A Robust Design of Electric Vehicle Frequency Regulation Service

Enxin Yao, Vincent W.S. Wong, and Robert Schober Department of Electrical and Computer Engineering The University of British Columbia, Vancouver, Canada e-mail: enxinyao, vincentw, rschober @ece.ubc.ca { }

Abstract—Electric vehicles (EVs) have the potential to provide provide frequency regulation service by varying the charging frequency regulation service to an independent system operator power around a set point. The works in [7], [8] consider (ISO) by changing their real-time charging or discharging power capacity-based frequency regulation compensation, where the according to an automatic generation control (AGC) signal. Recently, the Federal Energy Regulatory Commission has issued revenue only depends on the frequency regulation capacity. an order to ISOs to introduce a performance-based compensation The Federal Energy regulatory Commission (FERC) issued paradigm. In this new compensation scheme, the ISOs make pay- Order 755 [9] in Oct. 2011, which requires ISOs to introduce ments to the EVs for providing the frequency regulation capacity performance-based compensation for the frequency regulation and the real-time dispatch when following the AGC signal. In service. The ISOs are required to make payments for frequency this paper, we propose a robust EV frequency regulation (REF) regulation capacity and frequency regulation performance. algorithm to determine the hourly regulation capacity for each EV considering the randomness of the AGC signal. The pro- Performance is evaluated based on the real-time dispatch (i.e., posed REF algorithm is based on a robust optimization problem changing charging or discharging power around a baseline) formulation. It enables EVs to follow the random AGC signal when EVs follow the AGC signal. Performance-based fre- reliably and to improve the EV frequency regulation revenue. quency regulation compensation has been implemented by Simulation results show that the proposed REF algorithm obtains some ISOs, e.g., the Pennsylvania Jersey Maryland Intercon- a higher revenue compared with a benchmark algorithm from the literature under a performance-based compensation paradigm. nection (PJM). Considering the trend towards performance- based compensation, it is desirable that the EVs follow the I. INTRODUCTION random AGC signal reliably to improve their revenue. Electric vehicles (EVs) are among the potential candidates Each EV has limited battery capacity. An EV cannot follow to replace combustion engine vehicles for reducing the emis- the AGC signal to charge if it is fully charged and it cannot sion of CO2 and other greenhouse gases. When EVs are discharge if its battery is depleted. The challenge of using EVs connected with the grid, they can be regarded as energy to provide reliable frequency regulation service has caught storage systems and can be coordinated to provide frequency the attention in both the academia [10], [11] and ISOs [2]. regulation service to independent system operators (ISOs), The works in [2], [10], [11] focus on designing a special such as the California ISO (CAISO). Frequency regulation AGC signal for which the regulation up and regulation down service helps ISOs keep the utility frequency within an components are almost identical for each hour. As stated in acceptable range by maintaining the instantaneous balance [11], the AGC signal approach is only applicable for ISOs between generation and load. The pilot projects in [1], [2] which request EVs to have the same amount for regulation show that EVs are able to provide frequency regulation service up and regulation down capacity, such as PJM. However, this by following an automatic generation control (AGC) signal approach is difficult to apply for ISOs which have separate issued by the ISO. Fast ramping resources such as EVs can regulation up and regulation down markets, such as CAISO reduce the overall frequency regulation capacity requirement and Electric Reliability Council of Texas (ERCOT). In those for ISOs and lead to lower costs for ISOs and consumers [3]. markets, the EVs may have different regulation up and reg- The market-based EV frequency regulation service has ulation down capacity and the AGC signal approach cannot received significant attention [4]–[8]. The ISO purchases the be applied. On the other hand, it is desirable to provide hourly frequency regulation capacity from EVs on the market reliable EV frequency regulation service in order to improve and the participating EVs are obliged to follow the AGC the performance-based frequency regulation revenue. signal and provide hourly regulation capacity. An aggregator In this paper, we tackle the above issue and propose a robust is typically used to serve as an agent between the ISO and the design of the EV frequency regulation service by taking into fleet of EVs. The works in [4]–[6] propose algorithms for the account the randomness of the AGC signal. The contributions aggregator to distribute regulation tasks among EVs in real- of this paper are as follows: time. An EV frequency regulation algorithm is proposed in We propose a robust EV frequency regulation (REF) algo- • [7] for the aggregator to determine the charging schedule of rithm to improve the revenue of EVs for performance- each EV and the hourly regulation capacity. A unidirectional based compensation. frequency regulation algorithm is proposed in [8] for EVs to We use robust optimization approach to formulate an EV • frequency regulation problem. We transform the formu- ISO lated combinatorial optimization problem into a linear AGC signal Regulation capacities program based on duality. Aggregator We evaluate the performance of the proposed REF algo- Two-way communication link • rithm with a historical AGC signal from PJM. A com- parison with the benchmark OptMaxReg algorithm in [8] EV shows that the proposed REF algorithm improves the Battery performance-based frequency regulation revenue. 12 N Fig. 1. The block diagram for EVs to provide frequency regulation service This paper is organized as follows. We introduce the system by following an AGC signal from the ISO. The AGC signal is generated by model in Section II. We present the problem formulation the ISO in real-time and is random. and the REF algorithm in Section III. Numerical results are their charging power by adjusting the pilot signal duty cycle presented in Section IV. Conclusions are given in Section V. [4]. We analyze the AGC signal on an hourly basis as ISOs II. SYSTEM MODEL typically purchase the hourly capacity. We denote the hourly The considered EV frequency regulation scheme is illus- regulation up component and the hourly regulation down component of the AGC signal within hour h by f u(h) and trated in Fig. 1. A two-way communication system is im- d plemented to enable information exchange between the ISO, f (h), respectively. We have aggregator, and EVs. The aggregator coordinates the EVs to 1 f u(h)= [ q(h, t)]+ , (1) provide frequency regulation service to the ISO. First, the T − t X2T aggregator aggregates the hourly frequency regulation capacity 1 of the EVs. The ISO purchases the hourly capacity and the f d(h)= [q(h, t)]+ , (2) T aggregator enters into a contract with the ISO to provide t X2T frequency regulation service. During operation, the aggregator where [x]+ = max x, 0 . Let e (h) denote the charged energy retrieves the AGC signal from the ISO every few seconds { } i for EV i within hour h. Then, for EV i , hour h , (e.g., 2-6 seconds, depending on the ISO’s requirement) and 2 N 2 H e (h) can be represented as broadcasts the AGC signal to EVs. The EVs are obliged to i provide frequency regulation service by changing their real- 1 u + d + ei(h)= xi(h) v (h)[ q(h, t)] +v (h)[q(h, t)] time charging or discharging power based on the AGC signal. T − i − i t 2T We denote the operation hours by = 1,...,H . The set X � u u d d � H { } = xi(h) v (h)f (h)+v (h)f (h). (3) of EVs is denoted by = 1,...,N . Each EV i has a − i i N { } 2 N baseline charging power x (h), regulation up capacity vu(h), d d u u i i The terms vi (h)f (h) and vi (h)f (h) represent the charged d and regulation down capacity vi (h) for each hour h . The and discharged energy due to following AGC signal during u 2 H d aggregator determines the values of xi(h), vi (h), and vi (h) hour h, respectively. Note that we assume EV i needs to follow to maximize the frequency regulation revenue. the AGC signal in every time slot because EV is a fast ramping The AGC signal is generated by the ISO according to the resource and it has a zero lost opportunity cost [10], [13]. real-time unbalance between generation and load in the grid. The performance-based compensation paradigm comprises The AGC signal is updated in short intervals. We divide one payments of two parts. The first part of payment is the hour into multiple time slots. Each time slot corresponds to reward for providing the regulation up and regulation down the duration of one interval of the AGC signal, e.g., one time capacity. The ISO pays pu(h) to each EV for providing slot lasts 4 seconds for PJM. Let = 1,...,T denote the d T { } regulation up capacity and price p (h) for providing regulation set of time slots in one hour, i.e., the duration of one time slot down capacity at hour h. The second part of payment is of the AGC signal is 1 . The AGC signal in time slot t T 2 T performance related. The performance is typically evaluated of hour h is denoted by q(h, t) [ 1, 1]. 2 H 2 − based on the real-time dispatch when EVs follow the AGC The AGC signal indicates the amount by which EV i 2 N signal. We denote the performance price (e.g., the regulation should increase or decrease its charging power, compared to market performance clearing price (RMPCP) in PJM) at hour the baseline charging power xi(h). A negative AGC signal h as pp(h). Let mu(h) and md(h) denote the summation of (i.e., q(h, t) < 0) indicates the power generation in the grid is absolute changes of the regulation up and regulation down lower than the load. EV i provides regulation up by multiply- components of the AGC signal, respectively. We have u ing the AGC signal with its regulation up capacity vi (h) and + + decreasing the charging power accordingly. A positive AGC mu(h)= [ q(h, t)] [ q(h, t 1)] ,h , (4) − − − − 2 H signal (i.e., q(h, t) > 0) indicates the generation is higher t X2T � � than the load in the grid. EV i provides regulation down by � � md(h)= �[q(h, t)]+ [q(h, t 1)]+ ,h� , (5) multiplying the AGC signal with its regulation down capacity − − 2 H d t � � vi (h) and increasing the charging power accordingly. Note X2T � � that for chargers that comply with the Society of Automotive where x denotes� the absolute value of x. The� payment for the | | Engineers (SAE) J1772 standard [12], EVs are able to change performance at hour h is based on the performance price and the summation of absolute changes of the real time charging or 0.2 discharging power [13]. We assume the communication links 0.15 between the ISO, aggregator, and EVs are reliable. If EV i 0.1

0.05 follows the AGC signal at hour h, it receives a payment based mass function Joint probability on the performance as follows 0 0 0.05 + + 0.1 p u d 0.15 0.45 0.2 0.4 p (h) xi(h) vi (h)[ q(h, t)] + vi (h)[q(h, t)] 0.25 0.35 0.3 0.3 − − Regulation down 0.25 t 0.35 0.2 f d h 0.4 0.15 Regulation up 2T � component ( ) 0.45 0.1 X � � 0.05 component f u(h) � 0 x (h)� vu(h)[ q(h, t 1)]+ + vd(h)[q(h, t 1)]+ − i − i − − i − Fig. 2. Joint distribution of the hourly regulation up component and the hourly regulation down component of an AGC signal. The distribution is p⇣ u + + ⌘ � = p (h) v (h) [ q(h, t)] [ q(h, t 1)] � obtained by analyzing the AGC signal data for 2,160 hours from [14]. There i − − − − � t are two types of AGC signals in [14] and this figure is obtained by analyzing X2T ⇣ � � the RegA signal. The figure shows the randomness of the regulation up d +� + � + vi (h) [q(h, t)] � [q(h, t 1)] � component and regulation down component. Note the distribution can be − − different for other AGC signals. Our approach is applicable to any distribution. p �u u d d �⌘ = p (h) v�i (h)m (h)+vi (h)m (h) �. (6) � � component and regulation down component, respectively. ⇣u EVs have� to pay for charging energy� from the grid. We and ⇣d can be selected based on the historical AGC data. e denote the price for energy charging at hour h by p (h). Note An integer parameter ⌘ 0, 1,...,H is introduced to 2 { } that ISOs typically calculate revenue on an hourly basis. Let adjust the level of robustness. We aim to find a solution u d ri(vi (h),vi (h),xi(h)) represent the revenue for EV i at hour which enables EVs to follow AGC signal reliably when the h. It can be written as unknown parameters f u(h) and f d(h) change adversely in u d u u d d at most ⌘ hours and take expected values in other hours. ri(vi (h),vi (h),xi(h)) = p (h)vi (h)+p (h)vi (h) Note that the worst case of the unknown parameters (e.g., + pp(h) vu(h)mu(h)+vd(h)md(h) 1 i i i,h f u(h)=⇣u,fd(h)=0) does happen for certain hours and e u u d d p (h) xi(h) v (h)f (h)+v (h)f (h) . (7) our approach has no compromise on the range in which the − � − i i � unknown parameters change. However, it is unlikely that the The first term on the� right hand side of (7) is the payment� AGC signal will have the worst case consecutively in all for capacity, whereas the second term is the payment for operation hours. If we set ⌘ = H, we are considering the performance, and the third term is the cost for charging energy. case where the AGC signal changes adversely in all operation 1 is an indicator function which is equal to 1 when EV i i,h hours, which may lead to a conservative solution. On the other follows the AGC signal in hour h, and is equal to 0 otherwise. hand, if we set ⌘ =0, the randomness of the AGC signal is The limited battery capacity is a challenge for EVs to follow ignored and the solution is unreliable. Therefore, parameter ⌘ the AGC signal reliably. EVs cannot follow the AGC signal is introduced to enable EVs follow the AGC signal reliably to charge when it is fully charged and it cannot discharge without making the solution conservative. when its battery is depleted. EVs may fail to follow the AGC Let ⌧ represent an arbitrary hour in set 1,...,h . We use signal occasionally because the randomness of the AGC signal { } S u to denote the set for the hours when the unknown parameters regulation up and regulation down components (f (h) and u d d f (⌧) and f (⌧) change adversely. The cardinality of set f (h)) can lead to an uncertain charged or discharged energy, u d S u is denoted by . The expected value of f (⌧) and f (⌧) are see (3). We studied the statistical joint distribution of f (h) u|S| d d u d denoted by µ and µ , respectively. We consider two cases for and f (h), i.e., P(f (h),f (h)), by analyzing the AGC signal the uncertain parameters f u(⌧) and f d(⌧). In the first case, data from PJM [14], for the period from January 1, 2012 to u d u d d March 31, 2012. The results in Fig. 2 show that f u(h) and f (⌧) and f (⌧) change adversely (f (⌧)=0,f (⌧)=⇣ ) f d(h) may deviate significantly from their expected values. In to increase the state of charge (SOC) of EVs at the end of hour the next section, we use robust optimization to determine the h. Let si(0) denote the initial SOC of EV i at the beginning hourly regulation capacity which enables EVs to follow the of operation hours. We denote the battery capacity of EV i as Bi. We assume the charging efficiency for both charger and AGC signal reliably (i.e., 1i,h =1). battery is close to one. We have the following constraint III. ROBUST EV FREQUENCY REGULATION 1 u d d si(0) + max xi(⌧) vi (⌧)0+vi (⌧)⇣ In this section, a robust EV frequency regulation algorithm Bi − 1,...,h ⌘ ⌧ S✓{ } |S| 2S is proposed. The algorithm aims to obtain a reliable solution n o X � � u d 1 u u d d of v (h), v (h), and xi(h) to enable EVs to follow the AGC + � xi(⌧) v (⌧)µ + v (⌧)µ 1, (8) i i Bi � − i i  signal reliably. In particular, we use the robust optimization ⌧ 1,...,h 2{ X }\S� � approach [15], [16] to analyze the cases when the unknown where 1,...,h represents the relative complement of parameters f u(h) and f d(h) change adversely (i.e., take { }\S set with respect to set 1,...,h , i.e., the hours when values such that EVs tend to miss the AGC signal) within S { } the unknown parameters f u(⌧) and f d(⌧) take expected 0 f u(h) ⇣u, 0 f d(h) ⇣d. Constants ⇣u, ⇣d [0, 1]     2 values. The operator max will select at most ⌘ represent the maximum value of the AGC signal regulation up 1,...,h ⌘ {S✓{ }| |S| } hours from set 1,...,h . In the selected hours ⌧ , the upper bound and the expected revenue can be rewritten as { u } d 2 S p u u d d unknown parameters f (⌧) and f (⌧) change adversely (i.e., (1 P(1i,h))E(p (h))(vi (h)λ + vi (h)λ ), where P(1i,h) u d d − f (⌧)=0,f (⌧)=⇣ ) to increase the SOC of EV i, see represents the probability for 1i,h =1. Our simulation results the first term following the max in (8). In other hours (i.e., show the gap is a small value when we choose ⌘ > 0 to ⌧ 1,...,h ), the unknown parameters f u(⌧) and f d(⌧) improve robustness (e.g., the gap is less than 3% of the 2 { }\S take the expected values, see the last term on the left hand side optimal value when ⌘ =1). The gap decreases when we of (8). Constraint (8) can be equivalently written as choose ⌘ to be a larger value. We consider the case when the h aggregator aims to maximize the aggregate revenue of EVs in 1 u u d d si(0) + xi(⌧) vi (⌧)µ + vi (⌧)µ 10% Bi − the objective function (11a) and keeps a percentage (e.g., ) ⌧=1 u d p X ⇣ ⌘ of the aggregate revenue. E(p (h)), E(p (h)), E(p (h)), and 1 u u (9) e min vi (⌧)µ (p (h)) in (11a) represent the expected price for regulation − Bi − E 1,...,h ⌘ ⌧ S✓{ } |S| X2S ⇣ up capacity, regulation down capacity, regulation performance, d d d + vi (�⌧)(µ ⇣ �) 1 ,i,h . and charged energy at hour h. We consider the expected − �  2 N 2 H Eq. (9) is obtained⌘ by adding component values in the objective function because the uncertainty in the 1 u u d d revenue does not affect whether EVs follow the AGC signal xi(⌧) v (⌧)µ + v (⌧)µ on the last term of Bi ⌧ − i i left hand2S side in (8) and removing the same component on reliably or not. Constraints (11b) and (11c) guarantee that the P � � real-time charging power of EV i in hour h is within the the first term after max. max maximum and minimum charging power of EV i. Ei and In the second case, the unknown parameters change ad- min u u d Ei represent the maximum and minimum charging power of versely (i.e., f (⌧)=⇣ ,f (⌧)=0) to reduce the SOC of EV. EV i, respectively. Constraint (11d) guarantees the frequency The following constraint to keeps the SOC to be above zero regulation capacities are non-negative values. Constraint (11e) h d 1 u u d d depicts the charging demand of EV i. di and si represent the si(0) + xi(⌧) vi (⌧)µ + vi (⌧)µ Bi − expected departure hour and the expected SOC for EV i upon ⌧=1 ⇣ ⌘ 1 X u u u (10) departure, respectively. We assume EV i is connected to the + B min vi (⌧)(µ ⇣ ) i − grid from the beginning of operation hours until di. The values 1,...,h ⌘ ⌧ S✓{ } |S| X2S ⇣ d d d of si and di are set by EV owners. vi (�⌧)µ 0�,i ,h . − ≥ � 2 N 2 H Problem (11) is a non-convex combinatorial optimiza- Constraints (9) and⌘ (10) confine the SOC of EV i to be within [0, 1] at the end of hour h. Note the SOC within hour h is tion problem because constraints (9), (10) have non- bounded by the SOC at the end of hour in the considered cases convex components. We first analyze constraint (9). The optimal value of the non-convex component of (9) when the unknown parameters change adversely. Hence, in this u u d d d paper, we use constraints (9) and (10) to enable EVs follow min ⌧ vi (⌧)µ + vi (⌧)(µ ⇣ ) is 1,...,h ⌘ 2S − − AGC signal reliably (i.e., 1 =1). We denote the expected S✓{ } |S| i,h equivalent to the optimalP value� of the following problem� [16] value of mu(⌧) and md(⌧) as λu and λd, respectively. We � � � w vu(⌧)µu + vd(⌧) µd ⇣d formulate a robust EV frequency regulation problem as follows minimize ⌧ − i i − w⌧ , ⌧ 1,...,h ⌧ u u d d 2 { } X2H ⇣ � � ⌘ maximize E(p (h))vi (h)+E(p (h))vi (h) (12a) vu(h),vd(h),x (h), h i i i i 2H 2N ⇣ i ,h X Xp u u d d subject to 0 w⌧ 1, ⌧ 1,...,h , (12b) 2 N 2 H + E(p (h))(vi (h)λ + vi (h)λ )   2 { } e u u d d w⌧ ⌘, (12c) E(p (h)) xi(h) vi (h)µ + vi (h)µ  − − ⌧ (11a) X2H � � ⌘ x (h)+vd(h) Emax,i ,h , where w⌧ [0, 1] for ⌧ 1,...,h are variables in problem subject to i i i (11b) u 2 d 2 { } u  min 2 N 2 H (12). vi (⌧) and vi (⌧) take arbitrary finite values. Problem xi(h) vi (h) Ei ,i ,h , (11c) − ≥ 2 N 2 H (12) is a linear program which is both feasible (e.g., w⌧ = u d u u vi (h),vi (h) 0,i ,h , (11d) 0, ⌧ min v (⌧)µ + ≥ 2 N 2 H ) and bounded (e.g., ⌧ 1,...,h i di 1 d 2 Hd d 2{ } {− 1 − vi (⌧)(µ ⇣ ), 0 is a lower bound of the objective function). s (0) + x (⌧) vu(⌧)µu + vd(⌧)µd − } P i B i − i i From strong duality, the optimal values of problem (12) and i ⌧=1 X ⇣ ⌘ its dual problem are the same. The dual problem of (12) can sd,i , (11e) ≥ i 2 N be written as . Constraints (9) and (10) (11f) maximize z⌘ y⌧ (13a) y⌧ , ⌧ 1,...,h ,z − − 2 { } ⌧ 1,...,h The objective function (11a) represents an upper bound of 2{X } u u d d d the expected aggregate revenue of EVs. An upper bound is subject to y⌧ + z v (⌧)µ v (⌧) µ ⇣ , ≥ i − i − considered because using (7) as the objective function leads ⌧ 1,...,h� , (13b)� to an untractable problem. Note the upper bound is close 2 { } y ,z 0, ⌧ 1,...,h , (13c) to the expected revenue as we use constraints (9) and (10) ⌧ ≥ 2 { } to ensure 1i,h =1in most cases. The gap between the where y⌧ and z are dual variables for constraints (12b) and (12c), respectively. Now we replace the non-convex compo- initializes the parameters and then solves problem (14). The nents in constraint (9) with the objective function in problem results are sent to each EV and the aggregate results are sent (13) and add all constraints in (13) to problem (11). We to the ISO. use a similar approach to convert the non-convex component in constraint (10) in three steps. First, we convert the non- Algorithm 1 REF algorithm executed by the aggregator at the convex components in constraint (10) into a linear program by beginning of operation hours u u d d d e u d substituting vi (⌧)µ + vi (⌧)(µ ⇣ ) in problem (12) 1: Initialize ⌘, , ,p (h),p (h), and p (h),h u u− u d d − d N H max min 2 H with v (⌧)(µ ⇣ ) v (⌧)µ . Then, we convert the linear 2: Collect si ,di,si(0),Ei ,Ei , and Bi from each EV i i i u d 2 N � − − � 3: Solve problem (14) to obtain v (h),v (h), and xi(h),h program obtained in the first step into its dual problem by i i 2 H u u d d d for each EV i substituting vi (⌧)µ + vi (⌧)(µ ⇣ ) in problem (13) u d2 N − − 4: Send vi (h),vi (h), and xi(h) to each EV i and report (the substitution is the same as in the first step). Finally, we u d 2 N � � i vi (h), i vi (h), and i xi(h),h to the ISO substitute the non-convex component in constraint (10) with 2N 2N 2N 2 H the objective function of the dual problem obtained in the P P P second step and add the constraints of the dual problem to problem (11). We have the following problem IV. P ERFORMANCE EVALUATION u u p u e u maximize vi (h) p (h)+p (h)λ +p (h)µ In this section, we evaluate the performance of the proposed vu(h),vd(h),x (h), i i i h i z1 ,z2 , REF algorithm under the historical record of the AGC signal i,h i,h X2H X2N d � d p d e d� y1 (⌧),y2 (⌧) + vi (h) p (h)+p (h)λ p (h)µ from PJM. We conducted a statistical analysis of the historical i,h i,h − i , (h, ⌧) (h, ⌧) x (h)pe(h) 2 N 2 { i AGC signal data record [14] and obtained the parameters h , ⌧ 1,...,h − � � u d u d u | 2 H 2 { }} (14a) ⇣ =0.473, ⇣ =0.483, µ =0.116, µ =0.113, λ = 13.852, d h and λ = 13.805. We compare our proposed REF algorithm 1 u u d d with the OptMaxReg algorithm from [8]. The OptMaxReg subject to si(0) + xi(⌧) vi (⌧)µ + vi (⌧)µ Bi − algorithm assumes parameters f u(h) and f d(h) take expected ⌧=1 ⇣ ⌘ X h values and neglects their uncertainty. The performance-based 1 + z1 ⌘ + y1 (⌧) 1,i ,h , frequency regulation prices are obtained from [14], [17]. B i,h i,h  2 N 2 H i ⌧=1 The average price over the operation period for the hourly ⇣ X ⌘ (14b) energy consumption pe(h), regulation up capacity pu(h), 1 1 u u d d d pd(h) pp(h) zi,h + yi,h(⌧) vi (⌧)µ vi (⌧)(µ ⇣ ) , regulation down capacity , and the performance ≥ − − are $42.4/megawatt per hour (MWh), $4.9/MWh, $7.2/MWh, i , (h, ⌧) ⇣(h, ⌧) h , ⌧ 1,...,h⌘ , 2 N 2 { | 2 H 2 { }} and $3.2/MWh, respectively. (14c) We consider a fleet of 100 EVs. Each EV is connected with h 1 a bidirectional level-2 charger. The maximum charging power s (0) + x (⌧) vu(⌧)µu + vd(⌧)µd i B i − i i is 3.3 kW. EVs are able to discharge power to the grid. Note i ⌧=1 X ⇣ ⌘ that the battery capacity of EV may vary from several kWh 1 h z2 ⌘ + y2 (⌧) 0,i ,h , (e.g., 4.4 kWh for a Toyota Prius) to tens of kWh (e.g., 20 kWh − B i,h i,h ≥ 2 N 2 H i ⌧=1 for a Honda Fit). We assume the battery capacity is 12 kWh ⇣ X ⌘ (14d) for simulation purposes. We consider an overnight charging 2 2 u u u d d case where EVs charges in the night and is used for driving zi,h + yi,h(⌧) vi (⌧)(⇣ µ )+vi (⌧)µ , ≥ − on the next day. EV’s charging deadline di is generated based i , (h, ⌧) ⇣(h, ⌧) h , ⌧ 1,...,h⌘ , 2 N 2 { | 2 H 2 { }} on the 2009 national household travel survey [18]. (14e) For performance metrics, we determine the revenue and 1 2 1 2 zi,h,zi,h,yi,h(⌧),yi,h(⌧) 0,i , reliability of EV frequency regulation service by testing the ≥ 2 N u d (h, ⌧) (h, ⌧) h , ⌧ 1,...,h , (14f) algorithm outputs (i.e., xi(h),vi (h),vi (h)) with the historical 2 { | 2 H 2 { }} AGC signal. The EVs receive the payment for their perfor- Constraints (11b)-(11e), (14g) mance if they follow the AGC signal in hour h. If an EV i fails 1 1 2 2 where (zi,h,yi,h(⌧)) and (zi,h,yi,h(⌧)) are dual variables to follow the AGC signal in hour h, its frequency regulation corresponding to the non-convex components in constraints capacity in hour h is regarded as forced derated [19] (i.e., the (9) and (10), respectively. Linear constraints (14b) and (14d) capacity from EV i is discarded by the aggregator in hour replace the non-convex constraints (9) and (10) by substituting h). We denote i as the set of hours when the frequency D the combinatorial optimization components with the corre- regulation capacity from EV i is forced derated. The reliability sponding linear dual problems. Constraints (14c) and (14e) are of EV frequency regulation service is evaluated by 1 EFDH − H obtained from the constraints in the dual problems. Problem [11], [20], where the equivalent forced derated hours (EFDH) (14) is a linear program and can be solved efficiently. is [19] (vu(h)+vd(h)) The robust EV frequency regulation (REF) algorithm is i h i i i 2N 2D EFDH = 1 u d . presented in Algorithm 1. In the algorithm, the aggregator first H i h (vi (h)+vi (h)) P 2NP 2H P P 300 Proposed REF algorithm, Reliability the formulated combinatorial optimization problem into a Benchmark robust algorithm, Reliability linear program based on the duality. We performed numerical Proposed REF algorithm, Revenue 250 Benchmark robust algorithm, Revenue experiments on the historical record of the AGC signal from Proposed REF algorithm PJM. Simulation results show that our proposed algorithm 200

1 achieves a higher revenue compared with a benchmark algo-

150 rithm from the literature. For future work, an interesting topic Reliability 0.9 Revenue ($) is the extension of the proposed algorithm to the case when 100 0.8 the mobility behavior of EV is unknown. Benchmark robust algorithm 0.7 50 REFERENCES 0 1 2 3 4 5 6 7 8 9 10 Parameter η [1] W. Kempton, V. Udo, K. Huber, K. Komara, S. Letendre, S. Baker, Fig. 3. The reliability and revenue with respect to parameter ⌘. We choose D. Brunner, and N. Pearre, “A test of vehicle-to-grid (V2G) for ⌘ =1for our proposed algorithm and ⌘ =2for the benchmark algorithm. energy storage and frequency regulation in the PJM system,” Tech. Rep., Nov. 2008, available: www.udel.edu/V2G/resources/test-v2g-in- 300 Proposed REF algorithm, η=1 pjm-jan09.pdf. Benchmark robust algorithm, η=2 280 Benchmark OptMaxReg algorithm [2] H. Ni, “PJM advanced technology pilots for system frequency control,”

260 in Proc. of IEEE Innovative Smart Grid Technologies (ISGT), Washing- ton, DC, Jan. 2012. 240 [3] A. D. Papalexopoulos and P. E. Andrianesis, “Performance-based pricing

220 of frequency regulation in electricity markets,” IEEE Trans. on Power Systems, vol. 29, no. 1, pp. 441–449, Jan. 2014. Revenue ($) 200 [4] E. Sortomme and K. W. Cheung, “Intelligent dispatch of electric vehicles

180 performing vehicle-to-grid regulation,” in Proc. of IEEE Int’l Electric Vehicle Conf. (IEVC), Greenville, SC, Mar. 2012. 160 [5] J. Lin, K.-C. Leung, and V. O. K. Li, “Online scheduling for vehicle-to- 140 grid regulation service,” in Proc. of IEEE SmartGridComm, Vancouver, 10 11 12 13 14 15 16 17 18 19 20 Battery Capacity (kWh), Bi Canada, Oct. 2013. [6] S. Sun, M. Dong, and B. Liang, “Distributed regulation allocation with Fig. 4. Comparison of the proposed REF algorithm with OptMaxReg aggregator coordinated electric vehicles,” in Proc. of IEEE SmartGrid- algorithm [8] under performance-based compensation. Comm, Vancouver, Canada, Oct. 2013. Fig. 3 shows the revenue and reliability for the pro- [7] S. Han, S. Han, and K. Sezaki, “Development of an optimal vehicle-to- grid aggregator for frequency regulation,” IEEE Trans. on Smart Grid, posed REF algorithm. We compare with a benchmark ro- vol. 1, no. 1, pp. 65–72, June 2010. bust algorithm, which considers the case when f u(h) and [8] E. Sortomme and M. El-Sharkawi, “Optimal charging strategies for d u ⌘ u u ⌘ u u unidirectional vehicle-to-grid,” IEEE Trans. on Smart Grid, vol. 2, no. 1, f (h) change adversely in µ H µ ,µ + H (⇣ µ ) , ⌘ ⌘ − − pp. 131–138, Mar. 2011. µd µd,µd + (⇣d µd) for all hours. As shown in the − H H − ⇥ ⇤ [9] FERC Order 755: Frequency Regulation Compensation in the figure, both the revenue and reliability improve as ⌘ increases Organized Wholesale Power Markets, Oct. 2011, Available: ⇥ ⇤ at the beginning because EVs earn more revenue for the http://www.ferc.gov/whats-new/comm-meet/2011/102011/E-28.pdf. [10] C. Quinn, D. Zimmerle, and T. H. Bradley, “An evaluation of state-of- performance (i.e., 1i,h =1 for more hours). When ⌘ further charge limitations and actuation signal energy content on plug-in hybrid increases, the revenue decreases as the obtained solution is electric vehicle, vehicle-to-grid reliability, and economics,” IEEE Trans. more conservative which leads to lower revenue for the capac- on Smart Grid, vol. 3, no. 1, pp. 483–491, Mar. 2012. [11] C. Quinn, “Evaluation of distributed energy storage for ancillary service ity. Our proposed REF algorithm outperforms the benchmark provison,” Master’s thesis, Colorado State University, 2011. robust algorithm. Our proposed REF algorithm selects ⌘ =1 [12] SAE J1772 Standard on Electric Vehicle and Plug in Hy- to have 97% reliability and $224 frequency regulation revenue brid Electric Vehicle Conductive Charge Coupler, Available at: http://standards.sae.org/j1772 201001/. for each day. The benchmark robust algorithm selects ⌘ =2 to [13] PJM Manual 11: Energy and Ancillary Services Mar- achieve $192 revenue with 94% reliability. Note the revenue ket Operations, Revision 65, Jan. 2014, Available at: is obtained from (7). For the proposed REF algorithm, the gap http://www.pjm.com/ /media/documents/manuals/m11.ashx. [14] PJM Interconnection⇠ Market Based Frequency Regulation, between the revenue (7) and the considered objective function http://www.pjm.com/markets-and-operations/ancillary-services/mkt- (11a) is $2.13 when ⌘ =1. based-regulation.aspx. We compare the proposed REF algorithm with the bench- [15] A. Ben-Tal and A. Nemirovski, “Robust optimization methodology and applications,” Mathematical Programming, vol. 92, no. 3, pp. 453–480, mark robust algorithm, and the OptMaxReg algorithm [8] in May 2002. Fig. 4. The figure reveals that the proposed REF algorithm [16] D. Bertsimas and M. Sim, “The price of robustness,” Operations outperforms the OptMaxReg algorithm [8] with a higher rev- Research, vol. 52, no. 1, pp. 35–53, Feb. 2004. [17] CAISO open access same-time information system (OASIS), enue under performance-based compensation. This is because http://oasis.caiso.com/. the proposed algorithm enables EVs to follow the AGC signal [18] A. Santos, N. McGuckin, H. Nakamoto, D. Gray, and S. Liss, “Summary reliably and to earn more revenue for performance. of travel trends: 2009 national household travel survey,” U.S. Department of Transportation, Federal Highway Administration, Tech. Rep., 2011, available: http://nhts.ornl.gov/2009/pub/stt.pdf. V. C ONCLUSION [19] IEEE Standard Definitions for Use in Reporting Electric Gener- In this paper, we proposed a robust EV frequency regu- ating Unit Reliability, Availability, and Productivity, Available at: http://www.nerc.com/docs/pc/gadstf/ieee762tf/762-2006.pdf. lation algorithm to improve revenue under performance-based [20] Generation Availability Report, North American Electric compensation. A robust optimization problem is formulated to Reliability Corporation (NERC), Sept. 2012, Available: enable EVs to follow the AGC signal reliably. We transformed http://www.nerc.com/pa/RAPA/gads/Pages/Reports.aspx.