Continuous-time Model Predictive Control for Real-time Flexibility Scheduling of Plugin Electric Vehicles

10th IFAC Symposium on Control of Power and Energy Systems 10th IFAC Symposium on Control of Power and Energy Systems Tokyo, Japan, September 2018of Power and Energy Systems Control 10th IFAC Symposium on 4-6, Tokyo, Japan, September 4-6, 2018 Available online at www.sciencedirect.com 10th IFAC Symposium on 4-6, Control Tokyo, Japan, September 2018of Power and Energy Systems Tokyo, Japan, September 4-6, 2018

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IFAC PapersOnLine 51-28 (2018) 498–503

Continuous-time Model Predictive Control Continuous-time Model Predictive Control Continuous-time Model Predictive Control for Real-time Flexibility Scheduling of Continuous-time Model Predictive Control for Real-time Flexibility Scheduling of for Real-time Flexibility Scheduling of Plugin Electric Vehicles for Real-time Flexibility Scheduling of Plugin Electric Vehicles Plugin Electric Vehicles Plugin Electric Vehicles Roohallah Khatami ∗∗ Masood Parvania ∗∗

Roohallah Khatami ∗ ∗ ∗ ∗ Masood ∗ Roohallah Khatami Masood Parvania Parvania Avishan Bagherinezhad ∗ Avishan Bagherinezhad ∗ ∗ ∗ ∗ Roohallah Khatami Masood Parvania Avishan Bagherinezhad ∗ ∗ Avishan Bagherinezhad ∗ Department of Electrical and Computer Eng., University of Utah of Electrical and Computer Eng., University ∗ ∗ Department Department of Electrical andUSA Computer Eng., University of of Utah Utah Salt Lake City, UT 84112 (e-mails: {roohallah.khatami, ∗ Salt Lake City, UT 84112 USA (e-mails: {roohallah.khatami, Department of Electrical andUSA Computer Eng., University of Utah Salt Lake City, UT 84112 (e-mails: {roohallah.khatami, masood.parvania, avi.bagherinezhadsowmesaraee}@ utah.edu) masood.parvania, avi.bagherinezhadsowmesaraee}@ utah.edu) Salt Lake City, UT 84112 USA (e-mails: {roohallah.khatami, masood.parvania, avi.bagherinezhadsowmesaraee}@ utah.edu) masood.parvania, avi.bagherinezhadsowmesaraee}@ utah.edu) Abstract: This paper proposes a continuous-time model predictive control (MPC) for coAbstract: This paper a model control (MPC) for Abstract: Thischarging paper proposes proposes a continuous-time continuous-time model predictive predictive control (MPC)schedule for cocooptimizing flexibility plugin (PEVs) generation optimizing the the charging flexibilityaof ofcontinuous-time plugin electric electric vehicles vehicles (PEVs) and and generation schedule Abstract: This paper proposes model predictive control (MPC) for cooptimizing theunits charging flexibilitypower of plugin electric vehiclesA (PEVs) and generation schedule of generating in real-time systems operation. continuous-time queuing model of generating units in real-time power systems operation. A continuous-time queuing model optimizing the charging flexibility of plugin electric vehicles (PEVs) and generation schedule of generating to units in real-time power systems queuing model is developed aggregate and cluster a large operation. population Aofcontinuous-time PEVs, which represents their is developed to aggregate and cluster aa large population PEVs, which represents their of generating units in power systems Aof queuing is developed to aggregate cluster large operation. population ofcontinuous-time PEVs, which represents their aggregate flexibility to real-time the and power system operator. The proposed model integrates themodel most aggregate flexibility to power system The model integrates the most is developed to aggregate and cluster a operator. large population of PEVs, which represents their aggregate flexibility to the the power system operator. The proposed proposed model integrates the most recent about the system load PEV and the flexibility of recent information information about thepower system load and and PEV arrival, arrival, and utilizes utilizes theintegrates flexibilitythe of PEVs PEVs aggregate flexibility to the system operator. The proposed model most recent information the system load and PEV arrival, and utilizes the flexibility of PEVs and generating unitsabout to supply the ramping requirements of load, while ensuring the delay-based and units to the requirements of while ensuring the recent information the system load and PEV arrival, and utilizes the flexibility of PEVs and generating generating unitsabout to supply supply the ramping ramping requirements of load, load, while model ensuring the delay-based delay-based deadline-based service quality constraints of PEVs. The proposed is implemented on and deadline-based service quality constraints of PEVs. The proposed model is implemented on generating unitsservice to supply the using ramping requirements of load, while model ensuring the delay-based and deadline-based quality constraints of PEVs. The implemented on the IEEE Reliability Test System, PEV and load data ofproposed California. Theissimulation results the IEEE Reliability Test using PEV and load data California. The results and deadline-based service constraints of PEVs. Theof proposed model issimulation implemented on the IEEE Reliability Test System, System, using PEV and load data of California. The simulation results demonstrate effectiveness ofquality the proposed model to utilize the flexibility of PEVs in real-time demonstrate effectiveness of the proposed model to utilize the flexibility of PEVs in real-time the IEEE Reliability Test System, using PEV and load data of California. The simulation results demonstrate effectiveness of the considerably proposed model to utilize the requirements flexibility of PEVs in real-time power systems operation, which reduces ramping from conventional power systems operation, reduces ramping from demonstrate effectiveness of the considerably proposed model to utilize the requirements flexibility of PEVs in real-time power systems operation, which which considerably reduces ramping requirements from conventional conventional generating units. generating units. power systems operation, which considerably reduces ramping requirements from conventional generating units. © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. generating units. Keywords: Continuous-time model predictive control, queuing model, plugin electric vehicles. Keywords: Continuous-time Keywords: Continuous-time model model predictive predictive control, control, queuing queuing model, model, plugin plugin electric electric vehicles. vehicles. Keywords: Continuous-time model predictive control, queuing model, plugin electric vehicles. 1. INTRODUCTION of aggregated PEV charging demand from the reference 1. INTRODUCTION of aggregated PEV charging demand from the reference point instructed by the operator. 1. INTRODUCTION of aggregated PEV charging demand from the reference point instructed by operator. 1. INTRODUCTION of aggregated PEV charging demand from the reference point instructed by the the operator. Multiple researchbyworks have explored the application of point instructed the operator. Multiple research works have explored the application of The increasing population of plugin electric vehicles queuing theory in modeling flexible loads, including PEVs. research works have explored the application of The increasing population of plugin electric vehicles Multiple queuing theory in modeling flexible loads, including PEVs. (PEV)increasing on roads population offers a sizable source electric of demand side Multiple The of plugin vehicles research works have explored the application of queuing theory modeling flexible loads, including PEVs. Deferrable loadsin are modeled as deadlined tasks in Hao (PEV) on roads offers a sizable source of demand side The increasing population ofoperators plugin vehicles Deferrable loads are modeled as deadlined tasks in Hao (PEV) on to roads offers a sizable source electric demand side queuing flexibility power systems ifof appropriately theory in modeling flexible loads, including PEVs. Deferrable loads are modeled as deadlined tasks in Hao and Chen (2014), while Guo et al. (2013) used the queuing flexibility to power systems operators ifof appropriately and Chen (2014), while Guo et al. (2013) used the queuing (PEV) on to roads offers a in sizable source demand side harnessed optimized system operation and flexibility power systems operators appropriately loadsthe are modeled as (2013) deadlined in Hao theory to model arrival/departure process of deferrable and Chen (2014), while Guo et al. usedtasks the queuing harnessed and and optimized in power power systemif operation and Deferrable flexibility to power systems operators if appropriately theory to model the arrival/departure process of deferrable control problems Hosseini et al. (2014); Palomino and Parharnessed and optimized in power system operation and and Chen (2014), while Guo et al. (2013) used the queuing loads in power systems operation. Aprocess non-homogeneous to power model the arrival/departure of deferrable control problems Hosseini et al. (2014); Palomino and Parharnessed andInoptimized in al. power system operation and theory loads in systems operation. A non-homogeneous vania (2018). 2016, the et worldwide PEV stock surpassed control problems Hosseini (2014); Palomino and Partheory toprocess model the of deferrable loads in power systems operation. non-homogeneous Poisson is arrival/departure utilized in HafezAprocess and Bhattacharya vania (2018). In 2016, the worldwide PEV stock surpassed control problems Hosseini al. (2014); Palomino and ParPoisson process is utilized in HafezA and Bhattacharya vania (2018). 2016, the et worldwide PEV stock million surpassed 2 million cars In and is anticipated to reach 40-70 by Poisson loads in power systems operation. non-homogeneous (2016) to model the arrival of PEV considering process is utilized in Hafez and Bhattacharya 2 million cars In and is anticipated to reach 40-70 million by (2016) to model the arrival of PEV considering customer customer vania (2018). 2016, theU.S., worldwide PEV500,000 stock million surpassed 2025 IEA (2017). In there PEVs on 2 million cars and is the anticipated to were reach 40-70 by Poisson is utilized Hafez and Bhattacharya convenience and the response toofin charging prices. In Liu et al. (2016) toprocess model arrivalto PEV considering customer 2025 IEA (2017). In the U.S., there were 500,000 PEVs on 2 million cars and is anticipated to reach 40-70 million by convenience and response charging prices. In Liu et al. the roads in 2016,Inwhich is expected to reach 9-21 million 2025 IEA (2017). the U.S., there were 500,000 PEVs on (2016) model the arrival ofcharging PEV considering customer (2016), to delay-tolerant demands can jump to high-priority convenience and response to prices. In Liu et al. the roads in 2016, which is expected to reach 9-21 million 2025 IEADOE (2017). Inwhich the U.S., there were 500,000 PEVs on convenience (2016), delay-tolerant demands can to by 2030 (2017). the roads in 2016, is expected to reach 9-21 million and response to charging prices. In such Liu etthat al. (2016), delay-tolerant demands can jump jump to high-priority high-priority queues using an upgrade-by-probability scheme by 2030 DOE (2017). the roadsDOE in 2016, which is expected to reach 9-21 million (2016), queues using an upgrade-by-probability scheme such that by 2030 (2017). delay-tolerant demands can jump to high-priority it reduces their potential excessive delay. queues using an upgrade-by-probability scheme such that A wide variety of studies have been conducted on PEV it reduces their potential excessive delay. by 2030 variety DOE (2017). A wide of studies been conducted on PEV using anpotential upgrade-by-probability scheme such that it reduces their excessive studies have have PEV queues charging controlofstrategies and been their conducted impacts onon reliable A wide variety The aforementioned studies utilizedelay. discrete-time control charging control strategies and their impacts on reliable it reduces their potential excessive delay. A variety ofstrategies studiesofhave been conducted on PEV The aforementioned studies utilize discrete-time control charging control and their impacts reliable andwide economic operation power systems. A on congestion models for control, the deaforementioned studies utilizewhere discrete-time control and economic operation of and power systems. A on congestion models for PEV PEV charging charging control, where the control control decharging control strategies their impacts reliable The management-based technique predicated upon internet and economic operation of power systems. A congestion The aforementioned studies utilize discrete-time control cisions form piecewise constant trajectories. However, the models for PEV charging control, where the control demanagement-based technique predicated upon internet and economic operation of power systems. A congestion cisions form piecewise constant trajectories. However, the networking protocolstechnique is used predicated in Fan (2012) tointernet locally models management-based uponto forconstant PEV charging control, where control depiecewise control trajectories are the only zero-order cisions form piecewise constant trajectories. However, the networking protocols is in (2012) locally management-based technique predicated upon piecewise constant control trajectories are only zero-order networking protocols is used used in Fan Fan (2012) tointernet locally control the PEVs charging demand where the objective is cisions form piecewise constant trajectories. However, the piecewise constant control trajectories are only zero-order approximations of the actual continuous-time decisions, control the charging demand where the objective is continuous-time decisions, networking protocols is used inowners. Fan (2012) to approximations of the actual control the PEVs PEVs charging demand where is piecewise to maximize the utility of PHEV Inthe Luoobjective andlocally Chan constant trajectories are only zero-order and utilize the of of control the actual decisions, to maximize the utility of PHEV owners. In Luo and Chan and cannot cannot appropriately appropriately utilizecontinuous-time the ramping ramping flexibility flexibility of control PEVs charging demand where the is approximations (2013), athe novel power/voltage leveling factor is objective introduced to maximize the utility of PHEV owners. In Luo and Chan approximations of the actual continuous-time decisions, PEVcannot charging load in power systems operation. and appropriately utilize the ramping flexibility of (2013), a novel power/voltage leveling factor is introduced to maximize the utility of PHEV owners. In Luo and Chan PEV charging load in power systems operation. to minimize losses and improve thefactor voltage profile of and (2013), a novellosses power/voltage leveling is introduced appropriately utilize the ramping flexibility of PEVcannot charging load in power systems operation. to minimize and the voltage profile (2013), a novel power/voltage leveling factor is introduced In thischarging paper, we leverage oursystems recent operation. development on the to minimize losses and improve improve thecharging voltage profile of of PEV distribution buses through optimal control load in power In this paper, we leverage our recent development on the distribution buses through optimal control of application to minimize losses and improve thecharging voltage profile of of we continuous-time optimal control in on power this paper, leverage our recent development the distribution buses optimal charging control PEVs. In Soltani et through al. (2015), the price price elasticity of PEVs PEVs of In application of continuous-time optimal control in power PEVs. In Soltani et al. (2015), the elasticity of this paper, leverage our recent development application of we continuous-time optimal control in on power distribution buseset through optimal charging control of In systems Parvania and Scaglione (2016b); Khatami et the al. is modeled using a conditional random field, and utilized PEVs. In Soltani al. (2015), the price elasticity of PEVs systems Parvania and (2016b); Khatami et is modeled using et a conditional random field, andofutilized of continuous-time optimal in model power systems Parvania and Scaglione Scaglione (2016b);control Khatami et al. al. PEVs. In Soltani al. (2015), the price elasticity PEVs application (2018b,a), and propose a continuous-time MPC in modeled a stochastic economic dispatch problem to optimize is using a conditional random field, and utilized (2018b,a), and propose a continuous-time MPC model in a stochastic economic dispatch problem to systems Parvania and Scaglione (2016b); Khatami et al. is using a conditional random field, and utilized (2018b,a), for optimizing the charging control of PEVs in real-time and the propose a continuous-time MPC model in acharging stochastic economic dispatch problem to optimize optimize themodeled power of PEVs. Among real-time charging for optimizing charging control of PEVs in real-time the charging power of PEVs. Among real-time charging (2018b,a), and propose a continuous-time MPC model in acharging stochastic economic dispatch problem to optimize operation of power systems. control We firstofpropose, inreal-time Section for optimizing the charging PEVs in control methods, model predictive control (MPC) merits the power of PEVs. Among real-time charging operation of systems. We first Section control methods, model predictive control (MPC) merits for the charging PEVs inin of power power systems. We firstoftopropose, propose, inreal-time Section the charging power of PEVs. Among real-time charging 2, aoptimizing continuous-time queuingcontrol model aggregate a large in capturing the model intertemporal characteristics ofmerits PEV operation control methods, predictive control (MPC) 2, a continuous-time queuing model to aggregate a in capturing the intertemporal characteristics of PEV operation of power systems. We first propose, in Section 2, a continuous-time queuing model to aggregate a large large control methods, model predictive control (MPC) merits population of PEV loads, and cluster them into charging. In Di Giorgio et al. (2014), the objective is to population of PEV loads, and cluster them into flexible in capturing the intertemporal characteristics of PEV flexible charging. In et (2014), the objective is a continuous-time queuing model tocharacteristics aggregate a large in capturing theGiorgio intertemporal characteristics of PEV load queuesofwith different charging and population PEV loads, and cluster them into flexible charging. In Di Di Giorgio et ofal. al.PEV (2014), is to to 2, minimize the charging cost loadsthe andobjective the deviation load queues with different charging characteristics and minimize the charging cost ofal.PEV loads andobjective the deviation PEVdifferent loads, and clustercharacteristics them into flexible charging. In Di Giorgio et of (2014), is to population load queuesofwith charging and minimize the charging cost PEV loadsthe and the deviation load queues with charging characteristics and minimize charging cost of PEV loads and the deviation 2405-8963 ©the 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd.different All rights reserved.

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service quality constraints. In the proposed queuing model, the PEVs’ aggregated ramping trajectory is defined as time derivative of its aggregated charging power trajectory. The proposed continuous-time MPC model is developed in Section 3, which co-optimizes the charging flexibility of PEVs with the generation trajectories of generating units to supply the real-time variability and ramping requirements of the electricity load over receding control horizons. The proposed model effectively utilizes the ultimate ramping flexibility of PEVs by integrating the most recent load prediction, and updating the PEV queues with the real-time PEV arrival information at each MPC run. A function space solution method is proposed in Section 4 that is based on reducing the dimensionality of the continuous-time decision and parameter trajectories by modeling them in a finite-order function space spanned by the Bernstein polynomials. The numerical results are furnished in Section 5 using PEV and load data of California, and the conclusions are drawn in Section 6. 2. QUEUING SYSTEM FOR REAL-TIME PEV AGGREGATION In this section, we present a queuing model for aggregating a large population of PEV in real-time operation of power systems. The model in this section is based on our recent works presented in Khatami et al. (2018a,c). Consider a set of L aggregators that cluster and aggregate PEVs to provide the aggregate flexibility to the power system operator. Once a PEV is connected to a charger, a charge request is sent to the aggregator, who uses multiple queuing systems to cluster the PEV charge requests that include information on the amount of energy required, and the service quality requirements (i.e., delay or deadline constraints). The charge requests are clustered based on the location of chargers (e.g., residential, workplace, public), type of chargers (level 1-3), and service quality requirements. The PEV clusters are modeled by a M (t)/GI(t)/D(t) queue, where M (t) is a time-dependent Markovian (Poisson) distribution that models the arrival time of PEVs, GI(t) is a general time-dependent distribution that models the duration of charge, and D(t) is the variable that controls the charging of PEVs at time t. Consider the scheduling horizon Tτ = [τ, τ + T ] that is −1 divided into N intervals Tτ,n = [tn , tn+1 ), Tτ = ∪N n=0 Tτ,n of the same length ∆t = tn+1 − tn , where t0 = τ and tN = t + T . Let the non-negative stochastic process jl,c (Tτ,n ) represent the number of PEV charge requests received by aggregator l under cluster c over time interval Tτ,n , and the continuous-time curve pl,c,n (t−tn ) represent the total PEV charge load received during that time interval by aggregator l under cluster c. Therefore, the total PEV charging load request trajectory over Tτ , represented by Jl,c (t), is defined as: Jl,c (t) =

N −1  n=0

pl,c,n (t − tn ),

t ∈ Tτ .

(1)

The PEVs may stay idle and wait before charging starts. It would then take for the duration of service time for the PEVs to be fully charged. Assume that Ql,c (t) represents the backlog of the queue accumulated and not served until time t. In addition, Dl,c (t) controls the amount of 499

499

power that is supplied to the PEV cluster c represented by aggregator l at time t. Accordingly, the state equation of the proposed queuing model is formulated as: (2) Q˙ l,c (t) = Jl,c (t) − Dl,c (t), t ∈ Tτ ,

where the rate of change (derivative) of the queue backlog at time t equals the PEV charging requests received minus the PEV charging load supplied. The proposed queuing model is schematically shown in Fig. 1. PEV Charging Power Requests

Consumption Pattern

Total PEV Charging Load Request Trajectory

Backlog of the Queue

Controlled PEV Charging Power Trajectory

Fig. 1. Queuing model for PEV charging load Khatami et al. (2018a) 2.1 Service Quality Constraints The waiting time is determined based on the service quality preferences and charging characteristics of PEVs. Ensuring the service quality requirements (represented by either a deadline or a maximum charging delay) would positively impact the willingness of PEV owners to allow the aggregators control their PEV charge. As discussed below, the queuing model facilitates imposing both delaybased and deadline-based service quality constraints over the scheduling horizon Tτ , by translating them into maximum limits on the queue backlog variable Ql,c (t). In delay-based constraints, PEV owners specify a maximum charging delay time, e.g., one hour, by which the car should be charged. Let τl,c be the maximum delay time of requests queued by aggregator l under cluster c. The following delay-based service quality constraint (3) specifies that the queue backlog at any time t should not exceed the total charging requests received from τl,c times ago, in order for the aggregator to serve the PEV charging requests in the delay time τl,c Khatami et al. (2018c):  t Ql,c (t) ≤ Jl,c (t )dt , t ∈ Tτ − [τ, τ + τl,c ). (3) t−τl,c

The deadline-based constraints ensure that the PEVs would be charged before a predetermined deadline. Let tD l,c be the deadline of the charging requests queued by aggregator l under cluster c. The deadline-based constraint requires the queue backlog to be empty at the deadline, which is mathematically expressed as: Ql,c (t) = 0, t = tD l,c . 3. CONTINUOUS-TIME MODEL PREDICTIVE CONTROL OF PEV CHARGING In this section, we propose a continuous-time model predictive control (MPC) for co-optimizing the generation schedules and the PEV load flexibility in real-time power systems operation, with the goal of serving the realtime system load at minimum cost. The structure of the proposed continuous-time MPC is shown in Fig. 2. In Fig. 2, the power system operator receives the generation constraints and bids of generating units as well as the queued flexible PEV loads of aggregators, and utilizes the proposed continuous-time MPC model for co-optimizing

Aggregator L

Cluster 1

Aggregator 1

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Cluster C

the continuous-time PEV charging power and ramping trajectories with the power and ramping trajectories of Queue 1 Queue C generating units in real-time operation of power systems. The proposed model shapes the PEV charging power trajectories by optimally scheduling their departure from the queues, while satisfying delay-based and deadline-based service quality constraints of the PEVs. Dispatchable Generating Units

Aggregator L

Queue 1

Queue C

Aggregator 1

Queue 1

Time

Fig. 3. Continuous-time MPC model timeline ˙ as D(t) = dD(t) dt . The design objective of the proposed continuous-time MPC is to optimize the control trajec˙ ˙ tories G(t) and D(t) over Tτ to minimize the operation cost (4), subject to operation constraints (5)-(12).    C G(t) dt (4) min ˙ ˙ G(t), D(t)

Queue C

˙ Q(t) = J(t) − D(t), t ∈ Tτ ,

(5)

G(t) ≤ G(t) ≤ G(t), t ∈ Tτ ,

(7)

˙ ˙ ˙ G(t) ≤ G(t) ≤ G(t), t ∈ Tτ , ˙ ˙ ˙ ≤ D(t) ≤ D(t) t ∈ Tτ , D(t)

(9)

1TK G(t) Power System Operator

= 1TLC D(t) + DI (t), t ∈ Tτ ,

D(t) ≤ D(t) ≤ D(t), t ∈ Tτ ,

Continuous-time MPC-based PEV Charging Control Continuous-time PEV Charging Power and Ramping Trajectories

Tτ

Power and Ramping Trajectories of Generating Units

Q(t) ≤ Q(t) ≤ Q(t), t ∈ Tτ , G(τ ) = Gτ , D(τ ) = Dτ , Q(τ ) = Qτ .

Fig. 2. Schematic of the proposed model The proposed continuous-time MPC implements the receding horizon control idea and involves solving an optimal control problem for the next immediate control horizon Tτ = [τ, τ + T ], while utilizing the continuous-time state space model of generating units and PEV queues, proposed in Section 2. Two consecutive receding control horizons, represented by Tτ1 and Tτ2 , are shown in Fig. 3, which start respectively at τ1 and τ2 and end at τ1 +T and τ2 +T . The proposed continuous-time MPC optimizes the control decisions of generating units and PEV loads over each control horizon Tτ1 , Tτ2 , . . ., capturing the intertemporal flexibility and constraints of the PEV load queues in realtime operation of power systems, implementing only the first instant of control decisions at each run. 3.1 Problem Formulation Consider a set of L aggregators of PEVs and K generating units that are available to supply the flexibility requirement in real-time power systems operation. For each t ∈ Tτ , the set of K generating units are modeled by continuous-time generation trajectories G(t) = (G1 (t), ..., GK (t))T . The continuous-time ramping trajectories of generating units, defined as time derivatives of ˙ the generation trajectories, are represented by G(t) =  T ˙ ˙ G1 (t), . . . , GK (t) . The operation cost of generating unit k is denoted as C(Gk (t)). Further, L aggregators, each offering C clusters of PEV charging loads, are represented by a CL-dimensional controlled PEV load vector T th D(t) = (D1 (t), . . . , DLC (t))  T , with l sub-vector Dl (t) = D(l−1)C+1 (t), . . . , DlC (t) representing the clusters offered by aggregator l, which are modeled using the model in Section 2. Similar to the continuous-time ramping trajectory of generating units, we define the continuoustime ramping trajectory of controlled PEV charging load 500

(6) (8)

(10) (11) (12)

The objective functional (4) is equal  tototal operation cost of generating units C G(t) = K Ck Gk (t) over Tτ . The state equations of the queuing systems for PEV loads T is formulated in (5), where Q(t) = (Q1 (t), . . . , QLC (t)) T and J(t) = (J1 (t), . . . , JLC (t)) are respectively the vectors of queue backlog and PEV charging power request trajectories. The continuous-time power balance constraint of the system is formulated in (6), where DI (t) is the inflexible component of the system real-time load (load minus the aggregated PEV charging load requests), and 1K and 1LC are K- and LC-dimensional unit vectors. The continuous-time power and ramping constraints for generating units and PEV load aggregators are stated in (7)-(10), where the time-varying underlined and overlined terms respectively represent the minimum and maximum limits of the trajectories. The service quality constraints of PEV loads, as discussed in Section 2, are translated to a maximum limit on the queue backlog (11). Initial values of the state trajectories are enforced in (12), where Gτ , Dτ , Qτ are vectors of constant initial values at time τ . Note that the PEV charging load request vector J(t) in (5) is an input parameter for the optimal control problem (5)-(12), and is updated at each MPC run. 4. THE PROPOSED SOLUTION METHOD The proposed model in (4)-(12) is a continuous-time optimal control problem with infinite-dimensional decision space that is computationally intractable. Here we intend to leverage our previous works in Parvania and Scaglione (2016b,a), and develop a function space-based solution method for the proposed continuous-time optimal control problem (4)-(12). The proposed solution method is based on reducing the dimensionality of the continuous-time decision and parameter trajectories by modeling them in a finite-order function space spanned by the Bernstein poly-

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nomials. The Bernstein polynomials of degree Q include Q + 1 polynomials defined as:   Q q bq,Q (t) = t (1 − t)Q−q , t ∈ [0, 1], q = 0, ..., Q. (13) q Let us construct a set of basis functions in each interval n ∈ N of the receding scheduling horizon Tτ using the Bernstein polynomials of degree Q. Thus, the vector of (Q) (Q) basis functions e(Q) (t) = (e1 (t), . . . , eP (t))T spanning the whole Tτ contains P = (Q+1)N functions defined as:   t − tn (Q) (14) en(Q+1)+q (t) = bq,Q , t ∈ [tn , tn+1 ), Tn for n = 0, . . . , N −1; q = 0, . . . , Q. To reduce the notation, we define p ≡ n(Q+1)+q, where p goes from 0 to P − 1. Below we present the different components of the proposed models for generating units and PEV charging loads. 4.1 Modeling Trajectories in Bernstein Function Space Power Trajectories of Generating Units The Bernstein function space e(Q) (t) in (14) is utilized to model the generation trajectories of generating units as: G(t) = Ge(Q) (t), t ∈ Tτ , (15) where G is a K ×P matrix of coordinates of the generation trajectories in the function space. Details on modeling generation ramping trajectory as well as the operating constraints of generating units in the Bernstein function space are provided in Parvania and Scaglione (2016b). In the following, we utilize similar approach to model trajectories of the proposed PEV queuing model. PEV Charging Power Trajectories Let us project the C ×1 vector of controlled PEV charging power trajectories associated with the C clusters of aggregator l, Dl (t), in the function space e(Q) (t) as: Dl (t) = Dl eQ (t), t ∈ Tτ , (16) where Dl is the C × P matrix of associated Bernstein coefficients. We also project the C ×1 vector of PEV load request trajectories Jl (t) in the same function space: Q

Jl (t) = Jl e (t), t ∈ Tτ , (17) where Jl is the C ×P matrix of Bernstein coefficients. PEV Ramping Trajectories The time derivatives of the Bernstein polynomials of degree Q can be expressed as a linear combination of Bernstein polynomials of degree Q − 1 Dierckx (1995). This property is utilized to define the continuous-time ramping trajectories of PEVs in the space spanned by the Bernstein polynomials of degree Q−1, over the scheduling horizon Tτ , as follows: ˙ l e(Q−1) (t), (18) ˙ l (t) = Dl e˙ (Q) (t) = Dl Me(Q−1) (t) = D D where M is the P × (P − N ) matrix relating e˙ (Q) (t) ˙ l is the C × (P − N ) matrix of the and e(Q−1) (t), and D ramping Bernstein coefficients, which are linearly related to the Bernstein coefficients of corresponding PEV power trajectories as follows: ˙ l = Dl M. (19) D

PEV Queue Backlog Trajectories Integrals of the Bernstein polynomials of degree Q are linearly related to the Bernstein polynomials of degree Q + 1 Dierckx (1995), 501

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meaning that there exist a P × (P + N ) linear mapping N relating the integrals of the basis functions of degree Q to the basis functions of degree Q+1 as follows:  t e(Q) (t )dt = N e(Q+1) (t), (20) τ

where e(Q+1) (t) is a (P + N )-dimensional vector of Bernstein basis functions of degree Q+1. Integrating (5) over t and using (20), C × 1 vector of queue backlog trajectories of C clusters of aggregator l are calculated as:  t τ e(Q) (t )dt Ql (t) = Ql + (Jl − Dl ) τ

= Qτl + (Jl − Dl )N e(Q+1) (t)   = Qτl 1TP +N + (Jl − Dl )N e(Q+1) (t)

(21) = Ql e(Q+1) (t), t ∈ Tτ , where Qτl is the vector of constant initial backlog values calculated at the beginning of scheduling horizon Tτ from the solution of previous MPC run; Qτl 1TP+N in the third line is the projection of Qτl to the space spanned by e(Q+1) (t); and Ql is a C × (P + N ) matrix of Bernstein coefficients of the queue backlog trajectories defined as: Ql = Qτl 1TP +N + (Jl − Dl ) N . (22) Modeling Inequality Constraints The convex hull property of Bernstein polynomials is utilized here to efficiently apply the continuous-time inequality constraints (7)-(11). In the case of the service quality constraints (11), let R be the control polygon formed by the Bernstein coefficients Ql of the continuous-time queue backlog trajectory of aggregator l in the space spanned by e(Q+1) (t). The convex hull property of the Bernstein polynomials states that the queue backlog trajectory Ql,c (t) would never be outside of the convex hull of the control polygon R. Therefore, the continuous-time service quality constraints are imposed by limiting the associated Bernstein coefficients at each interval as follows: Ql,c ≤ Ql,c,n(Q+2)+q ≤ Ql,c , ∀q, n, c, l. (23) where Ql,c and Ql,c are respectively the minimum and maximum limits of queue backlog of aggregator l. The continuous-time inequality constraints on power and ramping trajectories can be similarly imposed by leveraging the convex hull property of the Bernstein polynomials. Continuity Constraints The optimality conditions of the proposed optimal control problem requires C 1 continuity of the generation and the PEV charging power trajectories in (15) and (16). In order to ensure the C 1 continuity requirement of PEV power trajectory at the interval connection points, we impose continuity constraints on the Bernstein coefficients of adjacent intervals as follows: Dl,c,n(Q+1)+Q = Dl,c,(n+1)(Q+1) , n = 0, . . . N − 2, (24)  1  1  Dl,c,n(Q+1)+Q −Dl,c,n(Q+1)+Q−1 = Dl,(n+1)(Q+1)+1 Tn Tn+1  (25) − Dl,(n+1)(Q+1) , n = 0, . . . N − 2.

Modeling Continuous-time Power Balance Constraint Let the inflexible net-load DI (t) in (6) be projected on the space spanned by the basis functions e(Q) (t) as: DI (t) = DI e(Q) (t),

t ∈ Tτ ,

(26)

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Fig. 4. Workplace PEV charging load requests where DI is a 1 × P vector of Bernstein coefficients. Substituting the Bernstein models of generation and the PEV charging power trajectories from (15) and (16), and the inflexible load trajectory from (26) in the continuoustime power balance constraint (6), and eliminating e(Q) (t) from both sides, we derive: 1TK G − 1TLC D = DI , (27) which converts the continuous-time power balance (6) into a set of algebraic equations on the Bernstein coefficients. In summary, Section 4 presented the proposed solution method that converts the continuous-time scheduling problem (4)-(12) into a LP problem with the Bernstein coordinates of decision trajectories as decision variables. 5. NUMERICAL RESULTS The numerical results are presented for the IEEE Reliability Test System Force (1999), where the day-ahead and real-time CAISO loads of 8a.m. Jan. 9, 2018 to 8a.m. the next day are scaled down to 2850 MW peak load of IEEERTS and used for the simulations OASIS (2018). We also use the California’s PEV arrival data for the residential (home), workplace, and public chargers from the National Household Travel Survey NHTS (2009) to form the PEV charging load request trajectories. In order to form these trajectories from (1), the NHTS arrival data serve as mean values of the Poisson distributions while the average charging power of the home, workplace and public charging stations are considered to be 8.8kW, 8.8kW, and 22kW. We use 9-hour receding control horizons for each MPC run, repeated every 15-minutes, and update the PEV demand request trajectories for each of the scheduling horizons by generating new random Poisson arrivals. As an example, the workplace PEV demand request trajectories are shown in Fig. 4 for the first five MPC runs. We assume that home PEV load is served immediately from hour 8 to 18, while after hour 18 the PEV charge control is allowed by the customers. For home PEV charging, a single deadline is imposed at the hour 8a.m. of the next day, and two deadlines are imposed for workplace charging at hour 18 of Jan. 9th, and hour 8 of the next day. We also impose a delay constraint of two hours on the public PEV charging. Two cases are studied here. In Case 1, the real-time power system operation without PEV charging control is investigated while in Case 2 the impact of PEV charging control flexibility on real-time operation is explored. In both cases, we first run a continuous-time day-ahead unit commitment problem as proposed in Parvania and Scaglione (2016b), and feed the commitment status of 502

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Fig. 5. Real-time load profiles with/without charge control generating units to the proposed real-time MPC model. The real-time MPC model is implemented and solved in GAMS, where the 9-hour receding scheduling horizons are divided to 15-minute intervals and the charging demand request trajectories at each interval are modeled with Bernstein polynomials of degree 3. The same function space is used for modeling the control decisions. The total day-ahead and real-time operation costs for Cases 1 and 2 equals to $451, 978.6 and $443, 272.9, where tapping the flexibility of PEV charging loads reduces the operation cost in Case 2 by $8, 705.7 as compared to Case 1. The continuous-time real-time load and ramping requirement of the system for Cases 1 and 2 are shown respectively in Figs. 5-(a) and (b). In Fig. 5-(a), charging control of PEVs levels out the load profile and reduces its peak. In addition, the proposed control method efficiently leverages the ramping flexibility of PEV charging load, which considerably reduces the ramping requirement of the system in Fig. 5-(b). More specifically, the PEV load control reduces average absolute value of the system ramping requirement from 111.9MW/h to 61.4MW/h, which considerably reduces the burden of providing ramping from the generating units. The PEV charging load and the queue backlog for three PEV charging types are shown in Fig. 6. In Fig. 6-(a), the PEV load request and the controlled charging power coincide from hour 8 to 18, and no queue is formed in this period. After the hour 18, the queue backlog trajectory starts forming to alleviate the peak load of the system, while it diminishes during the night hours as the queued PEV load is served by the system operator, and reaches zero at by the charging deadline of 8a.m. of the next day. As we observe in Fig. 6-(b), the controlled charging power trajectory exhibits several sub-hourly changes that reflects the tracking of real-time changes in workplace charging load requests delineated in Fig. 4. In Fig. 6-(b), the queue backlog trajectory reaches zero at the charging deadline hours 18 of Jan. 9th, and 8a.m. of the next day. The controlled power of the public chargers in Fig. 6-(c) shows more volatility due to the two-hour delay constraint.

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Fig. 6. PEV load with/without charge control for: a) Home charging, b) Workplace charging, c) Public charging. 6. CONCLUSION In this paper, a continuous-time MPC model is proposed for co-optimizing the charging flexibility of PEVs with the generation of generating units in real-time power systems operation. A function space based method is developed for accurate and scalable solution of the proposed continuoustime problem. The simulation results reveal the effectiveness of the proposed model in deploying the flexibility of PEV charging load for reducing the system operation cost and reducing the ramping requirements from generating units, while meeting the service quality constraints of the PEV owners. The future works include developing a the stochastic continuous-time MPC and models for pricing the PEV flexibility in real-time operation. REFERENCES Di Giorgio, A., Liberati, F., and Canale, S. (2014). Electric vehicles charging control in a smart grid: A model predictive control approach. Control Engineering Practice, 22, 147–162. Dierckx, P. (1995). Curve and surface fitting with splines. Oxford University Press. DOE (2017). National plug-in electric vehicle infrastructure analysis. Technical report. Fan, Z. (2012). A distributed demand response algorithm and its application to phev charging in smart grids. IEEE Transactions on Smart Grid, 3(3), 1280–1290. 503

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Force, I.R.T. (1999). The ieee reliability test system-1996. IEEE Trans. Power Systems, 14(3), 1010–1020. Guo, Y., Pan, M., Fang, Y., and Khargonekar, P.P. (2013). Decentralized coordination of energy utilization for residential households in the smart grid. IEEE Trans. Smart Grid, 4(3), 1341–1350. Hafez, O. and Bhattacharya, K. (2016). Queuing analysis based pev load modeling considering battery charging behavior and their impact on distribution system operation. IEEE Transactions on Smart Grid. Hao, H. and Chen, W. (2014). Characterizing flexibility of an aggregation of deferrable loads. In Proc. IEEE 53rd Annual Conference on Decision and Control (CDC), 4059–4064. Hosseini, S.S., Badri, A., and Parvania, M. (2014). A survey on mobile energy storage systems (mess): Applications, challenges and solutions. Renewable and Sustainable Energy Reviews, 40, 161–170. IEA (2017). Global ev outlook 2017. Technical report. Khatami, R., Heidarifar, M., Parvania, M., and Khargonekar, P. (2018a). Scheduling and pricing of load flexibility in power systems. IEEE Journal of Selected Topics in Signal Processing. Khatami, R., Parvania, M., and Khargonekar, P. (2018b). Scheduling and pricing of energy generation and storage in power systems. IEEE Transactions on Power Systems, 33(4), 4308–4322. Khatami, R., Parvania, M., and Oikonomou, K. (2018c). Continuous-time optimal charging control of plug-in electric vehicles. In Proc. 2018 IEEE Innovative Smart Grid Technologies Conference (ISGT), 1–5. Liu, Y., Yuen, C., Yu, R., Zhang, Y., and Xie, S. (2016). Queuing-based energy consumption management for heterogeneous residential demands in smart grid. IEEE Transactions on Smart Grid, 7(3), 1650–1659. Luo, X. and Chan, K.W. (2013). Real-time scheduling of electric vehicles charging in low-voltage residential distribution systems to minimise power losses and improve voltage profile. IET generation, transmission & distribution, 8(3), 516–529. NHTS (2009). URL http://nhts.ornl.gov/. National Household Travel Survey. OASIS (2018). URL http://oasis.caiso.com. California ISO Operan Access Same-Time Information System, Jan. 2018. Palomino, A. and Parvania, M. (2018). Probabilistic impact analysis of residential electric vehicle charging on distribution transformers. In Proc. 2018 North American Power Symposium (NAPS2018), 1–6. Parvania, M. and Scaglione, A. (2016a). Generation ramping valuation in day-ahead electricity markets. In Proc. 49th Hawaii International Conference on System Sciences (HICSS), 2335–2344. Parvania, M. and Scaglione, A. (2016b). Unit commitment with continuous-time generation and ramping trajectory models. IEEE Transactions on Power Systems, 31(4), 3169–3178. Soltani, N.Y., Kim, S.J., and Giannakis, G.B. (2015). Real-time load elasticity tracking and pricing for electric vehicle charging. IEEE Transactions on Smart Grid, 6(3), 1303–1313.