A Statistical Modelling and Analysis of PHEVs' Power Demand in Smart Grids
AA Statistical Modelling and Analysis of PHEVs’Power Demand in Smart Grids
Farshad Rassaei, Wee-Seng Soh and Kee-Chaing Chua
Department of Electrical and Computer EngineeringNational University of Singapore, SingaporeEmail: { f.rassaei, weeseng, eleckc } @nus.edu.sg Abstract —Electric vehicles (EVs) and particularly plug-inhybrid electric vehicles (PHEVs) are foreseen to become popularin the near future. Not only are they much more environmentallyfriendly than conventional internal combustion engine (ICE)vehicles, their fuel can also be catered from diverse energysources and resources. However, they add significant load on thepower grid as they become widespread. The characteristics of thisextra load follow the patterns of people’s driving behaviours.In particular, random parameters such as arrival time anddriven distance of the vehicles determine their expected demandprofile from the power grid. In this paper, we first present amodel for uncoordinated charging power demand of PHEVsbased on a stochastic process and accordingly we characterizethe EV’s expected daily power demand profile. Next, we adoptdifferent distributions for the EV’s charging time following someavailable empirical research data in the literature. Simulationresults show that the EV’s expected daily power demand profilesobtained under the uniform, Gaussian with positive support andRician distributions for charging time are identical when thefirst and second order statistics of these distributions are thesame. This gives us useful insights into the long-term planningfor upgrading power systems’ infrastructure to accommodatePHEVs. In addition, the results from this modelling can beincorporated into designing demand response (DR) algorithmsand evaluating the available DR techniques more accurately.
I. I
NTRODUCTION
The current state-of-the-art in information technology (IT)and data processing are going to be employed extensively insmart grids [1]. Widespread deployment of advanced meteringinfrastructure (AMI) enables real-time and two-way informa-tion exchange between demand side users and the electricutility. This evolution affects all different segments of the gridincluding generation side, transmission, distribution, as wellas the demand side.Traditionally, the utility designs and installs the power grid’sinfrastructure such that it can provide power to users’ adversedaily power demand profiles similar to that shown in Fig. 1.This power demand profile has a significant peak-to-averageratio (PAR) that can potentially reduce the power grids’efficiency and incur exorbitant costs for developing the powergrid’s infrastructure, i.e., increasing the power generation,transmission, and distribution capacity of the power grid. Thisextra capacity is just to serve the power demand of users duringpeak-time periods. Therefore, this drawback has motivatedintensive research on strategies that can utilize the existingpower grid more efficiently so that more consumers can beaccommodated and served without developing new costly D e m a nd ( k W ) Fig. 1. Annual mean daily power demand profile for domestic electricityuse in UK [5]. infrastructure. The main objective of these strategies is to makethe demand responsive [2]. Similar power efficiency concernshave become crucially important for supporting larger numberof tenants in green cloud data centres [3].Demand response (DR) is predicted to become even moreimportant as the use of new electricity-hungry appliances suchas plug-in electric vehicles (PEVs) or plug-in hybrid electricvehicles (PHEVs) is becoming more widespread. Typically, oncharging mode, they can double the average dwelling’s energyconsumption, with current PHEVs consuming 0.25-0.35 kWhof energy for one mile of driving [4].On the other hand, PEVs have several important advantagescompared to internal combustion engine (ICE) vehicles. Notonly do they have lower maintenance and operation costs, theyalso produce little or even no air pollution and greenhousegases in locales where they are being used [6]. Above that,they offer valuable flexibility as their fuel can be catered fromdiverse sources and resources, e.g., nuclear energy and windpower [7].However, in spite of their vast advantages, the market sizeof PEVs has been slower than expected as their adoption facesseveral barriers. One key reason is the extra cost of theirbatteries. In addition, the shortage of recharging infrastructurecauses range anxiety for pure electric vehicles’ drivers. But,plug-in hybrids resolve the latter problem for pure electricvehicles, by having a combustion engine which works as abackup when the batteries are depleted, yielding to comparable a r X i v : . [ c s . C E ] J u l riving ranges for PHEVs to conventional ICE cars [8].Although it makes sense to envisage the number of electriccars increasing, it is hard to see that the electricity infrastruc-ture capacity growing with the same rate concurrently. Thus,the ramification of introducing a large number of PHEVs intothe grid has become an important avenue for research in recentyears [9]. First, we need to ask how uncoordinated charging,i.e., the battery of the vehicle either starts charging as soon asplugged in or after a user-defined delay, can affect the existingpower grid. Next, we need to ask, considering this demand asa worst-case scenario, how we can satisfy it efficiently whenwe have information exchange capability and intelligence in asmart grid.There are several prior literature on modelling the impactof uncoordinated charging of PHEVs. However, most of themrequire much detailed information about passenger car travelbehaviour, e.g., [10] and [11]. Not only are the models mostlycomplicated and very test-oriented, but the sensitivity of thePHEVs’ charging load to different parameters is not also clear.Moreover, most of them do not provide expected daily powerdemand due to EVs, particularly when EVs are charged inhouseholds rather than in charging stations. For instance, [12]provides a spatial and temporal model of electric vehiclescharging demand for fast charging stations situated aroundhighway exits based on known traffic data. In [10], a utilizationmodel is proposed based on type-of-trip. The authors in [6]have used random simulation and statistical analysis to fit adistribution for the overall charging demand of PHEVs mainlyfor probabilistic power flow (PPF) calculations. In [13], thedaily load profile is modelled by using queuing theory andthe approach is suitable mainly for accurate short-time loadforecasting.Furthermore, since PHEVs are considered as the main com-ponent of the residential flexible electricity demand, numerousresearches have been carried out for PHEVs’ DR, e.g., [14]and [15]. Additionally, their storage capacity can be usedfor improving the power grid’s reliability, e.g., in terms offrequency control [16]. But, the main drawback in most ofthese demand response works is that they do not consider theinherent randomness of this demand in the first place.Therefore, in this paper, we present a stochastic model foruncoordinated charging power demand of a typical PHEV byformulating it as a stochastic process based on the arrivaltime and driven distance of the vehicles. Moreover, we derivePHEV’s expected daily power demand according to this modelfor arbitrary random distributions of arrival time and chargingtime. This gives us useful insights into the long-term planningfor upgrading the power systems’ infrastructure to accommo-date PHEVs. In addition, the results from this modelling canbe incorporated into designing DR algorithms and evaluatingthe available DR techniques more accurately.The rest of this paper is organized as follows. Section IIprovides the system model. Statistical analysis is addressed insection III. Numerical results and simulations are representedin section IV. Finally, section V concludes this paper. Energy Source Retailer
Customer 1 Customer 2 Customer N
Energy Market
Fig. 2. Basic model of a smart energy system comprised of multiple loadcustomers which share one energy source retailer or an aggregator.
Processing agent
Flexible load such as PHEV, washing machine, dishwasher, …
Inflexible load such as refrigerator, lighting, air conditioning, …
User i To the retailer
Fig. 3. Load segregation of a user according to power demand flexibility.
II. S
YSTEM M ODEL
In this section, we describe the energy system model andintroduce the layout of this study. Fig. 2 represents a basicpower system model where multiple energy customers shareone energy source retailer or an aggregator [2] and [17].Consumers’ total load consists of two different types of load;flexible load and inflexible load (see Fig. 3). Loads which need on-demand power supply (e.g., refrigerators) are consideredas inflexible, whereas loads that can tolerate some delays inpower supply (e.g., PHEVs) are assumed as flexible loads [17].Fig. 4 displays the demand flexibility of a flexible appliancefor different users. A certain job, ordered by user j , may taketime T j to be completed. Moreover, the users set not only thedesired job but also the deadline by which the job should beaccomplished. In this case, we may recognize the followingthree random variables for a generic flexible appliance: • Start Time shows the time when the user lets the powergrid connect to the appliance, and can potentially startdelivering energy. • Operating Time indicates the time interval required foraccomplishing a certain job, e.g., the ordered charging start time end time T user 2 Time of day start time end time T user N N start time end time T user 1 Fig. 4. Time setting for accomplishing a certain job on an appliance fordifferent users during a day. levels and modes (fast charging or slow charging) forPHEVs, which differs from one user to another. • End Time represents the deadline specified by the userfor accomplishing the task of the appliance.Hence, in general, we need to take into account this ran-domness when we investigate the overall behaviour of thesystem. Moreover, to design and analyse DR techniques moreaccurately, we should consider this stochasticity which comesfrom the patterns of people’s living behaviours and appliancespecifications.Therefore, in general, we can formulate the uncoordinatedpower consumption for an appliance operating a particular jobas follows: x ( t ) (cid:44) (cid:40) a t ≤ t < t + T otherwise (1)where we consider instantaneous power consumption as arandom variable a and assume that power consumption instandby mode is negligible. Additionally, T and t are theoperation time and the job’s start time, respectively. Theseparameters are random in general (see Fig. 5).In addition, here, we are mainly interested in knowing thedaily power consumption profiles, i.e., the power consumptionbehaviour throughout a typical 24-hour day. Therefore, wecalculate (1) in modulo 24-hours and then project the resultsonto a 24-hour day. In this case, some realizations of thestochastic process defined in (1) can be displayed as shown inFig. 6. This figure shows (1) for ten different users in a bargraph with one hour time granularity.Furthermore, a DR technique affects x ( t ) and changesits statistics. This process can be modelled as if x ( t ) ispassed through a system as shown in Fig. 7. Therefore,the information about the statistics of the input helps todesign the system such that the resulting random process y ( t ) fulfills the desired objectives of the DR techniques. The powerconsumption profile y ( t ) results from both the DR algorithm Time T x(t) a t Fig. 5. Demonstration of a typical form of x ( t ) . TimeUsers D e m a nd ( k W h ) Fig. 6. Some realizations of the stochastic process defined in (1) in modulo24-hours. and the particular statistics of the original power consumptionprofile.We can assume probability distribution functions (PDFs)for these two random variables, for instance, according tosynthesized models obtained from experimental data, e.g., in[18] for PHEVs. Here, focusing on PHEVs, we assume t and T have independent PDFs that can be found from empiricaldata. For example, for t , as the arrival time, a Gaussiandistribution is suggested in [18]: t ∼ N ( µ, σ ) (2)where µ and σ denote the mean and variance of the Gaussiandistribution, respectively. For PHEVs, there also exist differentcharging modes as described in Table I. The charging modemay be considered related to the other random variable in(1) which is a . But, we note that there is a tight correlationbetween a and T . This is obvious due to the fact that on fastcharging modes the charging time T is much shorter. statistics of original power profile statistics of final power profile DR system x(t) y(t) Fig. 7. Demand response technique modelled as a system.ABLE ID
IFFERENT T YPES OF C HARGING O UTLETS ( HTTP ://
WWW . TESLAMOTORS . COM /)OUTLET V/A kW MILES/1-HOUROF CHARGINGStandard 110 / 12 1.4 kW 3Newer Standard 110 / 15 1.8 kW 4Single Fast 240 / 40 10 kW 29Twin Fast 240 / 80 20 kW 58
III. S
TATISTICAL A NALYSIS
In this section, using the aforementioned definition of x ( t ) ,we calculate E [ x ( t )] which represents the expected value ofpower consumption for a certain appliance. This expectationcan be expressed by the following proposition for PHEVs(refer to the appendix for the proof). Proposition III.1.
Given f t ( · ) and f T ( · ) as the PDFs of theindependent random variables arrival time t and chargingtime T for a PHEV, the expected uncoordinated chargingpower demand can be expressed as: E [ x ( t )] = a × (cid:0) F t ( t ) ∗ [ δ ( t ) − f T ( t )] (cid:1) (3) in which, ∗ shows the convolution operation and δ ( t ) is theunit impulse function. Also, F ( · ) represents the cumulativedistribution function (CDF). We can calculate (3) for any given distribution analyticallyor numerically. Hereafter, we adopt different distributions forthe PHEV’s charging time T following some available empir-ical research data in the literature, as shown in Fig. 8, to studythe corresponding results of (3). We investigate four casesfor the distribution of T , namely, the uniform, exponential,Gaussian with positive support, and Rician distributions. Thesedistributions have different degrees of freedom (DoF) and allof them support T over [0 , + ∞ ) : • T: Uniform
In this case, we consider T to have uniformdistribution over the interval [ c, d ) . Then, E [ x ( t )] can beanalytically derived as stated in the following proposition(see the appendix for the proof). Proposition III.2.
Assuming t has a normal distribution withmean µ and variance σ and T has a uniform distribution overthe interval [ c, d ) , ≤ c < d , the expected uncoordinatedcharging power demand becomes: E [ x ( t )] = a × (cid:20) − Q ( t − µσ ) + σd − c ( c (cid:48) Q ( c (cid:48) ) − d (cid:48) Q ( d (cid:48) ) + f ( d (cid:48) ) − f ( c (cid:48) ) + d (cid:48) − c (cid:48) ) (cid:21) (4) where c (cid:48) = t − c − µσ , d (cid:48) = t − d − µσ . Also, Q ( x ) and f ( x ) aredefined as follows: T P r obab ili t y UniformExponentialGaussian with positive supportRician
Fig. 8. Uniform, exponential, Gaussian with positive support and Riciandistributions for T . Q ( x ) = 1 √ π ∞ (cid:90) x exp( − u du,f ( x ) = exp( − x ) √ π . • T: Exponential
The driven distance and hence the charg-ing time of an EV can be modelled by an exponentialdistribution [19]. For an exponentially distributed T withmean λ − , we have the following PDF: f T ( T ) = λ exp( − λT ) . (5) • T: Gaussian
When T has a Gaussian PDF with positivesupport as shown in Fig. 8, T has the following distribu-tion function: f T ( T ) = N ( T ; µ, σ | ≤ T < ∞ ) , (6) = 1 Q ( − µσ ) √ πσ exp( − ( T − µ ) σ ) , ≤ T < ∞ . (7) • T: Rician
Finally, we consider a Rician PDF for T havingthe following form: f ( T | ν, σ ) = Tσ exp( − ( T + σ )2 σ ) I ( T νσ ) (8)where ν ≥ and σ ≥ present the noncentralityparameter and scale parameter, respectively. I ( · ) is themodified Bessel function of the first kind with order zero.IV. S IMULATION R ESULTS
In this section, we consider Gaussian distribution for therandom variable t as the arrival time with µ = 19 and σ = 10 inspired from [18]. Furthermore, we considerfour cases for the distribution of the random variable T asdescribed in section III. First, we consider T to have a uniform D e m a nd ( k W ) UniformExponentialGaussian with Positive SupportRician
Fig. 9. A PHEV’s expected daily power demand profile for differentdistributions of charging time T . distribution over the interval [1 , . Thus, it will have µ = 6 and σ = 8 . . Second, we assume T to be exponentiallydistributed with mean µ = 6 . Third, we assume T to beGaussian distributed with positive support as presented in (7).In this case, we use the well-known accept-reject approachto generate the random values. Finally, we consider a Riciandistribution for T . In all cases (except for the exponentialdistribution), we set the parameters of the distributions suchthat they all have the same mean and variance. However, forthe exponential distribution case, we can only set either itsmean or variance to be the same as that of the others sincethis distribution has just one DoF. Based on an average 0.25kWh energy consumption for each mile of driving, we setall the parameters in (1). In addition, we assume a systemcomprising of N = 100 , PHEV users in our simulationsin order to obtain smooth curves representing the probabilisticexpectation.The results for the expected daily power demand of a typicalPHEV under the aforementioned settings are illustrated in Fig.9. As can be observed, the expected daily power demandresulting from the charging time distributions which possessthe same mean and variance tends to the same power profile.However, for the exponential distribution, since it has onlyone DoF, we see that its expected power demand differssignificantly from that of the others.Based on our proposed model and the obtained results,we observe that the expected uncoordinated charging powerdemand for a typical PHEV is much larger during 6 p.m. to1 a.m. compared to that during 8 a.m. to 2 p.m. in a one-dayframe.Next, in Fig. 10, we compare our obtained analytical resultfor the expected power demand according to a uniform distri-bution of T in proposition (III.2) with the simulation results ina one-day frame. As can be seen in this figure, the simulationresults follow proposition (III.2) closely, affirming the acquiredformulation. D e m a nd ( k W ) AnalyticalSimulation
Fig. 10. A PHEV’s expected daily power demand profile for a uniformdistribution of charging time T . V. C
ONCLUSION AND F UTURE W ORK
In this paper, we discussed the inherent randomness in thedemand for flexible appliances in general and for PHEVs inparticular. We considered random distributions for the arrivaltime and the charging time of PHEVs inspired by availableempirical data in the literature. Accordingly, we presented theuncoordinated charging power demand impact of a PHEV asa stochastic process based on these random variables. Next,we derived the expected daily power consumption profileaccording to this random process. Our simulation results showthat the EV’s expected daily power demand profiles obtainedunder the uniform, Gaussian with positive support and Riciandistributions for charging time are identical when the first andsecond order statistics of these distributions are the same. Ourobtained results introduce a simple description for the expectedpower demand of a typical PHEV and hence give us insightsinto the effect of adding each PHEV into the power system.The study presented in this paper can be extended and de-veloped in various ways. For example, the convergence of thedaily power demand for the aforementioned distributions needsto be proven. In addition, the results from this modelling canbe incorporated into designing DR algorithms and evaluatingthe available DR techniques more accurately.A
PPENDIX A. Proof of proposition III.1:
Since x ( t ) = 0 for t ≤ t − T and t ≤ t . Then, E [ x ( t )] becomes: E [ x ( t )] = a × P ( t ≤ t ≤ t + T )= a × P ( t − T ≤ t ≤ t ) . (9)Further, we can use the total probability theorem [20] toget [ x ( t )] = a × ∞ (cid:90) P ( t − T ≤ t ≤ t | T = T (cid:48) ) f T ( T (cid:48) ) dT (cid:48) = a × ∞ (cid:90) ( F t ( t ) − F t ( t − T (cid:48) )) f T ( T (cid:48) ) dT (cid:48) (10) = a × F t ( t ) − ∞ (cid:90) F t ( t − T (cid:48) ) f T ( T (cid:48) ) dT (cid:48) (11)for which we have taken into account the facts that ∞ (cid:82) f T ( T (cid:48) ) dT (cid:48) = 1 , and t and T are independent. Fur-thermore, we can express (11) in a more concise form byusing the definition of the convolution integral and theidentity f ( t ) ∗ δ ( t ) = f ( t ) as follows: E [ x ( t )] = a × ( F t ( t ) ∗ [ δ ( t ) − f T ( t )]) . (12)B. Proof of proposition III.2:
Since T is uniformlydistributed over the interval [ c, d ) , ≤ c < d , we canwrite (11) as follows: E [ x ( t )] = a × F t ( t ) − d − c d (cid:90) c F t ( t − T (cid:48) ) dT (cid:48) . (13)Then, by changing the integration variable from T (cid:48) to α = t − T (cid:48) , it can be rewritten as follows: E [ x ( t )] = a × F t ( t ) + 1 d − c t − d (cid:90) t − c F t ( α ) dα . (14)Further, we need to replace α with β = α − µσ to have E [ x ( t )] = a × F t ( t ) + σd − c t − d − µσ (cid:90) t − c − µσ F t ( β ) dβ (15)in order to be able to use the following formula for astandard normal random variable with CDF F ( · ) and PDF f ( · ) to calculate the last term in (15): (cid:90) F ( x ) dx = xF ( x ) + f ( x ) + c. (16)Also, we now set c (cid:48) = t − c − µσ and d (cid:48) = t − d − µσ forsimplicity to express (15) in the following form: E [ x ( t )] = a × (cid:20) − Q ( t − µσ ) + σd − c ( c (cid:48) Q ( c (cid:48) ) − d (cid:48) Q ( d (cid:48) ) + f ( d (cid:48) ) − f ( c (cid:48) ) + d (cid:48) − c (cid:48) ) (cid:21) (17)in which we used the equation F ( x ) = 1 − Q ( x ) . R EFERENCES[1] A. Ipakchi and F. Albuyeh, “Grid of the future,”
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