j

ij ij

ij ij

ij ij

i j ij

ij

ij

ij ij

i i j ij

ij

2 2

2

2 2(33)

Then, based on the assumptions that in the distribution system, thebus voltage angle difference along one branch is close to zero, and thebus voltage magnitude is around 1p.u., Eqs. (32) and (33) can besimplified as follows:

≈+

−+

+−

Pr x

r xV V

xx

r xδ δ

x·( )

·( )

ijij ij

ij ij

i j

ij

ij

ij ij

i j

ij2 2

2

2 2 (34)

≈ −+

−+

+−

Qr x

r xδ δ

xx

r xV V

x·( )

·( )

ijij ij

ij ij

i j

ij

ij

ij ij

i j

ij2 2

2

2 2 (35)

Which leads to the matrix formation of the relationship between buspower injection and bus voltage, as is shown in Eq. (29). δ and V canthus be calculated by multiplying the inverse of B and the LHS of Eq.(29). Since this paper mainly focuses on multi-microgrid cooperation,and the linearized power flow model is only used to calculate the net-work losses during the cooperation, more justification of the model isskipped to make the paper concentrated. Readers who are interested inthe power flow linearization method can refer to [33] for more details.

By obtaining the GSF and LF-D, the network losses can be calculatedusing Eqs. (17) and (18). Both equations are in the form of first-orderTaylor expansion series, where

∗−P t( )LossLj 1 and

∗−Q t( )LossLj 1 are the line losses

in the case where there is no generation output, i.e.= =∗ ∗P t Q t( ) ( ) 0i

GiG , and can be computed using the linearized power

flow model.Compared with [33], a considerable improvement of the model in

this paper is the development of voltage constraints (11)–(13), whichfurther guarantees that the multi-microgrid operation does not violatephysical constraints of a distribution system. In addition, according to[33], in the LOPF-D model, the loss factors for distribution are obtainedfrom a “cold-start” algorithm, which means that they are estimatedvalues with no knowledge of the present operating point of the system.This method can provide results accurate enough when there is novoltage constraints in the model. However, after adding Eqs. (11)–(13),it is discovered that the “cold-start” method becomes inapplicable. Theloss factors have to be repeatedly updated based on the current oper-ating point in order to reach a solution close enough to the standardACOPF result. Hence, it leads to the design of an iterative algorithm.The implementation of bus voltage constraints and the development ofthe associated algorithm is one of the main contributions of this paper.More details of the proposed iterative algorithm will be demonstratedin Section 4.

To conclude, the original nonconvex coalitional operation model ofmulti-microgrid system within the distribution system can now be re-presented as the following LOPF-D model: minimizing (1), subject to(2)-(18). Comparisons with the conventional ACOPF model in Section 5verifies the accuracy and high computation efficiency of the proposedLOPF-D model.

4. A Nucleolus-based fair cost allocation method

As has been stated in Section 2, the coalitional operation model ofmulti-microgrid system realizes a global optimum without consideringthe profits of local microgrids, which may violate the interest and au-tonomy of individual microgrid operators. In this section we introducea Nucleolus-based cost allocation method derived from cooperative

game theory to realize a fair cost share among coalition members, aswell as to improve local economic benefit.

4.1. A Nucleolus-based cost allocation method

According to definitions (1)–(3) in Section 2.1, a core solution to acooperative game must meet the requirements of individual rationality,efficiency and coalitional rationality. From a mathematical perspective,since all the three constraints are affine functions, finding the core ofthe game can be formulated as a convex optimization problem, wherethe feasible region is a polyhedron in high dimensional space, as isshown in Fig. 3. Under certain circumstances, multiple cores can exist,and additional measures are required to evaluate their optimality.While in other cases, the core set of the game may turn out to be empty.In this section we will present a Nucleolus-based core solution to themulti-microgrid cooperative operation case, as well as the associatedcalculation method.

In this paper, we use Nucleolus as the potentially feasible core so-lution to fair cost distribution among microgrids, as has been previouslyapplied in [21,34]. The Nucleolus allocation possesses several favorableproperties: its existence is unique, and if the core set of the game isnonempty, it always belongs to the core. The main idea behind theNucleolus allocation is to first find out the coalition that is most dis-satisfied with the current allocation of the total cost, which implies thatit may defect from the grand coalition. Then the grand coalition adjuststhe allocations to minimize the dissatisfaction of this coalition. Nu-cleolus refers to this adjusted allocation. By conducting a Nucleolus al-location, the dissatisfaction of any coalition is minimized, and the sta-bility of the grand coalition is ensured.

The dissatisfaction of a coalition under the current cost allocation ismeasured quantitatively as follows:

= − = =c x c x c x xg v x x x x v1 1( , ) ( ) [ , ,..., ,..., ] , ( )mT

1 2 MT T (36)

As we can see, in Eq. (36), the dissatisfaction is the difference be-tween the total cost allocated to a coalition c and the cost generatedwhen c operates separately from the grand coalition, where x is theallocation vector indicating the cost distributed to each microgrid, andv(1) is the total cost of the grand coalition, when every cm is equal to 1.The expression is meaningful because a positive value of g(c,x) in-dicates that staying in the grand coalition will cost the coalition c morethan the case in which it operates independently, therefore it may de-fect from the grand coalition. The adjustment model of the cost allo-cation based on dissatisfaction measurement can then be expressed asfollows:

Fig. 3. The core solution in a cooperative game (the red spot within the poly-hedron). (For interpretation of the references to colour in this figure legend, thereader is referred to the web version of this article.)

Y. Du et al. Applied Energy 222 (2018) 383–395

388

=f δminc x, (37)

⩾ = −c x c x cs t δ g v. . ( , ) ( )T (38)

< ⩽ − = ∈c c c c c c c10 M 1 [ , ,..., ,... ], {0,1}m M mT

1 2 (39)

In Eq. (37), the decision variables are c and x. An auxiliary variableδ is introduced to ensure that the maximum level of dissatisfactionamong all coalitions is minimized, as is shown in Eq. (38). In this way,the final allocation x satisfies the coalitional rationality. Eq. (39) en-compasses all the coalitions except for the empty set and the grandcoalition.

4.2. Finding Nucleolus solution via benders decomposition algorithm

The difficult part of Eqs. (37)–(39) is that the number of coalitionsgrows exponentially with an increasing number of microgrid players,which makes it impossible to obtain all the values of v(c) in (36). Toovercome this combinatorial explosion obstacle, we adopt a BendersDecomposition (BD) method, which has been previously applied in[35]. To begin with, Eq. (37) can be rewritten as follows:

−c x v cmin{max ( )}x c

T(40)

Which is a min–max problem, and can be further decomposed intothe following sub-problem and master problem:

Sub-problem:

= − ⋯⋯∗∈ ⧹

x c x cg v UB( ) max ( )c C

k1{ }

T ( )

(41)

< ⩽ −−

cs t 1. . 0 M 1(2) (18)

T

(42)

Master-problem:

= ⋯⋯f δ LBminkx

k( ) ( )(43)

⩾ + ∂ − ∀ = −∗ ∗x x x xs t δ g g s k. . ( ) ( ) ( ) , 1,2,..., 1s s s( ) ( )T

( ) (44)

⩽ =x v c c( ), [0,0,...,1,...,0]m m m

c c c c, ,..., ,...,m M1 2 (45)

=x v1 1( )T (46)

In the sub-problem (41), cost allocation x is fixed, and the variableis the binary coalition index cm. The objective is the difference betweenthe allocated cost and the operating-alone cost, which stands for thedissatisfaction of coalition c. Eq. (41) aims to find the coalition c that ismost dissatisfied under the current cost allocation x, and determines anupper bound of the optimum.

In the master-problem (43), the dissatisfaction g(x) from sub-pro-blem (41) is used as input, and the variable is the cost allocation x. Theobjective is to minimize the maximum dissatisfaction among all thepreviously found coalitions c, and it determines a lower bound of theoptimum. The algorithm iterates until the upper bound and the lowerbound converge to the same value. Since the master problem is linear,the convergence of the algorithm is guaranteed. Eq. (44) is the Benderscut generated from the sub-problem in the form of first-order approx-imation of a Taylor series expansion, where ∂g∗(x(s))T is the partialderivative of g∗ to x(s) in the sth iteration, and is equal to c s

T( ) . Eqs. (45)

and (46) are the representations of individual rationality and efficiency,respectively.

It can be observed that via Benders Decomposition method, there isno need to calculate v(c) for all coalitions c, but only have to focus onthe coalition that is most likely to refuse to cooperate, which con-siderably reduces computational efforts. Although the algorithm willtake several iterations to converge, the number of iterations is stillmuch less than the number of coalitions if applying an enumerationmethod. The computation efficiency of Benders Decomposition method

will be further proved in the case study in Section 5.It should be further pointed out that although the above Benders

Decomposition method has already been applied in [35], the majordifference between the current work and Ref. [35] is that an economicdispatch model is involved in this work, since each microgrid individualcontains RESs, dispatchable distribute generators, and energy storagedevices, among which the dispatch of the latter two are variables,which adds to higher complexity of the model, and leads to a multi-variable, high dimensional, and nonconvex problem. While Ref. [35]mainly focuses on renewable energy generation. In summary, in thispaper we include a more complicated economic dispatch model withinthe fair allocation model to suit to multi-microgrid cooperation case,and efficiently combine optimal power flow linearization techniquewith Benders Decomposition to tackle the original high dimensional,nonconvex problem.

Another explanation that has to be made is that although thereexists other core solutions in the cooperative game, such as Shapleyvalue [36] and Nash-Harsanyi solution [37], these solutions have someundesirable properties like nonlinearity and nonconvexity, and cannotbe efficiently decomposed as in the case of Nucleolus. Hence in thispaper we mainly consider the Nucleolus allocation as the way of a faircost distribution.

4.3. A panorama of solution process for fair cost allocation

It can be noticed that the above sub-problem (41) also includes theconstraints from LOPF-D model, i.e. constraints (2)–(18), since the sub-problem involves economic dispatch of coalition c to find v(c), whichrequires an iterative algorithm to solve, as is stated in Section 3. Themajor difference between the model described by Eq. (41) and themulti-microgrid coalitional operation model (1) is that in the formeronly microgrid coalition c is connected with the distribution system,while in the latter case all microgrids, namely the grand coalition isconnected with the distribution system. This leads to the result that inEq. (41), c is 0–1 binary variable, while in model (1) all cm indices are 1,since all microgrids are included in the grand coalition. Another trivialdifference is that Eq. (41) computes the difference between the costallocated to the coalition c when it stays in the grand coalition, and thetotal operation cost of c when it operates alone, while Eq. (1) calculatesthe total operation cost of grand coalition.

The combination of LOPF-D constraints with BendersDecomposition method leads to a double-loop iteration algorithm tosolve the fair cost allocation problem, as is shown in Fig. 4.

As can be seen from the figure, the first iteration takes place in thesub-problem of fair cost allocation model, with a known cost allocationx. The loss factors are constantly updated based the current operatingpoints, until the objective function g(x)= cTx – v(c) reaches a fixedvalue. After the first iteration converges, we get the coalition c that hasthe highest dissatisfaction with the current cost allocation x. We input g(x) into the master problem and readjust the cost allocation x tominimize this dissatisfaction. If the output from the master problem,which is the minimized dissatisfaction, does not equal the result fromthe sub-problem, we input the updated cost allocation x calculated fromthe master problem into the sub-problem, and repeat the above steps.The iterative calculation of the sub-problem and the master problemconstitutes the second loop. If solving the multi-microgrid coalitionaloperation problem (1)–(18), only the first loop is required, since theproblem only calculates the total operation cost and there is no de-composition of problem.

5. Case study

In this paper, the above established cooperative operation model ofthe multi-microgrid system and the associated cost allocation methodare tested on both IEEE 33-bus distribution system and IEEE 123-busdistribution system. The first case is used to verify the economic

Y. Du et al. Applied Energy 222 (2018) 383–395

389

effectiveness of multi-microgrid cooperation as well as the stability ofthe formulated grand coalition. The second test case is intended todemonstrate the computation efficiency of Benders Decompositionmethod for cost allocation.

5.1. Simulation results of IEEE 33-bus distribution system

In this test case, ten microgrids are connected to the distributionsystem at different buses, and are formed as one grand coalition underex-ante binding contracts, where there exist both electrical energy ex-change and thermal energy change among coalition members. Themulti-microgrid system topology has already been shown in Fig. 1. Thepower base of the system is 1 MVA. The voltage base is 12.66 kV. Theparameters for DGs and energy storages are given in Table 1. The re-newable energy generators, i.e. wind turbines and PVs, are assumed towork at MPPT mode with zero cost. Wind speed data and solar irra-diation data are acquired from [38]. Load data is acquired from [39].

5.1.1. Comparison between LOPF-D and ACOPFAs discussed in Section 3, the coalitional operation model of a mult-

microgrid system is established as a LOPF-D problem. To ensure theaccuracy of the proposed LOPF-D model, we compare the results of thecoalitional operation model (1)–(18) with a conventional ACOPFmodel. We simulate the coalitional operation of the multi-microgridsystem for 7 consecutive days, with the time interval set as 1 h, which is168 h in total. To fully validate that the benefits from coalitional op-eration for each microgrid is not occasional, and can be maintained inthe long term, we choose the time horizon as 168 h instead of the 24 hused in the daily schedule. The comparison of the results from LOPF-Dand ACOPF are shown in Table 2.

In Table 2, the total cost refers to the operation cost for the simu-lated 168 h. The relative error of bus voltage, active and reactive net-work losses are the maximum relative error among all the buses andover all the time intervals. As seen from the table, the results fromLOPF-D is very close to the one from ACOPF, which verifies the accu-racy of the former. Furthermore, it should be noted that the calculationtime of LOPF-D model is 4 times faster than the ACOPF model, whichsubstantiates its high computation efficiency.

5.1.2. Optimal cost allocation among microgridsWe run the above multi-microgrid coalitional operation model and

fair cost allocation model in two different scenarios: a winter scenarioand a spring scenario. The two are different from each other in para-meters including solar radiation, wind speed, electrical load andthermal load. The simulation length for both scenarios are 7 days, or168 h. The aim is to verify the economy of multi-microgrid coalitionover a long-time scale in diversified real-world scenarios. The final costsavings for each microgrid member in the two scenarios are presentedin Table 3:

The 1st loop: acquire the operation cost for sub-coalition c and its dissatisfaction with the current cost allocation x

The 2nd loop: Iteratively calculate the sub-problem and master problem from BD to decide the optimal cost allocation

Initializationk 1, x(1)=v(1)·M-1·1

UB(k) 10e4, LB(k) 0

No

Yes

Yes

Set the initial values for loss factors for distribution and generation shift factors

h = 1

Solve the sub-problem (41), subject to (2)-(18), (42) to

find the sub-coalition c with highest dissatisfaction

No

Set UB(k) to g(h)(x), record the value of x(h), g(h)(x) and

g(h)(x)

Solve the master problem (43)-(46), Set LB(k) to f (k),

update x

|UB(k)-LB(k)| 2?

|g(h)(x)-g(h-1)(x)| 1?

Update the loss factors based on the current

operating point

h = h + 1

x = x(k)k = k + 1

Algorithm terminatesOutput the optimal allocation x

Fig. 4. Flow chart of fair cost allocation process.

Table 1Parameters of DG and energy storage.

DG P MT bmin{ , }

(kW)

P MT bmax{ , } (kW) η (%) νh Cfuel ($/kWh)

MT 0 60 33 2.69 0.042Boiler 0 100 85 –

EnergyStorage

Capacity(kWh)

P P/thcmax

thdmax

(kW)

η (%) Initialstate(kWh)

Δt(h)

TES 100 50 98 50 1

Table 2Comparison of results from LOPF-D and ACOPF (33-bus System).

Relative errors (%)

Total cost: 0.0083 Bus voltage: 0.4339Active network losses: 0.4761 Reactive network losses error: 0.4241

Calculation time (s)

LOPF-D: 11.9291 ACOPF: 57.7869

Table 3Final cost savings based on Nucleolus method (33-bus System).

No. Nucleolus allocation ($) Independent operation ($) Cost saving (%) No. Nucleolus allocation ($) Independent operation ($) Cost saving (%)

Winter1 340.6064 390.4244 12.7599 6 357.8588 408.4880 12.39432 −100.7641 −81.1947 24.1017 7 554.0574 565.9426 2.10013 345.8551 350.5885 1.3501 8 388.7359 407.0190 4.49204 211.5046 241.3738 12.3747 9 434.3381 466.2974 6.85395 182.0589 197.0653 7.6149 10 299.9779 315.1441 4.8125

Spring1 174.4132 204.9736 14.91 6 108.6317 126.5169 14.142 −47.6777 -28.422 67.75 7 177.4422 178.7913 0.753 313.0457 314.3863 0.43 8 241.6237 244.3965 1.134 278.315 278.6241 0.11 9 141.3924 141.3924 0.005 203.6429 229.6581 11.33 10 225.1895 225.1895 0.00

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390

As is shown in the table, under the coalitional operation, each mi-crogrid receives some cost saving in different scenarios, which sub-stantiates the economic efficiency of multi-microgrid cooperation.Notice that in both winter and spring scenario cases, microgrid 2 has anegative cost under both Nucleolus allocation and independent opera-tion. This is because when microgrid 2 has extra power supply, it can

sell power back to the distribution system to support power consump-tion in its neighboring microgrids and make profits.

5.1.3. Economy analysis of multi-microgrid cooperationWe summarize two reasons behind the operation economy of the

multi-microgrid coalitional operation. First of all, mutual power

Time(h)

00

100

24

PMT(kWh)

200

68

300

10

Day

1214161820 7622 54324 21

Time(h)

00

100

24

200

PMT(kWh)

6

300

810

Day

1214161820 7622 54324 21

00

200

24

Psolar(kWh) 400

68

600

10

Day

1214161820 76

Time(h)

22 54324 21

Fig. 5. Comparison of MT generation in coalitional operation and independent operation.

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170

Time(h)

0

0.2

0.4

0.6

0.8

1

Pgrid(p.u.)

Pgrid

PmD

PWT

Fig. 6. DSO power exchange with transmission system.

Fig. 7. Daily dynamic changes of TES state of charge in coalitional operation case.

Y. Du et al. Applied Energy 222 (2018) 383–395

391

exchange among local microgrids increases the utilization efficiency ofthe zero-cost renewable energy and consequently reduces generationcost.

We first take a look at the micro turbine generation of microgrids indifferent situations. Fig. 5 demonstrates the micro turbine generation ofmicrogrids in the winter scenario in both coalitional operation case andindependent operation case, as well as the solar thermal generation. Inthe independent operation case, each microgrid has to supply the localthermal load on their own, with no power exchange with other mi-crogrids. As can be observed from the three figures, during the timeperiod with high solar thermal energy (i.e., 9 a.m.–4p.m.), the coali-tional operation case shows much less micro turbine generation thanthe independent case (the former is 0). This is because when forming asone grand coalition, thermal energy is transferred from microgrids withsurplus solar panel generation to the ones with higher thermal load inthe grand coalition, while in the independent case, since there is nolocal energy exchange, the more expensive micro turbine has to beapplied to provide thermal energy.

The utilization of wind power in the multi-microgrid coalitionaloperation case is further demonstrated in Fig. 6. As can be observedfrom the figure, the power exchange between DSO and transmissionsystem, Pgrid, follows the tendency of the microgrid demand Pm

D. Inaddition, when the wind power generation is high, i.e. in the 17th,84th, 85th and 98th hour, DSO purchased less power from the

transmission system; while during other periods with lower wind powergeneration, more power is sent to the DSO to support the microgridpower demand. The correlation coefficients between DSO power ex-change and microgrid demand is 0.6124, and the correlation coefficientbetween DSO power exchange and wind power is −0.7253, whichvalidates the above observations. Hence it can be concluded that theinterconnections among microgrids in the coalitional operation casemakes it possible for wind power share, which in return reduces powerpurchase cost.

The second reason for the cost saving effects is related to the energystorage devices in the microgrids, and is shown in Figs. 7 and 8:

The above two figures demonstrate the daily dynamic changes offour thermal energy storages owned by MG1-MG4 in the winter sce-nario and in both coalitional operation case and independent operationcase. As is shown in the figure, the energy storages reaches theirmaximum capacity more often in the coalitional operation case than inthe independent operation case, which is especially evidently shown inTES4. This is because when operating cooperatively, the surplusthermal solar power of the microgrid without energy storage can befully stored by the energy storage of another microgrid via local powerexchange, therefore makes full use of zero-cost renewable energy andsaves generation cost. While in the independent operation case, sinceeach microgrid has a large amount of thermal load to supply, there isless extra thermal power to be stored.

Fig. 8. Daily dynamic changes of TES state of charge in independent operation case.

Fig. 9. Bus voltage increase due to multi-microgrid penetration (winter scenario).

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Further comments on the above analysis is that although in Fig. 1the 10 microgrids are connected with the distribution system at specificlocations, if randomly located, the economy of cooperative operationwill still take place because the above two reasons behind the operationeconomy improvement does not relate to microgrid locations. This hasbeen proved by simulations with microgrids arbitrarily located in dif-ferent buses. For the sake of conciseness, the simulation results, whichare much similar to Table 3 are skipped in the paper.

5.1.4. Impacts of multi-microgrid penetration on distribution systemWe first investigate the impact of multi-microgrid penetration on

the system losses. Network losses rate is implemented here, which isdefined as the ratio of network work losses to the total power genera-tion. In the winter scenario case, when there is no microgrids in thedistribution system, the network losses rate is 4.65%; with 10 micro-grids, the maximum hourly network losses rate over the entire simu-lation time span (168 h in total) is 4.39%. Hence it can be concludedthat multi-microgrid penetration can help reduce network losses. This isbecause local energy consumption can be supplied by DGs in micro-grids, therefore avoids long-distance power transmission and decreasesnetwork losses.

We further consider the impact of multi-microgrid penetration onsystem voltage level. Fig. 9(b) demonstrates the percentage of busvoltage increase in the multi-microgrid coalitional operation casecompared with the original voltage level of the distribution system inthe winter scenario. The grey plane in the figure is the zero-level. Asseen from the figure, the majority of bus voltages experienced a voltageincrease during the entire simulation time span, with the maximum

voltage increase percentage reaching 1.11%. Notice that at some buses,the voltage increase is below zero, which indicates a voltage decrease.This is because of the penetration of the microgrid demand. Still,Fig. 9(a) shows that all the bus voltage levels are within the feasibleregion [0.95p.u., 1.05p.u], which indicates that multi-microgrid pene-tration can provide reliable voltage support to the distribution system.

5.2. Simulation results of IEEE 123-bus distribution system

One of the key contributions of our paper lies in the application ofBenders Decomposition method to overcome the combinatorial explo-sion in fair cost allocation among microgrid members. To fully verifythe efficiency of the applied algorithm, we further test the coalitionaloperation model and the fair cost allocation model on a larger-scale123-bus distribution system, with 30 microgrids involved in the co-operation, which leads to a tremendous number of coalitions, i.e. 230 –1 in total. The topology of the 123-bus system is shown in Fig. 10:

In Fig. 10, 30 microgrids are connected to IEEE-123 bus distributionsystem and operate as one grand coalition. Similarly to the 33-bus testcase, in this test case the multi-microgrid cooperation is simulated intwo scenarios with a time span of 168 h. The comparison with ACOPFand final cost allocations are shown in Tables 4 and 5:

As can be seen from Table 4, the LOPF-D has adequate accuracyeven on a larger system with far more variables, and the computationspeed is more than 20 times faster than the ACOPF. In Table 5, in thespring scenario, MG9-10, MG19-20, MG29-30 receive zero cost saving.Although those microgrids cannot receive any cost savings in the springscenario, they indeed benefit from coalitional operation in winter sce-nario (the cost saving percentage is greater than 0). If viewed from along-term perspective, they would still be willing to stay in the grandcoalition. For all the other microgrids, they receive some cost savings inboth winter scenario and spring scenario by cooperation.

The computation efficiency of the Benders Decomposition methodin finding Nucleolus cost allocation is demonstrated as follows: in thesub-problem (41), the number of constraints is 115,839 and the totalnumber of variables is 113,431, including 30 binary variables; in themaster-problem (43), the total number of variables is 30, the number ofconstraints grows with each iteration. The simulation is carried out on ahybrid platform, MATLAB 2016a plus GAMS 24.7, where MATLAB is

Fig. 10. IEEE 123-bus system with 30 microgrids.

Table 4Comparison of results from LOPF-D and ACOPF (123-bus system).

Relative errors (%)

Total cost: 0.0096 Bus voltage: 0.6196Active network losses: 0.7584 Reactive network losses error: 0.7467

Calculation time (s)

LOPF-D: 15.267 ACOPF: 355.7584

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used for creating input data profiles and recording computation results.The above min–max problem is solved by CPLEX on GAMS. The hard-ware environment is a laptop with Intel®Core™ i5-6300U 2.4 GHz CPU,and 4.00 GB RAM. Computation time and iterations for calculating theNucleolus cost allocation in each scenario is provided in Table 6:

In the 123-bus test case, the total number of coalitions is2^30–1= 1.0737× 109. From Table 6, we can see that only a smallnumber of coalitions needs to be checked before reaching the core so-lution, i.e. 278 in the winter scenario and 178 in the spring scenario.Therefore, we may safely conclude that the Benders Decompositionmethod holds considerable computation efficiency and can fairly beapplied to large-scale system simulation.

6. Conclusion

This paper highlights the potential advantages of cooperationamong multiple microgrids with different types of distributed energyresources at the distribution system level. The main contributions ofthis paper are summarized as follows:

(1). Inspired by cooperative game theory, a coalitional operation modelof grid-connected microgrids is constructed to minimize the totaloperation cost. Simulation results verify that via local power ex-change among microgrids, the utilization efficiency of renewableenergy and also energy storage devices can be increased, which

contribute to the generation cost reduction;(2). To secure the stability of the multi-microgrid coalition, a fair al-

location of the total operation cost among microgrids is furtherproposed, which is a core solution to the cooperative game.Benders decomposition algorithm is applied to obtain the coresolution with high computational efficiency. In this way, the localoptimum is achieved to guarantee the benefit of each microgridplayer;

(3). We apply a linearized optimal power flow for distribution (LOPF-D) model in both multi-microgrid coalitional operation model andfair cost allocation model to include distribution network losses,since at distribution level the losses accounts for a significant part.Comparison with the ACOPF model verifies the accuracy of theproposed LOPF-D model as well as its computation efficiency.

The proposed work and conclusions hold considerable implicationsfor real-world applications: (1) with the increasing penetration of theuncertain and intermittent renewable energy into the distributionsystem, a coalitional operation mode of multi-microgrid provides a vi-able solution to efficiently consume the surplus green power in case ofover generation, which can reduce the fuel cost of dispatchable gen-erators and also the dependence on the main grid for power supply; (2)the propose Nucleolus cost allocation method can realize local optimumand guard the interest of individual microgrid, which constitutes astrong prerequisite to fully realize a fair and profitable multi-microgridcooperation. Hence, the proposed model and solution can be well ex-plored for a practical application.

Acknowledgement

This work is partly supported in part by SGCC Science andTechnology Program and in part by CURENT, a US NSF/DOEEngineering Research Center under the NSF award EEC-1041877.

Table 5Final cost savings based on Nucleolus method (123-bus system).

No. Nucleolus allocation ($) Independent operation ($) Cost saving (%) No. Nucleolus allocation ($) Independent operation ($) Cost saving (%)

Winter1 364.5519 401.2525 9.1465 16 358.9493 401.7175 10.64632 −78.8023 −72.5693 8.5891 17 555.9820 559.5865 0.64413 367.8911 375.0287 1.9032 18 388.8479 404.5199 3.87424 234.4722 265.9025 11.8202 19 434.9585 468.4101 7.14155 186.4434 200.3160 6.9254 20 299.8734 314.5705 4.67216 359.0958 403.6232 11.0319 21 362.1107 393.5861 7.99717 556.2037 560.5060 0.7676 22 −78.1103 −73.9583 5.61398 388.2792 405.1344 4.1604 23 368.6778 372.2232 0.95259 436.2354 468.2867 6.8444 24 235.1628 263.7281 10.831310 301.5989 317.1347 4.8988 25 186.5898 197.8237 5.678711 363.6436 396.8117 8.3586 26 359.7733 398.0551 9.617212 −78.2742 −73.3835 6.6646 27 555.0086 556.3440 0.240013 367.4488 373.4819 1.6154 28 387.3975 402.9131 3.850914 235.3319 265.7271 11.4385 29 434.9293 465.7893 6.625315 184.9453 197.1955 6.2122 30 302.4844 315.0619 3.9921

Spring1 195.8652 212.6706 7.9021 16 109.0740 124.9667 12.71762 −26.9551 −22.5069 19.7638 17 176.8997 178.4936 0.89303 334.1948 336.9596 0.8205 18 240.5403 243.7097 1.30054 298.5242 300.4783 0.6503 19 141.3924 141.3924 05 206.3893 231.4749 10.8373 20 225.1895 225.1895 06 109.0474 125.4055 13.0442 21 195.8315 209.8261 6.66967 176.8823 178.5370 0.9268 22 −26.9405 −22.9849 17.20928 240.4132 243.8768 1.4202 23 334.2929 335.2857 0.29619 141.3924 141.3924 0 24 298.3843 299.0838 0.233910 225.1895 225.1895 0 25 206.3761 230.0033 10.272511 196.3429 211.0214 6.9559 26 108.8652 124.1300 12.297512 −26.8926 −22.7858 18.0239 27 176.8709 178.3374 0.822313 334.1976 336.0424 0.5490 28 240.2067 243.2668 1.257914 298.3681 300.3638 0.6644 29 141.3924 141.3924 015 206.0727 229.6383 10.2621 30 225.1895 225.1895 0

Table 6Computation efficiency of benders decomposition method (123-bus system).

Simulation scenario Number of iterations in bendersdecomposition

Computation time (s)

Winter 278 12,316Spring 178 7482

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