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The LyX User's Guide - Electrical and Computer Engineering

1. 4 92h Given rt and T5 the optimal value of the regulation service is r P ToP 738 1 kW h To evaluate the system we manually generate 100 zero mean random traces with To 45 min c 0 5 and maximum and minimum 1 and 1 AT respectively Then we multiply these traces by r P and lift them up by m to obtain final regulation signal trajectories We use these signals to charge the system of electric vehicles for a time duration with length Tj P We seek to determine whether the regulation signals can be safely absorbed by the system during this interval Moreover we wish to determine whether the residual capacity of the system at time t Tj Pt allows the system to be fully charged by time T We found that the generated trajectories almost always up to the predicted error lay within the f7 and Pen boundaries and the regulation service was safely provided over the interval 0 To Moreover residual capacity of the system at t Tj Pt was always large enough to allow the system to be fully charged by time t T using an input power less than Pr These results confirm our predictions in Section 5 From Theorem 4 it is easy to see that the optimal value of regulation service in the deterministic setting is 600 kW h This value is realized by mort rt 150 kW and PS 4 h B Dynamic optimization In this subsection we study how the value of reg ulation service increases by using dynamic optimization Figure 8 sh
2. value of S t at time t JAt 1 lt l k can be written as S t Sy At z 4 j l where the r s are the regulation signals that are unknown The value of the system residual capacity at time T must be less than or equal to C Mathematically this constraint can be written as S To lt C To guarantee that all vehicles will be fully charged by the end of the charging process constraint ii described above we need to ensure that there is enough time to fill the remaining capacity of the vehicle batteries C S T during the second phase of the charging process which has duration T To knowing that there is a constraint on the peak power of the main line Pr This constraint can be written as lt T To Using the above constrains the stochastic optimization problem that the utility should solve to select the regulation parameters m r and Ty is 22 max r To 5 m r To lo s t meds 0 To lt T Note that the regulation signals z s are stochastic unless we conserva tively model them using the maximal and minimal deterministic regulation sequences as will be discussed later and therefore S T5 is a random vari able in the optimization problem 5 As a result in general we need to use the statistical properties of S t when studying this optimization problem In the following we will study two particular cases of interest The first corresponds to the case where we assume that t
3. E Sortomme M A El Sharkawi IEEE Trans Smart Grid 2 2011 131 138 E Sortomme M A El Sharkawi IEEE Trans Smart Grid 3 2012 70 79 E Sortomme M A El Sharkawi IEEE Trans Smart Grid 3 2012 351 359 S Han S Han K Sezaki IEEE Trans on Smart Grid 1 2010 65 72 S Han S Han K Sezaki Stochastic analysis on the energy constraint of V2G frequency regulation IEEE Vehicle Power and Propulsion Con ference VPPC 2010 S Jang S Han S Han K Sezaki Optimal Decision on Contract Size for V2G Aggregator regarding Frequency Regulation 12th International 57 Conference on Optimization of Electrical and Electronic Equipment OPTIM 2010 22 Kashyap A Callaway D Controlling distributed energy constrained resources for power system ancillary services IEEE 11th Interna tional Conference on Probabilistic Methods Applied to Power Systems PMAPS 2010 23 N Rotering M Ilic IEEE Trans Power Syst 26 2011 1021 1029 APPENDICES Appendix A In this appendix we obtain the mean and the standard deviation of the system total residual capacity S t The value of S t at time t l At is given by l S t So X z At pe j l Using the linearity of expectation we can obtain the mean of S t as l E S t So p Elx At So ml At So mt 0 lt t lt T 30 j l 58 To obtain the variance of S t we note that x m 4 rvj 1 lt j lt k Here k is
4. as illustrated in Fig ting we used m 8 the two choices of m TL and m Pe are expected to perform similarly on average when T is close to Tj Pt We noted that the average values of the regulation service value in the stochastic and deterministic settings were improved to 913 9 kW h and 848 9 kW h respectively As suggested by these numbers the value of the regulation service can considerably increase by us ing the dynamic optimization and the difference between the stochastic and the deterministic settings reduces in the dynamic setting Figure I2 shows a geometric representation of the charging process in the stochastic setting 7 Conclusion and Future Work In this paper we studied how a fleet of electric vehicles can provide frequency regulation service while getting charged overnight Examples of potential fleets that can be used for this purpose include the DHL fleet of vehicles and the fleet of city transit buses We studied a scenario in which the fleet is charged by a variable rate signal overnight In this scenario we found the optimal values of the average charge rate and the maximum allowed deviation from the average Our objective function was to maximize 52 the value of the regulation service that can be offered to the grid Our results on a representative system indicate that if Pr the peak power of the fleet main line is at least twice as large as Pc the system can achieve much of the achievable reg
5. he contributions of this work are four fold 1 We introduce a framework to model regulation signals in a system that provides load side frequency regulation service This framework captures the performance boost gained from exploiting the opposing effect of regulation up and down signals It also helps to precisely characterize the uncertainty in the system residual capacity 2 We use an analytical approach to obtain the optimal values of the reg ulation parameters We study the system in the cases where regulation signals are deterministic worst case and stochastic We use our approach to study both a single shot optimization scenario carried at the start of the charging period and a dynamic optimiza tion scenario where the optimal control strategy is re evaluated sev eral times over the duration of the charging interval We show that most of the gains from dynamic optimization can be obtained by re evaluating the optimization problem at the midpoint of the charging interval Moreover the optimal value of the regulation service in the worst case deterministic setting nearly matches the stochastic setting with dynamic optimization We obtain an analytical condition under which a multi component sys tem can be modelled as a single large storage system thus achieving its maximum performance We also study how to satisfy this condition in practical systems and how the estimates of the system performance should be modified when
6. m and r lead to either Tj lt To or Ti gt To and show that both these assumptions lead to contradiction First consider the case where T lt T5 In this case we have Ty min Ti T5 Ti In this case it can be shown that for optimal values of m and r we must have m r 0 Otherwise if m r gt 0 we can fix the value of r and replace m by m m such that m r still remains greater than zero Now let Tf and T5 denote new crossing points associated with m and r It is not hard to see that TT will be larger than T3 To see this we note that the value of fme t is less than f P t for all t 61 fue t fob t So m t arag t So mt arag t t m m 0 The above inequality holds because m lt m and the time parameter t is always positive Since fj r t lt fre t for all t the crossing point fj with the top horizontal line i e 77 always lies at the right of the crossing pint fzP f with the same line i e T Thus the value of TT would be greater than T Because of the continuous nature of the modification in m we can choose m such that TT remains less than T7 Under this assumption the value of T defined as min TT T7 will be equal to TT Now it can be seen that in the new setting the value of r has not changed and T is greater than Ti As a result we have rTT gt rT Since Tj TT and T Ti this implies rTg gt rTo This is in contradiction with t
7. o 300r 200 i i i i i i i 0 7 0 8 0 9 1 14 1 2 1 3 1 4 TS power ratio unitless Figure 4 Optimal value and duration of the regulation service as a function of system residual capacity 750 T T T F Optimal 65 r y r r r m P e Optimal 700 A m P 2 al e mP Det Py Amp Det P 2 Bal Dat Pe z 650r S 5 Det P 2 E z 8 8 sl 600 2 8 gt 5 8 45 E B45 1 3 580 E E lE 4 soo 8 8 3 5 4 450 at Abo i i i i i m i i i i i i i i 07 0 8 0 9 i at 12 13 14 07 08 09 1 if 12 13 14 185 16 power ratio unitless power ratio unitless a Optimal value of the regulation service b Optimal duration of the regulation ser vice 76 Figure 5 Optimal value of the regulation service as a function of the correlation time of the regulation signals 1200 r r r e P 10 P 10 11008 P 10 F A P 105 1000 8 E 9 5 900r Ek 3 D 9 800L amp o 700 600 x 0 02 04 06 08 1 12 14 16 18 2 correlation time hour Figure 6 Optimal value of the regulation service as a function of the variance of the base regulation signal o 1200 T h E kW 1000 4 900 4 800 4 optimal regulation service 700 a 600 i i i 0 1 0 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 variance of the base signal 9 unitless TT regulation service K
8. 1600 1400r 1200r 1000r residual capacity kWh co o o 600 80
9. 20 kWh storage unit We as sume the fleet is available for charging from 10 PM to 6 AM thus the total charging period is 8 hours Using the notations introduced in the previous subsection we have n 80 C 20 kWh and T 8h If all vehicle batteries are empty at the beginning of the charging period the system needs a total energy of 80 x 20 1600 kWh to get fully charged by the next day s morning Given the length of the charging period 8 hours the input peak power is expected to be Pr gt 1600 kWh 8h 200 kW Because 36 we plan to provide regulation service the input peak power should be chosen greater than 200 kW Larger values of the input peak power are expected to result in higher values of regulation service We will study the effect of input peak power on the optimal value of regulation service later in this subsection We assume that the autocorrelation function of the regulation signals has a triangular shape As we will see in Section this is consistent with the results obtained from real world data We define the correlation time Tc of a triangular autocorrelation function as the time lag beyond which the autocorrelation function vanishes i e the time lag beyond which the incoming regulation signals are uncorrelated Based on the results obtained from the real world data see subsection 6 2 2 we set Tc 45min and 0 0 5 unitless We take the probability of error in the system to be P 10 Finally we assume
10. That is because regulation signals are added to each other over time and their uncertainties compound Kamboj et al La hh study the formation of electric vehicle coalitions and propose heuristic algorithms for determin ing the average charge rate and regulation service bids Although algorithms proposed in these studies are reasonable choices they are not guaranteed The residual capacity of a storage unit is defined as the total energy stored in the unit It is measured in units of energy kWh to result in the optimal value of regulation service in the system This is especially the case in unidirectional settings where charging of electric vehi cles and provision of regulation service should be done simultaneously and it is not clear how to select system parameters to effectively achieve both these goals In this study we aim to develop a mathematical framework that can be used to obtain rigorous values of the system parameters given the constraints imposed by both utility and owners of electric vehicle Another limitation of the above studies is that they all compute regulation service bids solely based on constraints of the individual battery units In practice however the main line connecting the aggregator to the utility also has a maximum tolerable power In this work we present a framework to model regulation signals and we study it both in a worst case deterministic setting and a stochastic setting 1 3 Contributions
11. approach can be the best solution In this figure we also show how the choice of the error probability impacts the optimal regulation service Clearly the higher the probability the higher the regulation service Finally in Fig 6 we study the effect of the variance of the base signal c on the optimal value of the regulation service for different values of the error probability As shown the optimal value of the regulation service decreases quickly as o increases That is because the variance of the system residual capacity S t increases as o increases see Eq 8 and the resulting in creased uncertainty leads to more conservative estimates of the value of the regulation service achievable in the system As in Fig 5 the diagram in Fig 6 presents only values of the regulation service greater than or equal to 600 kW h 5 4 Setting the physical parameters Pr and p In this section we study how to set parameters Pr and p in a system of electric vehicles that is planned to provide the regulation service This is a one time parameter design and should be performed when the infrastructure 41 of the fleet is designed We denote the obtained values of parameters by P2 des respectively and p As mentioned in Section 5 in order to get a good value for the regulation service we should have Q lt 1 irrespective of the value of the initial residual capacity So Hence the value of Pr should be set to be twice as large
12. are obtained m r The value of To is then equal to min 7 T5 e Deterministic with m Po In this method similar to the first case above the mean charge rate m is set to Pc and we allow for deviations of at most r min Fo Pr Po above or below this mean but we account for the worst cases where all regulation signals are equal to each other and equal to either Po r or Po r e Deterministic with m P In this method similar to the second case above the mean charge rate m and the deviation parameter r are 38 both equal to Pr 2 but again we account for the cases where all signals can be equal to each other and equal to either Pr or 0 Note that The orem 4 predicts that the value of the regulation service obtained from this method and the previous one are both optimal in a deterministic regime and equal to each other As illustrated in Fig 3 the amount of regulation service provided by taking into account the stochastic nature of the signals is in general greater than the value of the regulation service provided by the deterministic approaches Moreover not surprisingly the values of regulation service for the two deter ministic case are equal to each other As illustrated in Fig B Q 1 yields nearly the highest regulation service This implies that the utility should choose to design the system so that Pj is at least equal to 207 We will discuss this issue later in the present section Next we st
13. assumed to be a decreasing function of t and T gt T5 the right hand side of the above equality is greater than one and so is the left hand side As a result we have r T3 gt rT gt Since Tj Ty and To Th this implies that r 7j gt rTo This result is again in contradiction with 66 the assumption that the values of m and r are the optimal solutions to the problem Appendix C In this appendix we prove Theorem 3 In proving this theorem we assume regulation signals and the system residual capacity can be approx imated by continuous signals This is a reasonable assumption as the time span between regulation signals At is much smaller than other time scales in the problem With this approximation Eq H becomes At t t S t Sot At S sj S s r dro Se mter v r dr 35 0 0 j 1 In this continuous form variance of S t is given by o2 t r ce t where oX t nu amp 2 f rel di 36 To show the above equation one can expand variance of S t in 25 via a double integral and use change of variables to obtain the final result An alternative way to obtain Eq 36 is to show that Eq 36 is equivalent to Eq when At t Now we show that if R 7 gt 0 for all 7 then function g t gon will be a decreasing function of t Since g t is always positive we can equivalently show that f t g t zo is a decreasing function of t To show that 67 f t is a decreasing function of t
14. lem which includes only variables m and To Pr T To To max Be ao 4 s t mod gt 0 To lt T T To P m Ja oclo 2 0 15 T To P m CoOL lt p 16 S mTy FR C 17 To gain insights on the above optimization problem it is useful to plot constraints and l7 in the m To plane see Fig 2 To do so we first note that inequality 15 corresponds to the area above the curve C1 defined by m G 18 Similarly we note that the inequality presented in I6 corresponds to the 30 area below the curve C5 defined by T To Pr 2a oo To m P 19 At Ty T the curves C and C5 take the value 0 and Pz respectively Moreover it can be seen that these curves are symmetric around the line m P 2 in the m To plane and thus always intersect at a point with height P 2 Fig B Finally let the curve represented by Eq L7z be C3 We rewrite its equation as C So PL To T 2 T 2 m TT 20 Let Po Note that this quantity represents the average power required to fill all the batteries by the end of the charging period To study the behavior of the system we find it useful to define the power ratio Q as follows Po q P 2 21 In plotting the curve C3 we distinguish between the following three regimes i Q lt 1 or Po lt ii Q gt 1 or Po gt 5 and iii Q 1 or Po PL One can see that the convexity of this curv
15. load to take part in the frequency regulation market it needs to have a consumption rate that is significant with respect to the power fluctuations in the grid A coalition of multiple loads may be required to achieve this minimum required input power The integration of electric vehicles with the power grid to achieve a mutual benefit is termed vehicle to grid V2G 2 This integration may take two forms bidirectional and unidirectional In bidirectional integration electric vehicles can both receive energy from the grid and send energy to it de pending on the consumption profile of other loads in the network In the unidirectional case electric vehicles can only receive energy from the grid In this case electric vehicles can still be used for providing both regulation up and regulation down services This is because the input power to the elec tric vehicles can be decreased or increased depending on the status of other loads in the network Due to the relatively high loss currently associated with AC DC and DC AC power transfers the bidirectional case is thought to be infeasible in practice In contrast in the unidirectional case no extra infrastructure other than the communication and control system is required 16 In this work we therefore focus on unidirectional charging of electric vehicles 1 2 Relationship to prior work The use of electric vehicles for providing frequency regulation service has received much attention
16. mTo aroo To C 11 To loAt C Star lt T T 12 Pr In this problem T is an integer multiplier of At In the following we will 26 relax this constraint when solving the optimization problem 10 and allow To to take any real value within the interval 0 T This is a reasonable assump tion because At is much smaller than other time scales in the problem This problem can be complex depending on the function oo t which is a function of the autocorrelation function of v Before providing analytical insights and solutions to this problem for different examples we first provide some geometric interpretation 5 1 2 Geometric representation We find it useful to illustrate the constraints presented in 10 using a geo metric representation Such a geometric representation is given in Fig In this diagram the vertical axis denotes the system residual capacity and the horizontal axis represents time Each path within the presented rectangle illustrates a sample trajectory for charging the system based on given tra jectories of regulation signals All feasible trajectories must start from the bottom left corner of the rectangle 0 So and end at the top right corner T C Because of the limitation on the peak power of the system denoted by Pz all feasible trajectories must lie within the two parallel lines with slope Py one crossing point 0 So and the other one point T C Consider a variable rate reg
17. of the charging period The value of the regulation service to a utility depends on two factors the magnitude r of power variations from the mean value m and the duration To over which the regulation service is provided In current regulation service markets the value of regulation service is measured in MW h For example if a generator agrees to vary its output up to 5 MW above or below its mean output for 1 hour the value of the resulting regulation service will be 5 MW h Motivated by these considerations in our work the utility will select for the fleet the two regulation parameters m and r to maximize the objective function which is the product of the magnitude of the maximum variation in the regulation signal i e r and the length of the interval over which regulation service is provided i e To Batteries are by far the most expensive part of electric vehicles and their 12 charging and discharging need to obey a set of tight constraints to deliver the expected lifetime Typically the battery management system in the battery pack will tell the charger how much power it is willing to accept i e a maximum limit at a given time depending on the current state of charge temperature age etc As a result our assumption that any regulation signal can be obeyed is very strong By making it we obtain results that should be considered as a bound on the performance that can be achieved by practical systems 3 Model This sec
18. the batteries of the vehicles are initially charged at 2596 on average This leads the system initial residual capacity to be 2596 x 80 x 20kWh 400 kWh and the required average power Po to be C Sy 1600 kWh 400 kWh Poo a 150 kW Figure 3 plots the value of the regulation service rTo as a function of the input peak power Pr In plotting Fig we keep Po constant at 150 kW and vary Pzr in the range 200 400 kW We find it useful to normalize the horizontal axis and plot the value of regulation service as a function of the 37 power ratio Q defined in Figure 3 compares the optimal value of the regulation service obtained when assuming a Gaussian cumulative sum of the regulation signals with a triangular autocorrelation function with the value of the regulation service obtained from the following four sub optimal methods e Stochastic with m Pc In this method the mean charge rate m is taken equal to Po and the value of r is then equal to min Po Pr Po Given the values of m and r the functions f t and f t can be evaluated and from those values of T and T5 are obtained The value of Ty is then equal to min T To e Stochastic with m DL In this method the mean charge rate m and deviation parameter r are both taken equal to PL As in the previous case given the values of m and r the functions f t and m r low t are evaluated and from those values of T and T
19. the proper operation of the system In general it is interesting to extend the model presented here to the settings where the base signal v t can have a nonzero mean A Single shot optimization In this subsection we use the real world regu lation traces described above to assess the value of the regulation service in a system using single shot optimization To study the system in the stochas tic setting we use the parameters c and To obtained from the training set to solve the optimization problem for the the previously described system of electric vehicles As before we assume that the system is initially 2596 charged and the tolerable error in the system is P 10 With these parameters we obtain m P r t 150 kW and 7p 4 89 h leading to an optimal regulation service value of r P T P 733 36 kW h 50 Next we multiplied the traces in the test set by r P and lifted them by m to obtain trajectories for charging the system The test set consisted of eight days For two days in this set the regulation data was available for only part of the day We divided the regulation traces in each test day into blocks of approximately 8 hours This data taken together provided a total of 22 blocks of regulation traces We used these traces to charge the system for a period of T P 4 89 hours and then switched to a deterministic phase to completed the charging process by the end of the charging period at T 8h Figure
20. the total number of regulation signals sent to the fleet With these notations we have As a result To obtain Var S vj we note that E Y UG p R t l S t So s At j l l 94 X m T rv At j l l So9 mlAt rAt NGC j l l os t Var S t rAt Var vj 31 j l jl jal ji Elv 0 Let E v v be the autocorrelation function of the zero mean signal v We have 59 E 5 vj 1 I E Var vj e Il n S n v j 1 l l E v7 2E vj 2 1 lt j lt jxn l IR 0 2 R i l lt j lt jgn 1R 0 2 1 8 0 1 R 0 2Y R0 25 iu 32 Plugging the value of varli 1 vj from Eq B2 into BI we obtain 820 GAL Var 5 yy 323 i 1 r At I R 0 pea no 2 xe l 1 r S AGD 0 2 RA 2 At SiR i i l l 1 P AH 2 RO 2 At S in i i 1 k 1 res 60 where a2 t is given by Eq Appendix B In this appendix we prove Theorem 2 Let m r To be the optimal solution to L0 we have dropped the superscript opt for simplicity To prove Theorem 2 we show that the values of T and T corresponding to m r co t i are equal to each other if g t is a decreasing function of t This is equivalent to saying that both inequalities L1 and 12 hold at their equality under the above condition We use proof by contradiction We assume the optimal solutions of of
21. values of faz r TT and fpe Ti are both equal to C and thus equal to each other p m r So m Tt ar oo T So mT aroo Th Using m r and m r and removing So from both sides we can 63 simplify the above equation as follows r Tt ar o TY rT oaroo T1 Now we rewrite the above equality as 00 T1 Ti ao 11 rTy lta T rT l a or px go T1 rTy lta TR rT 1 amm colt t Since is a assumed to be a decreasing function of t and TT gt T the right hand side of the above equality is greater than one and so is the left hand side As a result we have r T gt rT The facts that Tj T and To Ty imply r Tj gt rTo This is in contradiction with the assumption that the values of m and r are the optimal solutions to the problem Next we consider the case where T gt T for the optimal solutions m and r and again show this is in contradiction with the optimality of the solution In this case 7o min Ti T5 T It can be shown that under the above assumption we must have m r Py for the optimal values of m and r Otherwise similar to the procedure mentioned in the previous case we can fix the value of r and increase the value of m such that the objective function is increased 64 Next we assume m r Pr We show that under this assumption if we increase the mean value m which will be at the expense of reducing r the overall objectiv
22. we show that its derivative is negative for all t gt 0 Let h t o t Then f t O and the derivative of f t will t be W P h t 2t th t 2h t P i t f 37 The denominator in Eq 27 is always positive when t gt 0 Thus to show f t lt 0 it suffices to show that the numerator is negative for t gt 0 By taking derivative of Eq we obtain h t 2 f R r dr Substituting the values of h t and h t in the nominator of Eq 37 we obtain th t 2h t 2t R r ar At R r dr 2 i Ridi 9 Ea Rd 38 0 Note that 7 lt t for all 7 0 t and thus for positively valued functions R T TR T tR T T 0 t This implies that the right hand side of Eq is negative as desired Appendix D In this appendix we prove Theorem 3 We first note that the optimization problem 5 in the deterministic case becomes 68 max r To 39 m r To s t m r To gt 0 To lt T Wie gt 40 m r lt Pr 41 So 4 n r Ty C 42 Coren lt T Ty 43 We consider two cases Case 1 Q lt 1 We first show that in this case 75 gt T in any optimal solution of the problem To show this we use proof by contradiction We first note that if 75 lt T in an optimal solution then m r must equal to Pr Otherwise one can fix r and increase To here equal to T5 by slightly increasing the value of m such that 75 increases and still remains less t
23. 0 2002 16 26 W Kempton and J Tomic J Power Sources 144 2005 268 279 W Kempton and J Tomic J Power Sources 144 2005 280 294 J Tomic and W Kempton J Power Sources 168 2007 459 468 Guille C Gross G Energy Policy 37 2009 4379 4390 A Brooks Vehicle to grid demonstration project grid regulation an cillary service with a battery electric vehicle California Air Resources Board 2002 W Kempton et al A Test of Vehicle to grid V2G for Energy Stor age and Frequency Regulation in the PJM System Online Available http www magicconsortium org 2008 Enbala Power Networks M http www enbala com S B Peterson J F Whitacre and J Apt J Power Sources 195 2010 2377 2384 S B Peterson J Apt and J F Whitacre J Power Sources 195 2010 2385 2392 Brooks A Lu E Reicher D Spirakis C Weihl B IEEE Power and Energy Mag 8 2010 20 29 56 13 14 15 16 17 18 19 20 21 http www udel edu V2G Tools html Kamboj S Decker K Trnka K Pearre N Kern C and Kempton W Exploring the formation of Electric Vehicle Coalitions for Vehicle To Grid Power Regulation AAMAS workshop on Agent Technologies for Energy Systems 2010 Sachin Kamboj and Willett Kempton and Keith S Decker Deploying Power Grid Integrated Electric Vehicles as a Multi Agent System AA MAS workshop on Agent Technologies for Energy Systems 2011
24. 11 shows the geometric representation of the charging process in the system As presented the regulation service is successfully provided over the interval 0 TS Pt and there is always enough time for the system to get fully charged by the end of the charging period The optimal values of the regulation service and the the regulation service time in the deterministic setting are independent of the parameters o and To and remain at r P T P 600 kW h and T5 4h respectively B Dynamic optimization In this subsection we use the PJM regulation traces to study the fleet of vehicles in a dynamic setting To this end we again use the 22 blocks of regulation data obtained from the test data to assess the value of the regulation service We consider both stochastic and deterministic settings Remember from the previous subsection that an appropriate value for Ty in a dynamic setting can be a time duration slightly less than Tj P in the single shot setting Therefore we set the value of T to 4 88 h in the stochas 51 tic and to 3 99 h in the deterministic setting Note that these are slightly less than the values of Tj Pt obtained above for the corresponding single shot settings These values of T achieve much of the improvement in the value of regulation service that is obtainable from the dynamic optimization while adding only a single updating point to the system In the deterministic set PL at each update point However
25. Optimal Contracts For Providing Load Side Frequency Regulation Service Using Fleets of Electric Vehicles Hadi Zarkoob Srinivasan Keshav Catherine Rosenberg Stanford University 450 Serra Mall Stanford CA 94305 USA b University of Waterloo 200 University Ave West Waterloo ON N2L 3G1 Canada Abstract We focus on the charging process of a fleet of electric vehicles overnight for providing load side regulation service At the heart of this complex problem the goal is to transfer a certain amount of energy to the fleet by a given dead line however when and how fast the energy is sent is flexible We examine a unidirectional setting in the cases where regulation signals are determin istic worst case and stochastic We study both a single shot optimization scenario carried at the start of the charging period and a dynamic optimiza tion scenario where the optimal control strategy is re evaluated several times over the duration of the charging interval We show that most of the gains from dynamic optimization can be obtained by re evaluating the optimiza tion problem at the midpoint of the charging interval Moreover the optimal value of the regulation service in the worst case deterministic setting nearly matches the stochastic setting with dynamic optimization We validate our Corresponding author Tel 1 650 223 3955 Email addresses hzarkoob stanford edu Hadi Zarkoob keshav uwaterloo ca Srinivasan Keshav ca
26. W h Figure 7 Geometric representation of a typical dynamic charging process residual capacity time Figure 8 Regulation service as a function of the number of updates 1050 TT I Stochastic Deterministic with m P 2 1000 Deterministic with m P 950r 900r 850r 800r 750r 700r 650r si 1 2 3 4 4 92 6 7 8 time span between updates hour 78 Figure 9 QQ plot of the PJM regulation traces versus standard normal quantiles of regulation data 1 standard normal quantiles 0 1 Figure 10 Autocorrelation functions of the regulation traces obtained from PJM The data is provided for four different days Training day 1 Training day 2 5 10 15 20 25 30 35 40 45 50 Lag min Training day 4 19 1 0 8 4 0 8 0 6 4 0 6 04 4 0 4 0 2 0 2 0 0 0 5 10 15 20 25 30 35 40 45 50 0 Lag min Training day 3 L5 o 0 8 j 0 8 0 6 j 0 6 0 4 4 0 4 0 2 i 0 2 0 0 0 5 10 15 20 25 30 35 40 45 50 0 Lag min 79 5 10 15 20 25 30 35 40 45 50 Lag min Figure 11 Testing the single shot charging scheme using PJM regulation traces 1600 1400 1200 1000 residual capacity KWh eo eo o 600 400 4 time hour Figure 12 Testing the dynamic charging scheme using the PJM regulation traces
27. an always be provided We compared the regulation service obtained from the stochastic and deterministic settings in both single shot and dynamic optimizations We noted that although stochastic setting is expected to result in higher values of the regulation service this difference in performance reduces with dynamic optimization This implies that even without knowing the stochastics of the system one can obtain high values of regulation service using a dynamic deterministic scheme The results presented in this work can be extended in multiple directions First we can study the optimal value of the regulation service when the equivalence condition between the distributed storage system and the single large storage unit condition does not hold In this situation we discussed replacing Pr by a modified value provided in Eq 28 However this is not guaranteed to provide the optimal value of the regulation service One should note that although it might be possible to improve the value of the regulation service by letting Pr go beyond the modified value presented above this choice may lead to situations when the system would fail to absorb incoming signals in some cases as a result of the limitations in individual power lines This means that in such situations in addition to the ordinary sources of error studied in this work namely the error that occurs when the storage units become fully charged before reaching end of the regulation service
28. as rp It is known that both overcharging and overdischarging of batteries of electric vehicles negatively impact their lifetime As a result batteries of electric vehicles are usually charged only up to a fraction of their total ca pacity e g 9096 of the total capacity Also they are discharged only up to a minimum charge level e g 1096 of the total capacity In the above formulation C is the maximum charge level up to which each individual storage unit can be charged This value is always less than the nominal ca pacity of the storage units Since in this work we study a unidirectional setting for charging electric vehicles the batteries of the electric vehicles are never discharged As a result we do not need to be concerned with the min imum charge level of the storage units We only need to assume that the initial residual capacities of the storage units are such that they satisfy the minimum setpoint constraint 4 Equivalence Assumption Before formulating our optimization problem we derive a condition under which we can model the fleet of electric vehicles as a single large storage unit with peak power Pr total capacity C nC and initial charge So 77 4 Si 15 Theorem 1 below summarizes the results in this section This theorem provides a sufficient and necessary condition under which the distributed storage system composed of n identical units of size C each having a max imum charging rate of p and the single
29. as the value of Po Larger values of Pr increase the price of the connection line but they do not significantly increase the value of regulation service see Fig B The maximum value of Po corresponds to the case when all storage units are fully discharged at the beginning of the charging process In that case the value of Po equals to ne To ensure that the full potential of the system is used for providing the regulation service irrespective of the value of So P es should be set based on this worst case scenarid nC pss 25 L T 25 Now we study how to set the value of p As mentioned in Section 4 the performance of a distributed storage system is upper bounded by a single big storage unit This upper bound can be achieved if the value of p is sufficiently large The minimum value of p that achieves this upper bound can be obtained using condition Hence 3 Alternatively if an accurate estimate of Po is available one may use this value to determine Pj 5 42 des IV 2 Friss 1 8 T 6 where 9 Lai O 0 is the average normalized initial residual capacity of the individual units A conservative value for p that ensures the validity of 26 can be obtained by replacing Rmax by C in this inequality This results in the following lower bound for p 27 Hence the value of p depends on the average residual capacity of the individual storage units Although the residual capacity of the storage u
30. ce as a function of the variance of the base regulation signal ay Fig 7 Geometric representation of a typical dynamic charging process Fig 8 Regulation service as a function of the number of updates Fig 9 QQ plot of the PJM regulation traces versus standard normal Fig 10 Autocorrelation functions of the regulation traces obtained from PJM The data is provided for four different days Fig 11 Testing the single shot charging scheme using PJM regulation traces Fig 12 Testing the dynamic charging scheme using the PJM regulation 72 traces T3 Figure 1 Geometric representation of a typical single shot charging process Cc mem regulation signals deterministic charging mean charge rate system hard limit i i extreme charge rates Statistical limits Eg c o c S T 2 k rr o S E 0 T T Ty T time 74 Figure 2 Three different regimes for solving the optimization problem 14 Pi eee hae M agate areas So Pc pM 2 M SEE P 2 0 0 T t T To b Phase ii Q 1 Wm rene Na vay ener eee nea ere c c Phase iii Q 1 Figure 3 Optimal value of the regulation service as a function of line peak power Pr 800 7 Optimal E mPa 700 v h A m P 2 Er Det Fe Det P 2 600r o Es 2 o a 6 500r E 3 Oo 2 w 400r g a
31. cles with a single owner who contracts with an electric utility such as a local distribution company or a third party aggregator to provide frequency regulation service From the owner s per spective it is critical that the vehicles are fully charged before a certain deadline typically overnight However the owner is insensitive to the ac tual charging rate as long as this condition is met Therefore we propose a contract where control over vehicle charging is ceded to the utility or aggre gator which can charge the vehicles at will as long as the vehicles are fully charged before the deadline In return the fleet owner obtains a monetary reward to offset operating costs In this work we do not study the form or 11 value of this monetary compensation Examples of fleets that can be used for this purpose include fleet of vehicles used for courier services or a fleet of city transit buses The utility uses the storage units of the electric vehicles to maintain the balance between generation and load in the grid More precisely the fleet has a charge controller that responds to a regulation signal to control the charging rate of the electric vehicles The value of this signal reflects stochastic fluctuations in other loads in the grid This naturally implies that the utility needs to stop obeying the regulation signals at some point and switch to a deterministic phase to ensure that all the storage units are fully charged by the end
32. during the recent years and it has been studied by researchers from different perspectives 14 23 Authors in lid use dynamic programming to obtain optimal regulation signal bids The algo rithms proposed in these studies aim to maximize the value of regulation service while providing electric vehicles with the desired target residual ca pacitiedi One limitation of the above studies is that the variation in system residual capacity due to regulation up and down signals is assumed to be negligible In other words it is assumed that the accumulative sum of reg ulation up and down signals is ideally equal to zero As we will see in this study regulation up and down signals are random in nature and even if they have zero mean they have a non zero variance Accounting for this variance is important in obtaining reliable estimates of the system perfor mance Sortomme and El Sharkawi study unidirectional frameworks for providing frequency regulation service and extend their analysis to bidi rectional settings in ld The formulation presented in these studies allows regulation up and regulation down signals to have unequal expected values However these formulations also do not account for the variance of the sys tem residual capacity due to the random nature of regulation up and down signals It should be noted that as time goes on the uncertainty in the sys tem residual capacity increases due to the random nature of the regulation signals
33. e D shows the QQ plot of the regulation traces in the training set versus standard normal It shows that the regulation signals well match the distribution of a normal random variable truncated at 1 We evaluated the variance of the signals in the unified data stream to obtain an estimate for the value of o The obtained estimate for a was 0 5069 We then use the autocorrelation function of the regulation signals to es timate the correlation time of the regulation traces Figure shows the autocorrelation functions of the regulation data plotted for the four days in 49 the training set As illustrated in this figure all autocorrelation functions exhibit a triangle like shape As described in Subsection 5 3 we define the correlation time of a triangular autocorrelation function as the lag time be yond which the autocorrelation function becomes zero As shown in Fig the correlation time Tc of the regulation signals is approximately equal to 45 min in all training days We use the values of o 0 5069 and To 45 min in the remaining of this section We also note that the mean of the regula tion signals in the training set was equal to u 0 10 instead of zero To account for this difference we lift the regulation signals in the test data by the mean value u obtained from the training set This change leads the base regulation signal to lay within the interval 0 9 1 1 rather than 1 1 We noted that this change did not affect
34. e depends on the regime in which the system operates The feasible solutions for the opti mization problem are those points on C3 that lay above C and below 31 Co After finding the feasible region we turn our attention to the objective function in the optimization problem The objective function in this problem depends only on 7o From Fig 2 it can be seen that the feasible values for To lay in the interval T7 T where T is the first coordinate of the intersection between curves C3 and C1 or C5 depending on the regime in which the system is operating Therefore to solve the optimization problem one needs to find the value of T in the interval Tr T that maximizes the objective function Our computations shows that in many practical cases the objective function I4 is strictly decreasing over the interval Tz T and thus the optimal value of T will be the starting point of this interval i e Ty A special case Uncorrelated regulation signals We now study the solution to I4 in a special case when the incoming regulation signals are uncorrelated As we will see here the solution in this case can be obtained by solving a third order polynomial equation As discussed earlier in the case of uncorrelated regulation signals o2 t tAtc With this value for o t the curves C1 and C intersect at Ty Pr 2 where VAT kAt V kAt T 7 22 Here k ao is a constant Note that in prac
35. e function will be increased To show this let m gt m be the new mean value and r P m be the associated deviation parameter We show f t gt fie t for all t To see this we note S jura t F So m t ar oo t So mt arao t m m r Sotm t a Pr m og t So mt a Pr m oo t t m m ago t m m t aoo t m m gt 0 Since FRX t gt flew t the intersection between fWY t and the line with slope Pr at the right hand side of the geometric representation lies at low m r the right of the intersection between t and the same line Thus we have T5 gt T Again note that because of the continuous nature of the above change in m the value of m can be chosen such that 77 remains less than Tt Under this assumption the value of T defined as min T7 T7 will be equal to T7 Now we show that r T7 gt rT To show this we note that the values of Gaver umm P T 17 and Come T T gt are both equal to zero and Ty 65 thus equal to each other So we have C So T m Ty ar ao 17 r m 33 C So mT aroo T2 P T T3 34 Using m P r and m P r and simplifying the above equality we obtain r T5 ar oo T3 rT arag 15 The above equality can be written as follows rT toMy ma a or pu wee 22 rT 1 oe Since olf is a
36. han Ti This contradicts the assumption that the solution is optimal Now we note that Q lt 1 and m r Pr imply Tj T To show this we note that if m r Py then T will be given by the intersection between the line with slope Py at the left hand side of Fig 1 and the horizontal line at the top of the diagram Since Q lt 1 this intersection is always at the left hand side of the line with slope Pj at the right hand side of the figure and thus Tj lt Th This contradicts the original assumption that T lt T Note that T5 gt T implies To T and that the inequality in H2 holds 69 at equality Next we show that if T5 gt 7 then m r 0 This is simple to verify Because if m r gt 0 one can increase the value of Ty which is equal to Tj in this case by fixing the value of r and slightly decreasing the value of m such that 7 increases and still remains less than 75 Again this contradicts optimality of the solution Given m r 0 we can replace r by m in and arrive at m lt A 44 This inequality provides the upper limit of m provided in the first part of Theorem 4 To obtain the lower limit we note that inequality 12 holds at equality when Q lt 1 and thus C So T 45 0 2m 45 Plugging the value of T in 43 we obtain Po 1 es 46 2 P 21 2 Finally from Eq 45 and the fact that r m it can be seen that at any optimal solution the value of object
37. hat g t T is a decreasing function of t 28 Proof See appendix B The geometric interpretation of the above theorem is that at the optimal solution the values of the first coordinates of T and T in Fig I are equal c t t to each other and equal to T provided that g t is a decreasing function of t In the special case when the incoming regulation signals are uncorrelated golt y 2 At it can be seen that g t 9 Vi is a decreasing function of t Theorem 3 below indicates that the same result is indeed true for all signals with a positively valued autocorrelation function R EN t P 7 E Theorem 3 g t T is a decreasing function of t if the corresponding autocorrelation function R is positively valued Proof See appendix C Examples of positively valued autocorrelation functions include triangu lar and exponential autocorrelation functions In Subsection 6 2 2 we obtain the autocorrelation function of real world PJM Interconnection regulation traces As we will see there the autocorrelation function of these signals has a triangular form and thus the above theorem can be applied Given Theorem 2 one can replace inequalities 11 and 12 in the optimiza tion problem IO by equalities We can then evaluate the value of m to obtain 29 the following equation in terms of r and T T as After some simple algebra we arrive at the following optimization prob
38. he assumption that the original values of m and r were the optimal ones Now let m r 0 or equivalently r m so the optimal solution can no longer be modified by the operations like the one mentioned above It can be shown that even in this case the solution can be improved by decreasing m although decreasing m in this case will be at the price of decreasing r To show this again let m m be a new mean value in the system We define r m to be the new deviation parameter associated with m As 62 before let T and T7 be the crossing points in the new setting We show that Tr gt T To prove this we again show that fme r t lt fPP t for all t Fme rlt fob t So m t ar ag t So mt aroo t So m t am oo t So mt amoo t t ago t m m lt 0 The above inequality comes from the fact that m lt m and the time parameter t and baseline standard deviation oo t are positive It can be seen that since fre r t lt f5 t the intersection between f t and the horizontal line at the top of the rectangle lies at the right of the intersection between fj t and the same line That is Tf is greater than Tj Again because of the continuous nature of the above modification in the mean value we can choose m such that TT remains less than 77 and thus the value of Tj defined as min TT T7 will be equal to TT Now we show that r TT gt rTi To show this we note that the
39. he cumulative sum of the regulation signals is Gaussian while the second is the deterministic worst 23 5 1 The Optimization Problem under the Gaussian Assumption 5 1 1 Formulation In a single shot stochastic process the values of m and r are not changed during the charging process Thus once the values of these parameters are set the system must be ready to deal with the worst case scenarios that may happen as a result of this selection Let jug t and o2 t denote the mean and variance of S t at time t Also let P be the maximum tolerable error in the system There are a priori two types of potential errors The first corresponds to the situation when the system gets fully charged before the end of the regulation service period and cannot provide regulation service any more and the second to the situation when the system is not fully charged by the end of the charging period We design the system in such a way that with probability 1 P neither error occurs in the system We assume that the cumulative sum of the regulation signals is Gaussian which translates into a Gaussian assumption on the system residual capacity Let us fix the value of P Under the Gaussian assumption the value of S t lies with probability 1 P within the interval us t J acs t s t acg t where a G 4 and G is the normal standard distribution function Using these notations the system optimization problem in its worst case i e ass
40. hydroelectric generators That is because unlike coal and Nomenclature m mean of regulation signals kW r maximum deviation of regulation signals from mean kW T total time available for charging the fleet h To time instant up to which regulation service is provided h Ti maximum possible duration of regulation service given the capacity constraint of storage units h T maximum possible duration of regulation service given the time constraint to fill all storage units by a given deadline h At time interval between incoming regulation signals h vps baseline normalized regulation signals kW zis regulation signals sent from grid kW p peak power of individual storage units kW Pr peak power of the main line connecting fleet to the grid kW n number of vehicles in the fleet C maximum capacity by which each individual storage unit can be charged kWh C maximum capacity by which the whole set of storage units can be charged C nC kWh Si initial residual capacity of the it storage unit i e at the beginning of the charging period kWh So sum of initial residual capacities of the storage units residua So 2c 5 kWh S t residual capacity of the whole system at time t kWh Ri remaining capacity of the it storage unit at the beginning of the charging period kWh Ri C S Po probability of error a standard interval multiplier given P o is determined such that u oc u ac i
41. ion to the following cubic polynomial 33 ax bz cxr 4 d 0 24 Fr q 2ao v At jat 2 P T g 2adyv At PUT d C 5 Tz Similarly in the second regime where Q gt 1 it can be seen that the opti mal value of Tp is given by y where y is the solution to the cubic polynomial ay by cy d 0 Once we obtained the optimal value of T for these 2 regimes we can obtain the optimal values of m using Eq PO and the optimal value of r using Eq Finally in the third regime where Q 1 the three curves Ci C5 and C3 all intersect at Tj Pr 2 It can be seen that in this case the optimal value of r equals zs m In this section therefore we have reduced the problem to a simple single variable optimization problem 34 5 2 Optimal solution in the deterministic case In this subsection we study the solution to optimization problem 5 in the deterministic worst case where regulation signals are all equal to each other and equal to either m 4 r or m r corresponding to the maximal and minimal regulation sequence respectively We find closed form solutions for this problem in all regimes i e for all values of Q The solution in this case provides a measure of the degree of improvement that can be obtained by taking a stochastic approach to the problem This setting is illustrated by the two lines crossing point 0 59 with slopes m r and m r in Fig We use the same definition f
42. is assumed independent of m and r and captures the intrinsic properties of the regulation signal When dealing with the stochastic case we will assume that the cumulative sum of x s can be modeled as a Gaussian random variable This is a reasonable assumption because as we will see in Section 6 2 regulation signals obey a truncated Gaussian distribution and the correlation time between them is typically considerably less than the length of the regulation period The charge controller charges the electric vehicles by obeying the regu lation signal sequence x until time T In the remaining time T To the utility switches to a deterministic phase and completes the charging process of all storage units in all vehicles by time T This two phase charging process is required because the utility needs to ensure that all storage units will be fully charged by the end of the charging period regardless of the uncertain residual capacity of the system at time t To We denote by n the number of vehicles in the fleet We use 5 to denote the initial residual capacity of the i vehicle i 1 n We denote the 14 maximum capacity by which each storage unit can be charged by C We represent the peak rate at which each individual storage unit can be charged by p and the power capacity of the power line connecting the fleet with the grid by Pr Using the above notations the value of the regulation service in the system can be represented
43. ivalent or not It can be seen that the distributed system and the single large storage unit are equivalent to each other only if the input power can be distributed among the storage units in such a way that the number of none full storage units times p remains greater than or equal to the incoming power during the whole charging process Theorem 1 below provides a necessary and sufficient condition for this relation to hold As we will see in the proof of this theorem Proportional Charging PC turns out to be an ideal scheme for distributing the incoming power among the storage units Theorem 1 Consider a distributed system composed of n storage units each with capacity C and peak power p Assume the i storage unit in this system has initial residual capacity S i e the capacity to fill is Ri C Sj 1x4 xn Also assume the system is connected to the grid through a power line with peak power Py The above distributed system is equivalent to a single large storage unit with total capacity C nC initial residual capacity So 35 4 Si and input peak power Pr if and only if the following condition holds Pisas lt X Ri 1 p Pr where Rmar max l5 Proof Recall that in our system a regulation signal always corresponds 17 to charging never to discharging It is not hard to see that any signal that can be obeyed by the distributed system can also be obeyed by the single large storage unit system because its cha
44. ive function is equal to Case 2 Q gt 1 Using procedures similar to the ones presented above it can be shown that when Q gt 1 T5 T and m r Pr The proofs are 70 similar and are not presented them for the sake of brevity Note that 75 lt T implies T T and thus the inequality in 43 holds at equality Setting r Pr m in HO implies ED Also we can set r Pr m in 43 to obtain ga PEOR 2 Py m Plugging the value of T in 42 we arrive at pesi m lt Po Pr 49 Inequalities 47 and H9 provide respectively the lower and upper limits of m in the second part of Theorem 4 From Eq it can be seen that the value of objective function at all optimal solutions is equal to 71 List of Figures Fig 1 Geometric representation of a typical single shot charging process Fig 2 Three different regimes for solving the optimization problem a Phase i Q 1 b Phase ii Q gt 1 c Phase iii Q 1 Fig 3 Optimal value of the regulation service as a function of line peak power Py Fig 4 Optimal value and duration of the regulation service as a function of system residual capacity a Optimal value of the regulation service b Optimal duration of the regulation service Fig 5 Optimal value of the regulation service as a function of the correlation time of the regulation signals Fig 6 Optimal value of the regulation servi
45. large storage unit are equivalent to each other Equivalence here means that both systems will behave the same irrespective of the regulation signals being sent In other words two systems are called equivalent if any signals that can be obeyed by one of them can be obeyed by the other one as well and vice versa What makes a distributed system different from its corresponding single storage unit is the limitation in the input power of individual storage units As an example consider a distributed system that has n 2 units with p 1 C 1 and with initial residual capacity both equal to 0 5 Let Py 2 We would like to replace this distributed system by a single central unit of size C 2 and initial residual capacity equal to 1 Whenever the charge controller receives a regulation signal it has to decide how to schedule the charging of the two units Imagine that it decides to charge unit 1 with higher priority Then at some point unit 1 will be fully charged and the regulation can only be performed on the second unit that can be charged with an input power p lt Pr Thus at this point the distributed system will not be equivalent to single unit any more It is easy to see that if the charge controller has treated each unit identically the equivalence would have held As illustrated by this example both the initial residual capacity of the storage units and the distribution mechanism determine whether the 16 two systems are equ
46. lence assumption earlier In Section 4 we have analytically obtained a condition under which this assumption is valid and see that this condition is likely to be satisfied in all practical systems where the initial residual capacity of the batteries is relatively low As previously mentioned the values of m r and T should be computed so that the value of regulation service r7 is maximized In solving this problem two key constraints must be taken into account i the system must be able to provide regulation service with the contracted deviation parameter r during the whole interval 0 To ii the storage will get fully charged by the end of the charging period T Note that these are two fundamental constraints in the system and similar constraints are expected to exist in other load side frequency regulation service providers To ensure that regulation service is successfully provided over the inter val 0 To constraint i described above the following constraints must be satisfied m r gt 0 and m r Pr The first inequality captures the uni directional nature of the charging process The second inequality accounts for the bounded size of the link connecting the fleet to the grid Another constraint is that the system must never get fully charged during the inter val 0 To otherwise it cannot respond to the incoming regulation signals anymore Let S t denote the residual capacity of the system at time t The 21
47. mentioned in Section B there are a range of optimal values for the mean charge parameter m We study two special cases where m is set to E and to Po The advantage of choosing n over Pc is that this choice makes the system parameters independent of the residual 46 capacity at each updating point We will see next that the value of the regulation service will significantly increase in a system with dynamic setting and that the difference between the deterministic and the stochastic settings reduces 6 9 Numerical Evaluations In this section we use simulation and real world regulation data to obtain results for different settings To this end we use the representative system described in Section 5 3 Sections 6 2 1 and 6 2 2 use simulation and real world data respectively to assess the performance of the system In each of these subsections we study both single shot and dynamic settings 6 2 1 Simulation data A Single shot optimization In this subsection we use simulation data to assess the value of the regulation service in a system using a single shot opti mization scheme To this end we study the representative system introduced in Section We assume the system is initially 25 charged Moreover we set Py 300kW To 45min o 0 5 and P 1078 Using the methodology presented in Section 5 1 3 we can show that the optimal values of the variables in the stochastic setting are m P 150 kW r P 150 kW and T5
48. n of regulation service h nuclear generators the output of these generators can be easily ad justed by changing the level of the input gas or water The problem with gas generators is that they burn fossil fuels thus causing economi cal and environmental problems Hydroelectric generators do not burn fossil fuels However a generator that is planned to provide regulation service has to work below its maximum capacity to create room for ma noeuvring its output As a result if a hydroelectric generator is used for providing regulation some capacity for generating clean energy will be lost e Generators typically achieve maximum efficiency when working at max imum capacity However a generator providing balancing service nec essarily works on average below its maximum capacity and thus does not achieve its maximum efficiency e Generating variable rate power leads to higher wear and tear of gen erators In contrast some loads may be insensitive or less sensitive to the variations in the input power and are therefore better candidates for providing regulation service e With the advent of new sources of renewable generation and variable load in the future grid the need for regulation will increase Being equipped with control units some loads in the future smart grid can actively participate in the regulation service market For a load to be able to participate in the frequency regulation market three conditio
49. nits may vary from day to day their average may have less variations Given an estimate for the value of 3 in the system we can use inequality 27 to obtain des the minimal value of p From this inequality it can be seen that if the des value of 3 varies between 0 50 the minimal value of p varies between 43 and 2e We are now ready to evaluate the values of parameters P and ps for the representative system introduced in Section Using Eq the input peak power in this system should be set at pde gnc 280x20kWh 400kW Moreover assuming an average initial residual capacity of 25 the individual peak powers should be set to a minimum of p POE 2okWh 6 66 kW This line power can be achieved by a level 2 charging infrastructure We conclude this section by considering the case when condition 1 does not hold e g when the initial residual capacity of the storage units is rel atively high small R and this initial charge is distributed heterogeneously among the individual units large Rmax In such cases one can replace the input peak power Pj by a modified value that can be obtained from condition as follows iB ia Fi ppet 245 28 This modified value of P can replace the value of Py in Section D to determine the charging parameters of the system One should note that if the value of 3 approaches 1 the right hand side of inequality 27 goes to infinity In such ca
50. ns must be met First it must have some level of flexibility in its consumption profile and therefore be relatively insensitive to variations in input power Second the load must be significant with respect to the power fluctuations in the grid because the utility does not want to deal with minor players Third the load should be controllable so that it can respond to reg ulation controlsignals Examples of such loads include industrial cold storage units industrial boilers large scale pumps and ventilators and storage units of fleets of electric vehicles Over the past years pilot programs have demon strated successful application of load side regulation service providers B This has even motivated commercial entities to monetize the aggregation of loads to provide regulation service In this paper we focus on the charging process of a fleet of electric ve hicles overnight as a representative system for providing load side regulation service This system provides a simple setting to study this complex problem At its heart the goal is to transfer a certain amount of energy to the fleet b a given deadline however when and how fast the energy is sent is flexible li Studying fleets also allows us to analyze a simple multi component systems each vehicle in the fleet is one component of the overall system Study ing multi component loads is important in the context of load side frequency regulation service because as mentioned earlier for a
51. or T and T5 as before when we replace the curve fP t resp fh t by the line of slope m r resp m r With these definitions the value of T will be again min Ti T5 One can follow a procedure similar to the one described in Appendix B to obtain the optimal values of m r and To in this deterministic setting Theorem 4 below gives the solutions in this case Theorem 4 Consider the optimization problem B in the deterministic case where regulation signals are all equal to each other and equal to either m rormc r e When Q lt 1 any value in the interval P Pr 2 is optimal for m where Ps 1 Dia Pr Pp In this case the optimal value of r is m and the optimal value of the 35 objective function rTo is e When Q gt 1 the optimal values of m lay in the interval Z P5 where P E E 5 2779 Fa Pu In this case the optimal value of r is equal to Pr m and the optimal value of the objective function rTo is DIL Proof See Appendix D As mentioned in the above theorem the optimal solutions in the deter ministic case are not unique Moreover one can see that the values P 2 and Pc always lay in the interval of optimal values for m 5 9 Results in the case of a representative system In this subsection we illustrate some of the results obtained in the pre vious subsections for a representative system We consider a fleet of electric vehicle consisting of 80 vehicles each with a
52. ows the expected value of the regulation service as a function of time span between updates in the stochastic setting and in the deterministic setting with both m 5 and m Po The system under study is the same system described in the previous section In plotting Fig 8 we generated 1000 random trajec tories for v recall that v is a zero mean random sequence with Tc 45 min oy 0 5 and maximum and minimum 1 and 1 respectively and use them 48 to charge the system while the value of T was varied between 0 and T As illustrated in this figure the resulting functions has a sawtooth shape When the number of updates increases the value of the regulation service increases sharply and perhaps surprisingly the deterministic setting outperforms the stochastic one What is also surprising is that the two deterministic settings do not perform identically when the number of updates increases As illus trated in the figure even a single additional update can greatly improve the value of the provided regulation service 6 2 2 Real world data In this section we use regulation traces collected in 12 consecutive days by the PJM Interconnection company to study the system of electric vehicles This data is available at id The regulation signals in this data set are normalized such that they lay within the range 1 We randomly divided the data into a training set consisting of four days and a test set consisting of eight days Figur
53. period and the error that occurs when there does not remain enough time for the system to become fully charged by the end of the charging period we need 54 to account for an extra source of error due to the limitations in the individual power lines Second we can generalize the above system to a bidirectional setting In that case constraint m r gt 0 in 5 should be replaced by m r gt Pr Third we can extend the above system to the setting where the base regulation signal v t can have a nonzero either negative or positive mean This would represent situations where the peak power of the regulation up and down signals are identical but the energy contents are not necessarily the same In this work we used a simple dynamic setting to illustrate the advantage of using dynamic optimization to improve the value of the regulation service provided by the system However as discussed in Section this set ting is not optimal It would be interesting to use techniques from dynamic programming to obtain optimal strategies for dynamic optimization in both stochastic and deterministic settings Acknowledgment We would like to thank the anonymous reviewer for his very helpful com ments References 1 C Quinn D Zimmerle and T H Bradley J Power Sources 195 2010 1500 1509 95 2 3 i4 5 6 7 8 9 10 11 12 S E Letendre and W Kempton Public Utilities Fortnightly 14
54. rging rate matches the peak input power Thus to show the above equivalence we just need to show that condition I is a sufficient and necessary condition under which any signal that can be obeyed by the single large storage unit can also be obeyed by the distributed snl First we prove that condition lis sufficient for this equivalence to hold Consider a regulation signal q t gt 0 over the interval 0 T that is known to be absorbable by the single large storage unit This implies that signal q t satisfies the following two constraints RQ eos cH ean Iu forall0 lt t lt T 2 J qd lt R 3 where R 77 R is the remaining capacity in the large storage unit We show that signal q t can also be sent to the distributed storage system if conditions and 3 hold and the PC scheme is used for distributing the incoming power among the individual units Under the PC scheme the fraction of input power that is sent to each storage unit is proportional to the We say a signal can be obeyed by a storage system if the peak power of the signal is at most as large as the peak input power of the storage unit and also the energy requested by the signal is at most as large as the remaining capacity of the storage unit 18 remaining capacity of that unit It is not hard to see that under this scheme the fraction of power sent to each storage unit does not change over time To show signal q t can also be obeyed by the distrib
55. s a 1 P confidence interval for the normal random variable N u o Iis t mean of total residual capacity at time t kWh og t standard deviation of total residual capacity at time t kWh Colt normalized standard deviation of total residual capacity at time t kWh R i autocorrelation function of the stationary baseline signal v kW Fi 1 E v v i Oy standard deviation of the stationary baseline signal v kW o R 0 py mean of the stationary baseline signal v kW Nomenclature Tc correlation time of the regulation signal x h P t upper bound on the system residual capacity at time kWh Wo f lower bound on the system residual capacity at time t kWh Po average power necessary for charging the fleet kW Pe Q Q is called power ratio and defined as Q P5 unitless K a constant defined as product of a and o squared kW x ac Tu time interval between update points in the dynamic charging h d number of update points in the dynamic charging m value of mean charge rate decided at time t iT kW r value of maximum deviation from mean m decided at time t iT KW qu optimal duration of frequency regulation service starting from t iT h T is realized duration of frequency regulation service starting from t iT h mort optimal value of the mean charge rate kW lt optimal value of the maximum deviation from mean kW T optimal duratio
56. ses the original condition in26 should be used to obtain the value of p which includes both 8 and Rmax 44 6 Dynamic Optimization 6 1 Introduction In the single shot optimization studied in Section 5 the optimal values of m and r are set at the beginning of the charging process and will not change until the end of the regulation service period In this system because of the one shot nature of the process the parameter values should be chosen conservatively so as to ensure the successful operation of the system One can extend the above system to a dynamic setting in which the values of m and r are updated multiple times during the charging process The updated values of m and r will be then determined based on the current state of the system at each updating point We expect this dynamic setting to improve the total value of the regulation service provided by the system However we should note that the time span between updating points in a dynamic setting cannot be too short Otherwise frequently altering the system parameters including the requested mean power may serve as a source of fluctuations in the grid rather than a mean of absorption of the fluctuations generated by other sources In this section we introduce a simple dynamic setting for charging the fleet of electric vehicles In particular we study a system in which the values of m and r are updated d 1 times at points t 0 Tu 2T dT At this point we as
57. signal and at the end of the charging period it will become fully charged If the big storage unit and the distributed system are equivalent to each other then the distributed system must also be able to absorb this signal One should note that if the distributed system absorbs the above signal then all individual storage units must get fully charged by the end of the charging period But the maximum energy that can be sent to each storage unit over the interval 0 roa is A Thus for all storage units we must have R lt Dj In particular we must have Rmax lt DE where Rmax is the maximum remaining capacity among the whole storage units This completes the proof 5 Optimal Single Shot Charging In this section we formulate the single shot optimization problem and derive the optimal parameters for the charging process We study a single shot charging scheme where the values of m r and To are determined at the beginning of the charging period and do not change until the end of the process Subsequently in Section 6 we study a dynamic setting in which the value of the charging parameters can be updated during the charging process In this section we assume that the fleet of electric vehicles can be mod 20 elled as a single large storage unit with input charge power upper bounded by P and a total capacity C nC The initial residual capacity of this system is denoted by Sp 77 S We have called this assumption the equiva
58. sume that the value of T is given and that the value of d will be determined during the charging period as will be described shortly We examine the system at each updating point and evaluate the three param 45 eters m r and To based on the current residual capacity of the system and the time remaining until the end of the charging period T Let m r and T denote respectively the values of m r and To at updating point t T If the value Tj is greater than T it means that the system can safely work until the next update point i 1 7 In that case we continue providing the regulation service until time t 4 i 1 7T and again check the system at time t On the other hand if the value of Ti evaluated at the updating point t is less than T we provide regulation service until t Tj and then switch to a deterministic phase to finish the charging process by time T i e in that case d i Figure 7 illustrates the geometric representation of a typical system with dynamic charging Let Ti denote the length of the jth regulation period during the charging reg process described above Note that T T4 ee To T and reg reg reg T a T The total value of the regulation service provided by the system can be then expressed as d Equivalent service gt rT 29 reg i 0 We study the dynamic setting in both the deterministic and the stochastic cases In the deterministic case as
59. th uwaterloo ca Catherine Rosenberg Preprint submitted to Journal of Power Sources April 1 2013 results using both simulation and real world data Keywords Vehicle to grid Load side frequency regulation service Ancillary service Variable rate charging 1 Introduction and State of the Art 1 1 Introduction The frequency regulation service one of the key ancillary services in the power grid balances generation and load taking control action as frequently as once every 30 seconds Without this balance alternating current frequency deviates from its standard value for example 60 Hz in North America which can hurt grid connected equipment and in the worst case perma nently damage generators Currently frequency regulation service is pro vided by a set of generators contracted to respond rapidly to control signals to increase or decrease their power If the amount of generated power is increased to compensate for the excess grid load the service is called regula tion up On the other hand if the amount of generated power is decreased to match the reduced load in the grid the service is called regulation down The balance between generation and load can be equivalently achieved by changing the aggregated load provided these loads have some level of flexibility in their consumption profile This approach has several benefits e Generators that provide frequency regulation service are typically nat ural gas or
60. the desired condition is not satisfied Although the results in this work are presented for the special case of charging a fleet of electric vehicles they address key aspects of load side frequency regulation service and can be applied to broader applications in this context such as large scale pumps and industrial boilers The organization of the paper is as follows In Section P we describe the characteristics of the fleet of electric vehicles that is studied in this work In Section 8 we explain mathematical models used to describe the incoming regulation signals as well as the system of electric vehicles In Section 4 we 10 analytically obtain a condition under which the assumption that a collection of storage units in the fleet can be modeled as a single large storage unit is valid In Section 5 we formulate a single shot optimization problem under this assumption and study this problem under a worst case deterministic setting and a stochastic setting In Section 6 we introduce a dynamic op timization scenario in which the parameters of the system may be updated multiple times during the charging process and study how the performance of the system improves as a result of this dynamic charging In Section 6 2 we present numerical results obtained from both simulation and real world data taken from PJM Interconnection broader 13 We conclude the paper in Section 7 2 System We study a fleet of electric vehi
61. tical systems where 32 At amp T the value of Ty in 22 will be close to T Next we study the objective function in 14 for this special case Using the above value for o2 t the objective function is f T oS T T Ts 23 It is easy to see that the above function has a global maximum at T T 3 and it is decreasing for values of T greater than T 3 As mentioned earlier to obtain the optimal solution to the optimization problem we must find the value of T in the interval T7 T that maximizes the objective function Recall that 77 is the first coordinate of the intersection between curves C3 and one of C or C2 depending on the regime in which the system is operating From Fig 2 it can be seen that in all regimes the value of Ty is greater than or equal to Ty As mentioned above in practical systems the value of T is close to T and since Ty T lt T the value of Tr will be close to T too As a result the whole interval Ty T lays on the right hand side of the point To T 3 and the objective function is strictly decreasing over the interval Tj T This implies that the optimal value of T will be the starting point of the interval Tr T i e Tr As mentioned in the previous subsection when Q 1 the first regime the value of 7 is given by the first coordinate of the intersection between curves C and C3 Using simple algebra we obtain the optimal value of To as x where x is the solut
62. tion presents a mathematical model for the system described in the previous section Let the charging process occur over the interval 0 T i e T is the dead line Recall that regulation signals are sent periodically by the grid to the charge controller typically every 30 seconds To model this phenomenon we assume that over this interval the grid sends k regulation signals denoted T1 p The j regulation signal z 1 lt j lt k is used to control fleet charging over the interval j 1 At At The utility which has been given control of the fleet during this interval has to determine the regulation pa rameters m and r where the regulation signals have mean m and may vary up to a maximum amount of r above or below m as well as the time T during which the regulation service is enabled The regulation signal sequence is bounded from below and above by two 13 worst case deterministic sequences The minimal regulation sequence is a se quence of k signals each with the value m r Symmetrically the mazimal regulation sequence is a sequence of k signals each with the value m r Unlike prior work which has only studied the charging problem in the pres ence of the worst case sequences we study the optimal fleet charging when we model the regulation signal sequence as x m rvj 1 j lt k where v denotes a zero mean stationary stochastic process with maximum 1 and minimum 1 Here the discrete time process v
63. udy the effect of the system initial residual capacity S on the optimal value of the regulation service To this end we keep the input peak power constant at Pr 300 kWh and vary the system initial residual capacity between 0 and 50 The results are presented in Fig 4 as a function of Q Lm Figure 4 a plots the optimal value of the regulation service as a function of the power ratio Q In this figure we have compared the proposed optimal method with the four sub optimal methods mentioned above Figure 4 b compares the length of the regulation period To for the same methods As illustrated in Fig 4a given the values of P and T the optimal value of regulation service is maximized when Q turns out to be one 39 or equivalently the initial residual capacity of the system turns out to be C D From Fig Hb it is clear that the system with the optimal parameters provides in general longer durations of regulation service Importantly it can be seen from Figs 3 and A a that the value of the regulation service obtained from the optimal method and the suboptimal stochastic method with m Po are close to each other in all three cases Q lt 1 Q 1 and Q gt 1 This implies that in practical applications the value of m Po can be used as a good approximation to the optimal value of m In Fig 5 we study the effect of the correlation time Tc of the incoming regulation signals on the optimal value of the regulation ser
64. ulation service value Fig 3 Also we observed that the optimal value of the average power that should be requested from the grid m is close to Pc in all three regimes Figs 3 and Hh This implies that Po can be used as a good approximation to m P in practice We also observe that the optimal value of the regulation service in the system is inversely related to the correlation time of the regulation signals Fig 6 This is expected when the correlation time of regulation signals is large regulation up and down signals slowly compensates the effects of each other and this leads to more conservative estimates of the regulation service We studied the relation between a distributed storage system and a single large storage unit We obtained conditions under which the distributed sys tem can be modeled as a single large storage unit thus achieving its upper bound performance We studied dynamic optimization settings in which the value of the sys tem parameters m and r are updated multiple times during the charging period We observe that such settings can significantly increase the value of the regulation service provided by the system In addition to a stochastic setting we studied the system in a worst case deterministic setting where regulation signals are all equal to each other and equal to the maximal or minimal regulation sequence The advantage of 53 this system is that there is no risk attached to it i e regulation c
65. ulation signal with mean m and maximum deviation r In this case the feasible trajectories are limited by the two lines crossing point 21 0 So with slopes m r and m r respectively Define fart us t 4 aes t So mt aroo t PSA us t aes t Syo mt aroo t m r As mentioned in the previous subsection all trajectories of the regulation low m r signal lay between f P t and t with a probability that can be con trolled by parameter a We denote by T the intersection between the curve uP t and the horizontal line y C or the line t T whichever comes first Also we denote by T5 the intersection between the curve not and the line with slope Pr crossing point T C or the line y C whichever comes first Given the values of m and r the optimal value of T is the smaller of the two first coordinates of T1 and T5 That is because the largest value of To that satisfies both constraints I and I2 is min T1 T5 5 1 3 Solution to the optimization problem In this section we reduce the nonlinear three variable optimization prob lem I0 to a simple one variable optimization problem over the interval 77 T see later which can be solved precisely We start by presenting the follow ing key theorem Theorem 2 Let m r t TOP be the optimal solution to problem IO At this optimal solution both inequalities HI and 12 are active i e hold at co t equality provided t
66. uming the worst case trajectories for S t can be expressed as 24 max r To 6 s t m r Tg 20 To lt T m rz20 m r lt Pr us To aos To lt C To lgAt C u ars To Ep esq L From Eq Hj it can be shown that the mean of S t at time t At is given by see Appendix A Ls t Sg ml At So mt 7 Also it is not hard to show that the variance of S t at time t JAt is given by o2 t r 0 t where o t can be expressed in terms of the autocorrelation function of the stochastic process v recall that the regulation signal sequence is 7 m ru for all 1 lt j lt k where v denotes a zero mean stochastic sequence in 1 1 as follows Appendix A i 1 oo t tAt R 0 2 3 R i 2 At X iR i 8 i 1 25 Here A i E v v denotes the autocorrelation function of v Note that o4 t only depends on the basic regulation signal v and not on parameter r In the special case when the incoming regulation signals are uncorrelated R i 0 for i gt 1 and thus o t simplifies to oa t tAtR 0 tAto 9 where o R 0 is the variance of the signal v In general however the signal v is correlated and it is important to take these correlations into account Substituting values of us t and os t from Eq J and Eq Blin the optimization problem 6 we obtain Ti 10 max r To 10 s t m r To gt 0 To lt T m r gt 0 m r lt P So
67. uted system we first note that the individual peak power of storage units p will be never violated during the charging process That is because the fraction of power sent to the i storage unit denoted by q t satisfies Fossa Gec oe ur B a Rn 7 o4 H Pd Ri The first inequality in the above expression comes from the fact that t P lt p Rmax is the largest value among all R s The second inequality comes from inequality 2 and the third inequality comes from condition Next we note that none of the storage units becomes fully charged until perhaps the end of the charging process To show this we note that the total energy send to the it storage unit by time T denoted by Ej satisfies T T R R T R B f ad f ERIE pas R t i yog EI t R The facts that the incoming power q t can be distributed among the individual units without their peak power being violated and none of the storage units become fully charged before the the end of the charging process imply that signal q t can be absorbed by the distributed storage system We next show that condition l is necessary to ensure that the any sig 19 nal that can be sent to the single big storage unit can also be sent to the distributed system To show this we consider the constant signal q t P R Pr over the interval 0 T where T and R is the remaining capacity of the big storage unit The big storage unit can absorb this
68. vice In plotting this figure the values of Pc and Py are kept constant at 150 kW and 300 kW respectively and the correlation time between the regulation signals is varied between 0 and 2 hours As shown in this figure the optimal value of the regulation service decreases as the correlation among the regulation signals increases This is because when regulation signals are not correlated or are correlated with small correlation times regulation up and regulation down signals cancel the effect of each other over short periods of time and the energy sent to the fleet remains close to its expected value However when regulation signals are correlated over longer intervals it may be the case that a sequence of consecutive up or down regulation signals appear in the system and lead the system residual capacity to largely deviate from its expected value In such situations the variance of the system residual capacity increases and the value of the regulation service decreases for a 40 given probability of error In Fig we have only presented values of the regulation service greater than or equal to 600 kW h That is because as shown in Fig Ma the deterministic approach that achieves zero error can yield a regulation service value of 600 kW h All other proposed methods are useful only if they offer regulation service values greater than this error free case This illustrates that depending on the values of the parameters a deterministic

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