1 Distributed Online Generalized Nash Equilibrium Tracking for Prosumer Energy Trading Games

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Distributed Online Generalized Nash Equilibrium

Tracking for Prosumer Energy Trading Games

Yongkai Xie, Zhaojian Wang, John Z.F. Pang, Bo Yang, and Xinping Guan

Abstract—With the proliferation of distributed generations,

traditional passive consumers in distribution networks are evolv-

ing into “prosumers”, which can both produce and consume

energy. Energy trading with the main grid or between prosumers

is inevitable if the energy surplus and shortage exist. To this

end, this paper investigates the peer-to-peer (P2P) energy trading

market, which is formulated as a generalized Nash game. We

ﬁrst prove the existence and uniqueness of the generalized Nash

equilibrium (GNE). Then, an distributed online algorithm is

proposed to track the GNE in the time-varying environment.

Its regret is proved to be bounded by a sublinear function of

learning time, which indicates that the online algorithm has an

acceptable accuracy in practice. Finally, numerical results with

six microgrids validate the performance of the algorithm.

Index Terms—Generalized Nash equilibrium, online optimiza-

tion, time-varying game, P2P energy trading market.

I. INTRODUCTION

The explosive growth of distributed generation in distri-

bution networks together with the advancement of commu-

nication and control technology at the consumer level have

gradually transformed the traditionally passive consumers into

“prosumers”, which can both produce and consume energy [1].

Then, energy trading with the main grid or between prosumers

is inevitable since energy surplus and shortage are bound to

exist [2]. In this situation, the peer-to-peer (P2P) market, which

operates in a distributed manner, is more popular due to the

ever-increasing number of prosumers, in which the various

prosumers can be self-organized to operate economically and

reliably under a given market mechanism [3]. In addition,

the increasing penetration and an aggravating volatility of

renewable generation calls for online market clearing methods.

In this paper, we intend to investigate the distributed online

energy trading market for prosumers.

For such P2P energy trading markets, they are usually

formulated as generalized Nash games, where each prosumer

maximizes its proﬁt with coupling constraints, e.g., global

power balance [1], [2], [4]–[7]. Then, clearing the resulting

This work was supported by the National Natural Science Foundation

of China (No. 62103265), and the “Chenguang Program” supported by

the Shanghai Education Development Foundation and Shanghai Municipal

Education Commission of China (20CG11). (Corresponding author: Zhaojian

Wang)

Y. Xie, Z. Wang, B. Yang, and X. Guan are with the Key Laboratory of

System Control, and Information Processing, Ministry of Education of China,

Department of Automation, Shanghai Jiao Tong University, Shanghai 200240,

China, (email:wangzhaojian@sjtu.edu.cn).

J.Z.F. Pang is with the Institute of High Performance Com-

puting (IHPC), A*STAR, Singapore 138632, Singapore, (email:

john_pang@ihpc.a-star.edu.sg).

P2P market corresponds to ﬁnding the generalized Nash equi-

librium (GNE) of the energy trading game. For example, in [1],

the energy sharing game among prosumers is formulated with

full information, and [2] further designs a fully distributed

algorithm based on Nesterov’s methods to seek the GNE

with only partial-decision information. In [4], a P2P energy

market is formulated as a generalized Nash game, where the

prosumers who share payments are mutually coupled and

inﬂuenced. Following this, [5] and [6] further consider system-

level grid constraints. Lastly, in [7], a P2P energy market of

prosumers is formulated as a generalized aggregative game

with global coupling constraints. The aforementioned works

have made great progress in the distributed GNE seeking for

the P2P energy trading market. However, they usually focus on

only one time section and provide ofﬂine solutions to solve the

game. Due to the volatility of renewable generations and the

complexity of load proﬁles, both current and future operation

status changes much more over time, requiring much faster

algorithms, i.e., online GNE tracking.

In this paper, we formulate a P2P energy trading market

among prosumers in the distribution network and propose a

distributed online algorithm to track the GNE of the market.

The major contributions are as follows.

•A P2P energy trading market is modeled as a generalized

Nash game with both individual and coupled time-varying

constraints. Moreover, we prove the uniqueness of the

GNE of this market at any time section.

•A novel distributed online algorithm is proposed to track

the GNE, where each prosumer can make decisions only

using local variables and neighboring information. This

reduces the communication burden and makes it easier to

implement in practice.

•We prove a sublinear regret bound, i.e., that the regret of

the online algorithm can be bounded by a sublinear func-

tion of learning time, indicating that the online algorithm

suffers minimal “loss in hindsight”.

The rest of this paper is organized as follows. In Section

II, the P2P energy trading game is formulated. Section III in-

troduces and analyzes the performance of a distributed online

algorithm to track the GNE of the game in a time-varying

environment. Numerical results are presented in Section IV to

verify the effectiveness of our algorithm. Finally, Section V

concludes the paper.

Notations: In this paper, Rn

+is the n-dimensional (nonpos-

itive) Euclidean space. For a column vector x∈Rn(matrix

Am×n∈Rm×n), its transpose is denoted by xT(AT). For a

matrix A,[A]i,j stands for the entry in the i-th row and j-th

arXiv:2210.02323v1 [math.OC] 5 Oct 2022

column of A. For vectors x, y ∈Rn,xTy=hx, yiis the inner

product of x, y, while ⊗represents the Kronecker product.

kxk=√xTxis the Euclidean norm. The identity matrix with

dimension nis denoted by In. Sometimes, we also omit n

to represent the identity matrix with the proper dimension.

0n,1nare all zero and all one vectors with dimension n,

respectively. The Cartesian product of the sets Ωi, i = 1,··· , n

is denoted by Qn

i=1 Ωi. Given a collection of yifor iin

a certain set Y, the vector composed of yiis deﬁned as

y=col(yi) := (y1, y2,··· , yn)T. The projection of xonto

a set Ωis deﬁned as PΩ(x) := arg miny∈Ωkx−yk.

II. PROBLEM FORMULATION

A. Network model

We consider a distribution network with a group of pro-

sumers, denoted by the set N={1,2, ..., N}. For each

prosumer, its load demand can be satisﬁed by its own gen-

eration and trading with the main grid or its neighboring

prosumers. The trading edge is denoted by E ⊆ N × N.

For a prosumer i, the set of its neighbors is denoted by

Ni={N1

1,...,NNi

1}with |Ni|=Ni. If j∈ Ni, prosumers

iand jcan trade and communicate directly. Otherwise, direct

trading and communication are not allowed. Then, the trading

network is modeled as an undirected graph G= (N,E). The

adjacency matrix of Gis denoted by Wwith elements wi,j . If

j∈ Ni, the weight wi,j satisﬁes wi,j =wj,i >0. Otherwise,

wi,j =wj,i = 0. The Laplacian matrix of the communication

graph is denoted by Land we have 1TL= 0, where 1is

an all-one vector. Moreover, the graph Gis assumed to be

connected. For the weights, we have the following assumption,

which implies that every row sum of Wis identical.

Assumption 1. The weight wi,i >0and Pj∈N wi,j =w0>

0for all i∈ N.

B. Prosumer model

The scenario is that each prosumer is equipped with dis-

patchable generation, a non-dispatchable load, and an energy

storage system (ESS). To maintain power balance, it can

generate electricity, charge or discharge from the ESS, and/or

trade with the main grid or neighboring prosumers. In this

paper, we focus on the time horizon T={1,2, ..., T }. Here,

we will introduce them in detail.

Dispatchable generation: The power generated by dispatch-

able generation units of prosumer iat time t, denoted by pg

i(t),

is limited by

pg,min

i≤pg

i(t)≤pg,max

i,∀i∈ N, t ∈ T (1)

where pg,min

iand pg,max

iare minimum and maximum local

generation, respectively. Its generation cost is as follows.

i(pg

i(t)) = ag

i(pg

i(t))2+bg

ipg

i(t)(2)

where ag

i>0and bg

iare constants.

Energy Storage Systems (ESS): The ESS proﬁle is con-

strained by the following dynamics.

0≤pc

i(t)≤pc,max

i,∀i∈ N, t ∈ T (3)

0≤pd

i(t)≤pd,max

i,∀i∈ N, t ∈ T (4)

si(t+ 1) = si(t) + ∆t

ecap

iηc

ipc

i(t)−1

ηd

i(t)(5)

smin

i≤si(t)≤smax

i,∀t∈ T (6)

where pc

i(t),pd

i(t), and si(t)are the charging, discharging

power, and state of charge (SoC) of the ESS i, respectively.

∆t,ecap

i,ηc

iand ηd

iare sampling time, ESS maximum

storage capacity, and (dis)charging efﬁciencies, respectively.

Moreover, smin

iand smax

i, with 0< smin

i< smax

i<1,

denote the minimum and maximum SoC, while pc,max

iand

pd,max

idenote the maximum (dis)charging power.

Each prosumer might also minimize the usage of its ESS

to reduce its degradation. Depending on the efﬁciency of a

storage unit, there are losses based on usage that usually grow

quadratically in power. For simplicity, we disregard the effects

due to SoC levels. As deﬁned in [8], the corresponding cost

function is

fes

ipc

i(t), pd

i(t)=ac

i(pc

i(t))2+ad

ipd

i(t)2(7)

where ac

iand ad

iare both positive constants.

Trading with the main grid: Let pmg

i(t)be the power

purchased from the main grid at time tand pmg(t) =

col {pmg

i(t)}i∈N . Similar to [9], we set grid cost as

Ct(pmg(t)) = cmg

tXi∈N pmg

i(t)2

(8)

where cmg

tis a time-varying cost coefﬁcient, since the energy

production varies along the time period according to the energy

demand and the availability of distributed energy sources.

Then the cost assigned to prosumer iis

fmg

i(pmg(t)) = pmg

i(t)

Pi∈N pmg

i(t)Ct(pmg(t))

=cmg

tpmg

i(t)Xi∈N pmg

i(t)(9)

Moreover, the total power exchanged with the main grid is

limited, i.e.,

pmg,min ≤Xi∈N pmg

i(t)≤pmg,max,∀t∈ T (10)

Trading with neighbors: The trading cost with neighboring

prosumers of prosumer iis

ftr

iptr

i,j (t)=X

j∈Nihatr ptr

i,j (t)2+dtr

i,j ptr

i,j (t)i(11)

where ptr

i,j (t)is the power purchased from prosumer jat time

t,dtr

i,j =dtr

j,i >0is the price and atr is a small positive

constant, which represents the tax cost incurred by using the

energy sharing platform.

Disregarding loss on the power lines, the sum of the trading

power of prosumer iand jat time tshould be 0.

ptr

i,j (t) + ptr

j,i(t)=0,∀(i, j)∈ E, t ∈ T (12)

Furthermore, trade between prosumers is limited by

ptr,min

i,j ≤ptr

i,j (t)≤ptr,max

i,j ,∀(i, j)∈ E, t ∈ T (13)

where ptr,min

i,j ≤0and ptr,max

i,j ≥0are the minimum and

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1DistributedOnlineGeneralizedNashEquilibriumTrackingforProsumerEnergyTradingGamesYongkaiXie,ZhaojianWang,JohnZ.F.Pang,BoYang,andXinpingGuanAbstractWiththeproliferationofdistributedgenerations,traditionalpassiveconsumersindistributionnetworksareevolv-ingintoprosumers,whichcanbothproduceandconsumee...

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1 Distributed Online Generalized Nash Equilibrium Tracking for Prosumer Energy Trading Games

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