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Behaviour-based Distributed Energy Managementfor Charging EVs in Photovoltaic Charging Station
Amro Alsabbagh, Dongxiang Yan, Songyang Han, Yandong Wang, Chengbin Ma∗Univ. of Michigan-Shanghai Jiao Tong Univ. Joint Institute,
Shanghai Jiao Tong University, Shanghai, P. R. [email protected], [email protected], [email protected], [email protected], [email protected]
Abstract—Behaviour-based distributed energy management forcharging electric vehicles (EVs) in photovoltaic (PV) chargingstation (CS) has been introduced in this paper. Based on theprovider or consumer of the power, CS and EVs are modeledas independent players with different preferences. The energydistribution problem is modeled as a noncooperative stackelberggame and the existence of equilibrium among players is proofedat each control instant. Update of Charging powers of EVs in adistributed fashion is implemented through utilising the learning-based consensus network. Static and dynamic analyses are shownin simulation. Moreover, different behaviours of the EVs’ driversto the discount on the power price offered by the station isalso showed. All the previous results proof the effectiveness andworkability of the proposed energy management.
Index Terms—Distributed energy management; PV chargingstation; Electric vehicle; Game theory; Consensus network.
I. INTRODUCTION
Environmental pollution concerns have motivated the focuson renewable energy sources (REs) and electric vehicles(EVs). Apart from their advantages, difficulties arise fromdealing with them. Fluctuations and intermittent of photovolta-ic generation and limitation of onboard battery capacity alongwith the heavy load of EVs penetration are the main concerns[1]. For the first point, adding energy storage system (ESS)could be a proper solution. While for the second, supporting adistributed infrastructure of charging points with the ability ofswitching from grid-tied mode (GTM) to islanded mode (IM)could be a reasonable choice. There are various conflicts anduncertainties related to such field of work. Unpredictable solarirradiation, onboard battery capacity, incoming of EVs withtheir initial SOCs are the main within others. So, developingenergy management in charging stations (CSs) to support sat-isfied charging service for costumers is inevitable. On contraryof relaxed mode of charging EVs in homes, fast chargingtechnology is one of the requirements in public charging areasto achieve a full-charged battery in a short time. This willincrease their penetration impacts, lower the solution keysand shorten the freedom in the system. Due to the competingbehaviour within the costumers and the conflict between themand the power provider, avoiding the rising complexity cannotbe bypassed. Taking all the above mentioned, evolving aproper energy strategy is a crucial challenge.
Energy management problem (EMP) for charging EVs has
been vastly studied in recent years [2]-[8]. Basically, EMP inthis field can be categorized into centralized and decentralized.Regarding the first, [2] has developed a control strategy inindustrial/commercial EV charging park. Reducing the entiredaily cost of charging PHEVs along with scheduling themwas the aim. [3] has discussed a rule-based energy strategy forcharging station. Controlling the energy flux and evaluating itsimpact on the main grid was the objective. A heuristic rule-based strategy to reduce the impact on the grid and to allocateEVs’ powers in a commercial building has introduced in [4].Concerning the decentralized, it has the main advantages ofminimizing the communication bandwidth and computationalefforts as well as allowing flexibility over reconfigurablesystem [5]. [6] has proposed a decentralized strategy for EVcharging station. The objective was to efficiently charge EVs.A simple prefixed way is utilised regardless the preferences ofEVs. The system works only in the grid-tied mode, thus thepros and cons of working in the islanded mode haven’t studied.[7] has conducted an optimal allocation of the availablecharging power in EV charging station. Charging powers ofEVs are designed to minimize the total charing cost regardlessof their own characteristics. [8] has designed a coordinateddistributed method to charge EVs in a smart community.Avoiding overloads and maximizing customer preferences wasthe objective. Although, due to the privacy and selfishness ofEVs cooperation cannot be guaranteed to be existed.
To the best knowledge of the authors, including compo-nents’ behaviours in the preferences will give better perfor-mance in the system and make it more close to the dailylife conditions. To this aim, the proposed power dispatchadopts different agents’ behaviours i.e., EV’s driver alongwith the station’s owner and reflects it in their preferences.Moreover, due to the uncertainties in the renewable energysystem and features and behaviours of EVs/drivers, choosingthe distributed management is much efficient in such theseapplications. Finally, because of the selfish attitude in chargingEVs, utilising the game theory tool is a perfect match tosolve the decision making problem here [9]. Accordingly,a noncooperative stackelberg game is employed to dispatchpower between selfish and individual players. Below are thepaper’s contribution points,
• Distributed: no centralized controller is required.• Flexible: breakdown-free against single point of failure.• Novel: inserting the behaviours of EVs’drivers and station’s
owner under waiting condition in the charging station.
II. SYSTEM STRUCTURE AND MODEL
The main components of the charging station are photo-voltaic System (PVS), battery energy storage system (BESS),grid system (GS), station load and a fleet of EVs (FEVs) whichare illustrated in Fig. 1. Each unit is connected to the DC-busthrough a compatible converter. To assure the voltage stabilityof the DC-bus, one of the connected devices supposed to workat the voltage control mode. Although the proposed chargingstation system has many components, from the service point ofview it can be conceptually divided into two parts, the provideri.e., the charging station and the consumer i.e., the fleet of EVs.
DC-bus
Commercial
DC/AC
PL
Load
DC/AC
Grid
DC/DC
BESS
PgPb
Energy Storage UtilityRenewable Sources
DC/DC
PVS
Ppv
EVn
DC/DC
Pevn
EV2
DC/DC
Pev2
EV1
DC/DC
Pev1
Pev
Fleet of Electric Vehicles
Fig. 1. System Structure of the charging station
A. Components and Objective Function of Charging Station
1) Photovoltaic system: which is composed of PV panelsand DC/DC converters. In general, PVS can work in threedifferent modes i.e, current, voltage and standby. In the currentmode and to fully utilise the irradiation profile, the maximumpower point tracking (MPPT) algorithm is implemented in itsrelated converters. The model of PV panel which relies on thetemperature and irradiation can be derived as in [10]. To makethe output power from PV more realistic, weather irradianceprofile should match the daily life one as much as possible.To this aim, beta distribution is selected to model the solarirradiance uncertainty [11].
2) Grid system: even though PVS is the main supplier of powerin CS, GS plays a role in filling the lack of power duringpeak hours or meanwhile abnormal weather i.e, cloudy days.The main grid is connected to DC-bus through bi-directionalDC/AC inverter with proper transformer. Similar to PVS, GScan also work in the three mentioned modes.
3) Station load: when the charging station has a specificworking time each day, its household load behaviour will
be similar to that one in a commercial company. Thus, areal-world office building profile is considered [12]. It shouldbe noted that any type of load profile such as residential,industrial, etc., can also be applied.
4) Battery energy storage system: composed of battery tankand bi-directional DC/DC converter. Its model can be achievedthrough its equivalent circuit model [13]. Similar to PVS andGS, BESS can work on the three discussed modes. The mainbenefit of BESS is to buffer the power and utilise it during theintermittent or lack of renewable energy along with filteringits dynamic. As a result, the dominant control loop for it ismaintaining the DC-bus voltage level.
5) Objective Function: as a profitable service, the preferencehere is to increase the revenue gained by selling power to EVs.This payoff can be indicated through the price of total powerderived either from PVS or supported by the grid. Althoughof its valuable benefit to fill in the shortage of generatedpower, power loss in the inverter and transformer could bethe drawback of the ancillary power from utility. This couldbe with higher impact or even inefficient when there is asmall number of plugged in EVs in CS. Another hurdle isthe changeable price from utility which may modify untimelythe incentives for the drivers to charge their EVs. As a result,CS would like to work on the islanded mode in such thesesituations. So, CS can select one of two working modes, i.e,IM or GTM, to maintain higher profit and lower maintenancecost. Thus, the economically/behavioural-based utility functionof the station is to maximize (1).
us =
{−|Pall| N ≤ Nth & SOCb ≥ SOCthb IM
C.∑Ni=1 Pi Otherwise GTM,
(1)
Pall = Ppv + Pb − Pl −∑Nj=1 Pi, (2)
where N is the number of plugged-in EVs, N th is thethreshold number of the plugged-in EVs, SOCb is the SOC ofBESS, SOCthb is the SOC threshold of BESS, C is the unitprice of power, and Ppv , Pb, Pl,
∑Nj=1 Pi are the powers of
PVS, load, BESS, and summed of EVs powers, respectively.As it can be seen, CS switches to IM when there is low numberof EVs under charging along with sufficient amount of powerin BESS. At this mode, CS tries to balance the total powerin the system without the assistance from grid. Meanwhile atGTM, CS seeks the profit maximization aim.To make CS more independent i.e., prolong its ability towork on the islanded mode, the following two tips shouldbe borne in mind. Number of PV modules is to meet the totalload demand. BESS’s capacity should hold the cumulativedifferences between the generated and consumed powers.
B. Fleet of Electric Vehicles
1) Model and uncertainties: since this paper focuses oncharging EVs, each EV can be considered as a battery withDC/DC converter i.e., on-board battery with charger. Theprocedure used for modeling BESS can also be used here[13]. As it is introduced before, the more data profile models
match the daily life ones, the more stable and efficient systemis achieved. Types of EVs i.e., capacities, coming to CSis highly diverse, i.e., passenger cars, vans, etc. Also, theirinitial SOCs, i.e., previous trip and EV’s driver incentivesare stochastic. As a result, it is more convenient to modelthese important uncertainties to enhance the performance ofthe system. Gaussian distribution is the most suitable functionto model these uncertainties [14]. Fig. 2 shows the distributionof both EVs’ capacities and initial SOCs.
(a) (b)
0 0.2 0.4 0.6 0.8 10
0.05
0.1
0.15
0.2
SOC
Pro
ba
bil
ity
0 20 40 60 80 1000.01
0.02
0.03
0.04
0.05
0.06
0.07
Capacity (kWh)
Pro
ba
bil
ity
Fig. 2. (a) Capacity distribution of coming EVs. (b) SOC distribution ofcoming EVs
2) Objective function of EV’s driver: this paper tries tomodel different behaviours of EV’s driver. Herein, it is sortedinto three types, namely, rich driver (RD), comfortable driver(CD), and poor driver (PD). The idea behind can be explainedin the interest and response of each. For the rich kind, thedriver has the thirst for charging his/her EV as much aspossible, while has less care about the discounted price offeredby CS. Meanwhile, the poor driver has the sensitive responseto the discounted price proposed from CS. Comparing withthe previous two kinds, the comfortable driver lies in betweenthem. Accordingly, the utility function can be divided into twoparts. The first indicates for the behaviour with respect to thecharging power, while the second cares about the discount onprice offered by the station. In either behaviour, the driverwants to maximize his/her charging power in the exist ofdiscount from the station, thus driver tries to maximize (3).
ui =
(− 1
2SOCiP
2i + P di Pi).(
eτ−1−1eτmax−1−1
+ 1) RD
(− 12SOCiP
2i + P di Pi).(
τ−1τmax−1
+ 1) CD
(− 12SOCiP
2i + P di Pi).(
ln(τ+e−1)−1ln(τmax+e−1)−1
+ 1) PD,
(3)
0 ≤ Pi ≤ Pmaxi (4)
where, SOCi is the state of charge of EV’s battery, P di isthe desired charging power of the on-board battery, Pi is thecharging power of EV, ψi is the incentive (the value of thesecond term of the utility), τ is the discount on the price, andPmaxi equals ψi.P di the maximum charging power of the on-board battery. The controller for each EV in each charging polereceives information from the battery management system ofEV i.e., SOC and maximum and charging station i.e., discount.For better understanding of the format of the utility functions,table I classifies the function form for the charging power andincentive parts of each driver behaviour.
TABLE IFUNCTION FORM OF CHARGING POWER AND INCENTIVE PARTS IN EACH
DRIVER UTILITY
Driver type Rich Comfortable Poor
Charging power part quadratic quadratic quadraticIncentive part exponential linear logarithmic
To catch the clue behind choosing the form of functions forthe two parts of each driver type with respect to their interests,Fig. 3 is dedicated for this purpose.
(a) (b)
0 10 20 30 40 50 60 70 80 90 100
Charging Power (%)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Uti
lity
Quadratic (RD,CD,PD)
0 10 20 30 40 50 60 70 80 90 100
Discount (%)
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Uti
lity
Exponential (RD)
Linear (CD)
Logarithmic (PD)
Fig. 3. (a) Driver’s preference with respect to charging power. (b) Drivers’preferences with respect to discount.
III. DISTRIBUTED ENERGY MANAGEMENT STRATEGY
From the control strategy point of view, the problem ismodeled as a noncooperative generalized stackelberg game.Number of coming EVs along with their SOCs may change ateach control instant (i.e., independent stages). In this game, thestation is designed to be the leader since it has the advantageof accessing all the charging poles, while EVs as followers.The proposed distributed energy management strategy relieson the generalized noncooperative game and the consensusnetwork-based learning algorithm. For better understanding oftheir functionalities in the system, it is more convenient toshow its envisioned hierarchical structure in Fig. 4. Here, twolayers are shown, the physical and the cyber layers besidesthe in-between control mapping. The first one represents thephysical system dynamics, while the second depicts suchcommunications between the nodes (i.e., components).
A. Generalized Noncooperative Game
Determining the charging powers of EVs in the station ateach time instant is the energy management problem in thispaper. It can conceptually be divided into two stages. The first,relies on the station, is to determine the available power forcharging EVs and the discount on price. The available powervalue will be the common constraint for all EVs. Afterwards,in the second stage, EVs will distributively set their chargingpowers.
1) First Stage: In either working mode of the station i.e., IMor GCM, it should support the available amount of power forcharging EVs i.e., Pava which lies between the lower and
Power Line
PVS BESS
EVnEV2
Physical Layer
Cyber Layer
Follower NodeLeader Node
Communication Link
EV1
Local ControllerInput SignalsOutput Signals
Managemernt Middlewareand Control
BESSPVS
mernt t t MiMiMiMiMiMiMiMi wareMaMaMaMaMaMaMaMaMana
LoadUtility
Charging Station
Fig. 4. The conceptual hierarchy scheme of the system.
upper bounds Pminava and Pmaxava , respectively.
Pava =
{Ppv + Pb − Pl GIM
Pg + Ppv + Pb − Pl GCM,(5)
Pminava ≤∑Ni=1 Pi ≤ Pava ≤ Pmaxava , (6)
Pg = P capg
N
N cap(1− SOCb), (7)
Pb = −(SOCb − SOCminb )Pmaxb , (8)
where Pg is the power supported from the grid, P capg is thecapacity power of the grid, N cap is the capacity of chargingpoles in the station, SOCb is the BESS state of charge,SOCminb is the minimum allowed state of charge of BESS,and Pmaxb is the maximum power of BESS. As it can be seen,there is always more power in GCM because of the help fromthe grid. Moreover, the power supported from the BESS isproportional to its state of charge. On the other hand, the realprice charged by the station is dynamics and related to thebasic price with the discount,
C =Cb
1 +NqNcapq
=Cbτ, (9)
where Nq is the number of EVs in the queue, C is thediscounted price, Cb is the basic price, τ is the discount, andN capq is the capacity of the queue. The discount supported
by the station comes from its condition as shown in Fig. 5.The station offers discount on the price when there are EVswaiting in the queue, i.e., no vacant charging poles (CPs), toincentivize the plugged-in EVs to charge by higher power toleave early i.e., allow vacant poles for waited EVs.
2) Second Stage: Since the game is treated as noncoopera-tive game thus, all players are selfish and try to increase theirown preferences. The leader i.e., station tries to maximize itsprofit through changing the basic price, while the followersi.e., EVs try to charge at higher powers with respect to theirincentives. The noncooperative power distribution among EVsis a generalized Nash equilibrium problem (GNEP). In thispaper, the distributed charging algorithm to find GNE is basedon Karush–Kuhn–Tucker (KKT) conditions of optimality and
Charging areaWaiting area
Charging Station
Vacant CPsCPs
Full CPs
Fig. 5. Working mechanism in the charging station
Lagrange multipliers method. The general optimization prob-lem for each player i.e., EV as well as KKT conditions areshown as follows,
OBJ : fmin = −uiS.t. g =
∑Ni=1 Pi − Pava ≤ 0,
(10)
Li(Pi, λi) = −ui + λig, (11)
∂Li∂Pi
= −∇Piui + λi∇Pig = 0, (12)
where λi is the Lagrange multiplier of each player. The mostsocially stable equilibrium is of interest and can be reachedby making all λis have the same value [15]. Since GNEPis convex—convexity of the objective function along witha linear inequality constraint, KKT necessary conditions aresufficient. The relaxed solution (λ = 0) and the constrainedone (λ 6= 0) at the most socially stable state can be derived asfollows under simplified case (given the discount a unit value),
P ∗i =
PdiSOCi
λ = 0
Pava −∑Nj=1
PdjSOCj
+ P di∑Nj=1
1SOCj
SOCi∑Nj=1
1SOCj
λ 6= 0,
(13)
Hence, the existence of GNE is proofed. After forming thederivative of the utility function (F ) and its jacobian (JF ),
F = −dui
dPi
=
(SOC1P1 − Pd1 )
.
.
.
(SOCnPn − Pdn)
, (14)
JF =
SOC1 0 · · · 0
.
.
....
. . ....
0 0 · · · SOCn
, (15)
it is evident that JF is positive definite on Pi, and so, F isstrictly monotone. Therefore, GNEP confesses a unique globalequilibrium solution.
B. Consensus Network
The aim in the proposed decentralized control is to let eachplayer updates its demand repetitively until a uniform value ofall λis is obtained. To this end, the concept of the consensusnetwork is utilised [8]. Because of the distributed nature, thereis an individual controller for each player who shares i.e.,communicates, only its own control variable (λi) with itsneighbours rather than revealing all its parameters. Herein, oneof the system’s nodes assists in reaching the power equilibriumstate by tuning its parameters corresponding to the power
mismatch i.e., violating (10). Without loss of generality, it willbe indexed as “1”. The proposed consensus-based distributedpower management (CDPM) algorithm for a single stage ofthe whole procedure is shown. Where, each local controllerat a node executes its belonging part. At the first step, aninitialization of all λis have been done with zero values togive maximum charging powers to EVs. Then, the consensusphase takes place which pursues to converge all the values ofλis to a single one. This can be achieved by updating eachnode’s λi utilizing the sum of weighted differences betweenthis node’s λi and its neighbors’ λs as in line 3. Where Niis the neighbors set of node i, and wi,j is the connectivitystrength between node i and j which should be chosen inthe range [0 1/n] to insure the intended convergence. Whenthe convergence is achieved, the power distribution amongthe players will be assigned accordingly within the localboundaries. Afterwards, the validity of the common constraintwill be checked. The algorithm will reach the Nash equilibriumthe time the constraint is satisfied. Otherwise, an iterativeprocedure takes place carrying modification on the λ1 as atranslation of the power mismatch as in line 13. It is worthto mention that the values of ε0 are ε1 are user defined, withbetter resolution at lower values sacrificing more iterations toreach convergence.
Algorithm CDPMI. Initialization1: λi(0) = 0 ; ∀i ∈ N
II. Consensus Phase2: while δλi > ε0 do ; ∀i ∈ N3: λi(k + 1) = λi(k) +
∑j∈Ni wi,j(λj(k)− λi(k))
4: Pi =Pdi − 1
ψiλi(k+1)
SOCi5: Pi = min(max(Pi, P
maxi ), 0)
6: end whileIII. Checking Constraint7: if ||
∑Ni=1 Pi − Pava|| ≤ ε1 then
8: Terminate9: else
10: Continue11: end ifV. Tuning 1’s Parameters12: P1(k + 1) = P1(k)− kp(
∑Ni=1 Pi − Pava)
13: λ1(k + 1) = ψ1(Pd1 − SOC1P1)
VI. Go back Step II
IV. SIMULATION RESULTS AND ANALYSIS
A. Specifications of the Charging Station and EVs
Most of the charging station parameters are listed in table II.The mentioned capacity of EV is just the mean value and thereal values follow the normal distribution as EVs’SOC withmean value of 0.4. The number of coming EVs is assumed tobe 100 per day according to Poisson distribution. Fast chargingtechnology is adopted in the charging station with tunable
charging rate 0.5 C to 1 C. Working time for the station isfrom 7 a.m. until 10 p.m i.e., 0∼900 min.
TABLE IISPECIFICATIONS OF THE CHARGING STATION AND EVS
Parameter PVS Grid BESS Charging/Waiting EVCapacity areas
Value 1200 1 2000 15/5 50kWp MW kWh pole/lot kWh
B. Static and Dynamic Power Dispatch
The workability of the game theory based control andconsensus algorithm during single stage is showed in the staticcase. The chosen instant time is at 60 min when six EVsare plugged in for charging. Fig. 6 shows the convergenceof λis under a given Pava. Here, λ1 is tuned according tothe mismatch power and λ2 to λ6 are identical. As it can beseen, the convergence is fast enough to be implemented inapplications which proofs the efficiency of both game theoryand the consensus network algorithm. Power distribution of
0 5 10 15 20 25
0
1000
2000
3000
4000
5000
6000
7000
1
λ
λ
λ
2
3
λ
λ
λ
4
5
6
Iteration
lambda
Fig. 6. Number of iterations for convergence of λis.
EVs with other parameters are shown in table III. Since thethe required powers of EVs are less than the available power,all EVs are charging at their maximum powers. It can be seenclearly that low-SOC with high capacity will be charged athigher power.
TABLE IIIPOWER DISTRIBUTION AT INSTANT TIME 60 MIN.
EV1 EV2 EV3 EV4 EV5 EV6
Capacity (kWh) 49 43 46 41 54 51SOC (%) 0.39 0.40 0.35 0.31 0.37 0.33Pi (kW) 24.4 21.3 22.8 20.2 26.9 25.4
Parameter Ppv Pb Pava ModeValue 114.67 (kW) 166.96 (kW) 291.34 (kW) IS
Showing the whole charging procedure of 100 EVs mean-while the working time of the station is relatively messy.Hereinafter, the game theory based power distribution of theprevious six EVs are presented which can be extended tothe rest. It can be concluded from Fig. 7 (a) that each EVis charging according to its preference under the specificSOC and Pmaxi at each control instant. The charging stateremains until it reaches the final demanded SOC. It is clearthat SOC rate is proportional to charging rate. In Fig. 7 (b)rather than showing the dynamics in the whole day, only the
station working time period is illustrated. First, the generatedpower of PVS is showed which matches the beta distributiondiscussed before. Then the condition of the coming EVs inthe charging and waiting areas are illustrated which followspoisson distribution is illustrated. Moreover, Due to the waitingEVs, the station offers a discount translated into differentincentive values in the three types of drivers. To illustrate theaffect of the incentive on the different type of EV drivers,Fig. 7 (c) is depicted. Here, EV1, EV2 and EV3 are chosen,similarly to EV1, EV3 and EV5 in part (a), to represent RD,CD and PD, respectively. Three scenarios are dedicated C1,C2 and C3 which adopt the incentive range of ’non’, ’lowestvalue’ and ’highest value’, respectively. As it can be seen, atC2, the increment in charging rate of EV3 is the highest thenEV2, EV1 which are compilable with the drivers’ behaviours.At C3, all EVs have the same rate of increment in charging,since it is the extreme high value of incentive. SOC responsehas a compilable results with what have just explained inthe power response. Showing the powers of BESS and Pava,BESS’s SOC, the station modes, scalability study and othersare of great interest which is kept for future extension of thepaper due to the current limitation.
V. CONCLUSION
This paper proposed a behaviour-based distributed energymanagement for charging EVs in PV. The power distributionproblem was modeled as a noncooperative stackelberg game.The station was designed as leader while EVs as followerswith different preferences. The existence of the equilibriumamong them was proofed at each control instant. The learning-based consensus network was utilised to reach the equilibriumin a distributed way. In the simulation section, static anddynamic analysis along with behaviours of the EVs’ driversto the discount on the power price offered by the station wereshowed. Effectiveness and workability of the proposed energymanagement were proofed through the results.
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Time (min)
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(a)
(b)
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500
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PV
Po
we
r (k
w)
0 100 200 300 400 500 600 700 800 900 10000
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m. E
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s S
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RD,incen
CD,incen
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(c)
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s P
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ers
(k
w)
EV1
EV2
EV3
20 40 60 80 100 120 140 160 1800
0.5
1
EV
s S
OC
s
C1C2C3
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