Power Output Smoothing for Renewable Energy System: Planning, …kcleung/papers/journals/power... · 2020-03-15 · CHEN et al.: POWER OUTPUT SMOOTHING FOR RENEWABLE ENERGY SYSTEM:

1034 IEEE SYSTEMS JOURNAL, VOL. 14, NO. 1, MARCH 2020

Power Output Smoothing for Renewable EnergySystem: Planning, Algorithms, and Analysis

Xiangyu Chen , Student Member, IEEE, Ka-Cheong Leung , Member, IEEE, andAlbert Y. S. Lam , Senior Member, IEEE

Abstract—The growing penetration of renewable energy sourcesin electricity generation will bring challenges to the power gridoperations due to the intermittency and fluctuation of renewables.In this article, we employ an energy storage system (ESS) in a grid-connected renewable energy system (RES) to serve the electricalload from the power grid. We study the control algorithms of theESS to smooth the renewable generation and analyze the tradeoffbetween the ESS size and the system performance, i.e., renewableutilization and operation cost. We provide performance guaranteesof the proposed algorithms and propose an optimization frame-work to configure the ESS. To analyze the reliability of the RES, ananalytical framework using the Markov modulated fluid queue isdevised. It is shown in our simulation studies that the devised poweroutput smoothing algorithm outperforms other existing algorithmsin terms of operation cost under different sizes of ESSs. We also findthat, to satisfy the renewable utilization requirement, the requiredESS size blows up with both the amplitude of fluctuation andthe intermittency level of the profile gap between the renewablegeneration and the load.

Index Terms—Energy storage system (ESS), Internet of Energy(IoE), power output smoothing (POS), renewable energy system(RES), smart grid.

NOMENCLATURE

t A particular time slot.T Total operation time in Δt.Δt Length of a time slot in minutes.D(t) Network load at Time Slot t in kW.R(t) Renewable generated power at time slot t in kW.B Size of the ESS in kWh.TL Total lifetime of the ESS in Δt.Pc(t) Charging power of the ESS at time slot t in kW.Pd(t) Discharging power of the ESS at time slot t in kW.P c Maximum charging power of the ESS in kW.P d Maximum discharging power of the ESS in kW.η Charging efficiency of the ESS.

Manuscript received November 7, 2018; revised February 20, 2019, July 13,2019, and October 27, 2019; accepted November 23, 2019. Date of publicationJanuary 1, 2020; date of current version March 2, 2020. This work was supportedin part by the Research Grants Council of the Hong Kong Special AdministrativeRegion, China, under Grant 17261416, and in part by the National NaturalScience Foundation of China, under Grant 51707170. This article is the extendedversion of the prior work [17]. (Corresponding author: Ka-Cheong Leung.)

X. Chen is with the Department of Electrical and Electronic Engineering, TheUniversity of Hong Kong, Hong Kong (e-mail: [email protected]).

K.-C. Leung is with the School of Computer Science and Technology, HarbinInstitute of Technology, Shenzhen 518055, China (e-mail: [email protected]).

A. Y. S. Lam is with the HKU Shenzhen Institute of Research and Innovation,Shenzhen 518057, China, with the Department of Electrical and ElectronicEngineering, The University of Hong Kong, Hong Kong, and also with theFano Labs, Hong Kong (e-mail: [email protected]).

Digital Object Identifier 10.1109/JSYST.2019.2958176

β Discharging efficiency of the ESS.S(t) State of charge of the ESS at time slot t in kWh.Pout(t) Actual power output from the RES at time slot t in kW.Pe(t) Power injection from other generation units at time slot

t in kW.Ps(t) Spilled power from the RES at time slot t in kW.p(t) Unit cost of buying electricity from the controllable

generation unit at time slot t in dollars/kWh.s(t) Unit spillage cost at time slot t in dollars/kWh.Pr Power output reference for the RES in kW.Cg Electricity cost stemmed from the controllable gener-

ation unit in dollars.Cs Renewable spillage cost in dollars.Ce Construction cost for ESS in dollars.Rc Contract revenue of the RES in dollars.C Maximum contracted power output from the RES in

kW.A Generator matrix for wind generation process.d(s) Energy rate of the ESS at the drift state s.M Number of the drift state for the energy rate of ESS.E Drift matrix for the energy rate of ESS.F Limiting probability vector of the SOC of the ESS.π Limiting probability vector of the continuous time

Markov chain in Section IV-A.λi ith eigenvalue for matrix pair (E,A).φi ith eigenvector for matrix pair (E,A).αi ith coefficient for the spectral representation of F.X(t) Length of the virtual queue.L(t) Defined Lyapunov function.Δ(t) Conditional Lyapunov drift.V Parameter that describes the weight of the penalty

function.Sth Threshold for SOC.Cavg Daily average cost in dollars.URG Renewable energy utilization.

I. INTRODUCTION

DUE to concerns on global warming, many countries haveestablished green policies to restrain the greehouse gas

emissions, such as employing more distributed renewable energysources in place of thermal power plants. With the integrationof modern communication and Internet of things technologies,distributed renewable energy sources with distributed and em-bedded intelligence are interfaced to the smart grid, which formsthe Internet of Energy (IoE) [1]. A major goal of the IoE isto provide reliable and cost-efficient electricity services to thedemand-side of the power grid [2]. To support the efficientand reliable operation of IoE, techniques to integrate renewableenergy sources are, thus, very important in the smart grid.

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CHEN et al.: POWER OUTPUT SMOOTHING FOR RENEWABLE ENERGY SYSTEM: PLANNING, ALGORITHMS, AND ANALYSIS 1035

In fact, how to effectively integrate distributed renewableenergy resources in the smart grid remains challenging. Therenewables cannot provide reliable power output to the powergrid due to its fluctuation [3] and intermittency [4] properties.Moreover, due to the stochastic natures of the renewables, thedaily profile of the renewable generated power is hard to be accu-rately predicted [5]. The uncertainty induced by the renewablescan cause power imbalance in the power grid, which may resultin frequency deviations and jeopardize the grid stability.

With the aforementioned challenges, research on the poweroutput smoothing (POS) techniques for renewable generationgains popularity in recent years. Basically, POS aims to flattenthe renewable generation profile so as to assist in the integrationof the renewable generation in the operation of IoE. How toimplement POS for a renewable energy system (RES) rests in theway to level off the fluctuations of renewable generation basedon the uncertainty of renewables. Energy storage system (ESS),which can reserve excessive renewable energy through chargingand compensate for energy shortage in the power grid throughdischarging, is considered to be a good candidate to help copewith the fluctuations of renewable generation. Several previousstudies investigated smoothing the power fluctuations of renew-able generation with the help of ESS [6]–[9], [11], [13], [14]. Tostabilize the photovoltaic (PV) power output, a model predictivecontrol (MPC) strategy was proposed in [7] to smooth the powerand voltage profiles. The proposed strategy requires no com-munication between the ESS and the PV system, since controlcommands can be sent along with local measurement data. In [8],a heuristic smoothing control method was proposed for an ESSstation to help reduce the fluctuations of the renewable generatedpower. The proposed method determined power output levelsbased on a dynamic filtering controller and, then, regulated thestate-of-charge (SOC) of each battery based on the proposedpower allocation method. In [14], an SOC feedback strategyfor ESS control was designed to smooth out the fluctuations ofwind power while keeping the SOC of the ESS within its properrange. The POS schemes based on filter design were devisedin [6] and [9]. In [9], Li-ion capacitors were deployed for filteringpower variations and control techniques were devised based onlow-pass filter (LPF). In [6], a general filter was designed toimprove the performance of wind power smoothing using LPF.A metaheuristic method was, then, proposed to calculate thecapacity and the power rating of the ESS. The aforementionedapproaches [6]–[9], [14] fail to provide theoretical performanceguarantee on POS. Moreover, they only aim at smoothing thereal-time fluctuations of renewable generation and do not reactto grid loads as well as electricity prices. Hence, the smoothedpower output still cannot be injected to the power grid norutilized for the operation of IoE.

To satisfy the demand of grid load and respond to electricityprices, some offline POS schemes have been devised to dispatchthe renewable generation based on the MPC framework [10]–[13]. MPC repetitively takes control actions from the solutionof an optimization problem with forward-looking objective andpredicts the future system state variables for the optimization.The combination of optimization process and prediction pro-cess aims to handle the uncertainty of renewables. In [11], acontrol approach based on stochastic MPC was proposed foran ESS so as to minimize the daily operation cost of an RES.In [10], stochastic MPC was applied to dynamic and adaptivereconfiguration of electrical distribution system with renewablepenetration so as to handle the uncertainty and variability ofrenewable energy resources. In [12], a control approach for

the joint charging management of plug-in electric vehicles andreconfiguration of electrical distribution system was proposedbased on stochastic MPC. Wang et al. [13] developed an op-timization and control method for a PV storage system basedon MPC. The proposed method includes an optimization pro-cess to provide energy arbitrage and smooth the PV output,and a prediction process based on the least square method.The MPC framework applied in [10]–[13] needs to performestimations for the future system states, e.g., wind speed forwind turbine [10]–[12], solar irradiance for PV panels [10]–[13], SOC of EVs [12], and electricity prices [13]. Therefore,effective and accurate forecast methods are still required to en-able the dynamic MPC-based scheduling. Moreover, except [6],the above mentioned techniques [7]–[9], [11], [13], [14] donot take the effect of ESS on the performance of POS intoaccount. Note that the configuration of ESS can impact onPOS to a large extent. Hence, ESS should be optimally con-figured and the method for the configuration of ESS should bedevised.

Some existing techniques considered the storage configura-tion problem in the RES [15], [16]. In [15], Anwar et al. proposeda two-stage strategy to configure ESS, in which the optimalpower output at each time instant was determined in the firststage and the minimum size of ESS was then determined in thesecond stage. Saez-de-Ibarra et al. [16] proposed a two-stageoptimization framework to determine the size of ESS in arenewable generation plant. The optimal operation profile ofthe battery for each day of the year was determined in the firststage and the optimal size of ESS was determined in the secondstage based on the results obtained in the first. The approachesproposed in [15] and[16] suffer the defects that the determinationof the size of ESS in the constructional level does not fully embedthe power scheduling in the operational level. Therefore, theoptimality of the approaches cannot be guaranteed. Moreover,the above works did not analyze the tradeoff between the ESSsize and the improvement in the system performance, say, systemreliability, when considering the generation and the load profiles.It should be noted that the profiles of generation and loadcan have impacts on this tradeoff, specifically: 1) How do thefluctuations of renewable generation and load impact on thistradeoff? 2) How shall we configure the ESS and power outputbased on the profiles of the generation and the load?

To answer the above questions, in this article, we proposePOS algorithms for an RES and analyze the tradeoff betweenthe ESS size and the improvement in system performance, i.e.,the reliability and utilization enhancements, as well as costreduction, on POS through theoretical analysis and simulation.We consider a grid-connected RES, with an ESS, which providesreliable power output to IoE. To utilize renewable generationto serve loads in distribution power network, a leaky bucket(LB)-inspired POS scheme is proposed to control the ESS. Inthe real operation, the configurations of ESS, i.e., the size of theESS as well as the maximum charging and discharging rates,can impact the overall system performance and should be jointlydetermined. To do this, we propose an optimization frameworkfor solving this joint storage configuration problem (JSCP) bytaking the generation profile into consideration. To minimize theoperation cost of the system, a price-adaptive power smoothingalgorithm is, then, proposed based on the Lyapunov optimizationto deal with time-varying electricity prices. We conduct simu-lations to study the effect of ESS size on system performanceunder different operation algorithms. Through theoretical studyand simulation, we also give some insights into how the ESS



should be configured to satisfy the performance requirementbased on different renewable generation profiles.

The preliminary conference version of this article can befound in [17]. In [17], we developed some control algorithmsof an ESS so as to smooth the power output of an RES. Inthis article, we provide theorems to analyze the performanceof the POS scheme proposed in [17]. We, then, extend theresults in [17] by conducting a theoretical analysis to analyzethe tradeoff between the ESS size and the system performance.We also perform simulations to verify such tradeoff. Moreover,a Lyapunov-based load-adaptive power smoothing algorithm isproposed in this article, which outperforms the heuristic-basedload-adaptive power smoothing (HB-LAPS) algorithm in [17]on operation cost. We also provide the performance guarantee forthe proposed algorithm theoretically. The specific contributionsof this article are as follows.

1) To utilize renewable generation to serve network loads indistribution power network, a POS scheme is proposed tocontrol the ESS. We prove that the proposed scheme canachieve the optimal renewable energy utilization.

2) We propose an optimization framework to configure theESS. The proposed framework fully embeds the schedul-ing scheme in the operational level, which differs from theexisting approaches [15], [16].

3) A price-adaptive power smoothing algorithm is devisedto minimize the operation cost of the RES based on theLyapunov optimization. Compared to [11] and [13], it re-quires no forecast information on generation and networkload, thereby is more practical to be implemented. Wealso provide the deterministic performance guarantee ofthe algorithm on operation cost.

4) In this article, the wind power generation process ismodeled as a Markov process. We conduct a theoreticalanalysis to analyze the reliability of an RES, measured bythe expected loss of power supply, based on the Markovmodulated fluid queue (MMFQ), which provides steady-state analysis for a system with Markovian arrivals. Tothe best of our knowledge, we are the first to conduct thetheoretical analysis on the expected loss of power supplyof an RES by using MMFQ.

The rest of this article is organized as follows. Section II intro-duces the system model. Section III describes the POS schemefor RES. An storage configuration problem is, then, formulatedand solved. In Section IV, a theoretical framework to analyze thetradeoff between the ESS size and reliability is devised. To adaptto the electricity price, a Lyapunov-based price-adaptive powersmoothing (LB-PAPS) algorithm is proposed in Section V. Theperformance evaluation results are discussed in Section VI andthe article concludes in Section VII.

II. SYSTEM MODEL

As shown in Fig. 1, an RES is considered, which is connectedto a power network with the power and information flows indi-cated. Consider that the RES operates over a scheduling horizondivided equally into T slots. Each time slot has a duration of Δtminutes. Each t ∈ 1, 2, . . ., T = T denotes a time slot in theoperation period. All system states only change at the beginningof a time slot. The total load of the power network is modeled asthe network loadD(t). Note thatD(t) is not a single load but theequivalent aggregated load in the power network interface. TheRES aims to provide stable power output to serve the networkload. The RES consists of an ESS, a storage controller, and a

Fig. 1. Architecture of the RES.

wind power plant. The renewable generated power at time slot tis denoted asR(t) kW. The ESS has a lifetime ofTL minutes, andits size is B kWh where B > 0. It gets charged or discharged atthe rate ofPc(t) ≤ P c orPd(t) ≤ P d at time slot t, respectively,where P c (P d) is the maximum charging (discharging) ratein kW, respectively. Note that the charging and dischargingprocesses of the ESS cannot happen at the same time. Theconstruction cost of ESS includes two parts, i.e., the cost in termsof the size of the ESS cB dollars/kWh and the cost in terms ofthe maximum charging rate as well as the maximum dischargingrate, cc dollars/kW, and cd dollars/kW, respectively. The SOCof the ESS at time t is denoted as S(t) kWh. 0 < η < 1 and0 < β < 1 represent the charging and discharging efficienciesof the ESS, respectively. The storage controller controls the ESSso as to smooth out the power output from the RES. The poweroutput from the RES at time slot t, Pout(t) kW, is given by

Pout(t) = R(t)− Pc(t) + Pd(t). (1)

In traditional power grid without the integration of renew-ables, controllable generation units are dispatched to serve thenetwork load. In our proposed power grid model with the inte-gration of renewables, the RES is employed to provide stablepower output to the network load D(t), with the help of thecontrollable generation units. When Pout < D(t), the poweroutput from the RES is insufficient to satisfy the network load.The controllable generation units, then, compensate for thepower shortage through injecting power of Pe(t) kW such that

Pe(t) = D(t)− Pout(t). (2)

The time-varying price of purchasing electrical energy from thecontrollable generation units is given by p(t) dollars/kWh.

When Pout(t) > D(t), which generally happens when theESS is full and the renewable generated energy is excessive,namely, R(t) > D(t), a portion of power output from the RESequal to Ps(t) = Pout(t)−D(t) is spilled and the unit spillagecost is given by s(t) dollars/kWh. We assume that s(t) ≤ p(t)since the electricity generation cost is much more expensive thanthe spillage cost in general.

III. POS CONTROL FOR RES

In this section, a POS scheme is devised for the RES tosmooth out the power fluctuations of the renewable generationto serve the network load. The proposed framework includes anoperation algorithm to control the charging/discharging schemeof the ESS and a planning problem to optimize the configurationsof the ESS. The principle of the POS scheme is first introduced,which defines the control scheme of the ESS in the operationallevel. A joint storage configuration problem (JSCP) is, then,discussed, in which it optimally configures the size and thecharging/discharging rates of the ESS as a planning problem.



Algorithm 1: POS Scheme.

Input: Real-time renewable generation R(t), SOC of theESS S(t), size of the ESS B, power output referencePr(t), P c, P d.

Output: Pc(t), Pd(t).1: for each time slot Δt do2: Calculate charging demand

Gc(t) = max0, R(t)− Pr(t).3: Calculate discharging demand

Gd(t) = max0, Pr(t)−R(t).4: Calculate maximum charging bound before the ESS is

full by current time slot Mc(t) =B−S(t)ηΔt .

5: Calculate maximum discharging bound before theESS is empty by current time slot Md(t) =

S(t)βΔt .

6: Calculate Pc(t) = minGc(t),Mc(t), P c.7: Calculate Pd(t) = minGd(t),Md(t), P d.8: Update S(t+ 1) = ηPc(t)Δt− Pd(t)

β Δt+ S(t).9: Return Pc(t), Pd(t).

10: end for

A. Design Principle

To implement POS, it is required to regulate the renewableenergy sources with uncertainty. By employing ESS, the REShas additional capabilities to manage its power output. A controlscheme of the ESS is, thus, needed to smooth the power outputfor power grid operation so as to keep power balance in the grid.When devising the control scheme, we found that the stochasticnature of renewable energy sources shares similarities withthose of network traffic sources. Actually, in network domain, awell-known algorithm, named the LB mechanism [22], has beendevised to regulate the network traffic sources (traffic shaping).Basically, the LB mechanism can smoothen the stochastic trafficsources by restricting the output traffic rate at a preset constantlevel and holding the awaiting packets for transmission in thenetwork buffer. Inspired by the LB mechanism, a control schemeis devised and applied to POS for the RES. In this scheme, theway how the ESS manages the energy can be analogized asthe way in which the LB handles the fluid. The RES servesthe network load with a power output reference that equals thenetwork load, i.e., Pr(t) = D(t) kW. The ESS is controlled tomatch Pout(t) to Pr(t) as close as possible.

An algorithm is devised to implement the POS schemeas Algorithm 1. In Steps 2–3, the charging/discharging de-mands are calculated so as to match Pout(t) to Pr(t). InSteps 4–5, the charging/discharging bounds are calculatedbased on the SOC constraints. In Steps 6–7, Pc(t) and Pd(t)are calculated by jointly considering the SOC constraints,the charging/discharging demands, and the maximum charg-ing/discharging rates. In Algorithm 1, the power output of RESis controlled to match the network load as close as possible.Moreover, the maximum charging/discharging rates, the SOCconstraints, and charging/discharging efficiencies are requiredto be considered in the POS scheme. Since POS scheme onlyrequires simple arithmetics and max/min calculations of currentstate variables, the time complexity of Algorithm 1 is O(1).

Define the renewable energy utilization URG in the RES asfollows:

URG =

∑Tt=1 minD(t), Pout(t)Δt

∑Tt=1 R(t)Δt

. (3)

Theorem 1: The devised POS scheme can achieve the opti-mal URG.

Proof: See Appendix A. In the proposed scheme, the maximum charging and dis-

charging rates in kW (P c and P d), and the size of the ESSB kWh, altogether have a great impact on the system perfor-mance. Therefore, these values are required to be configuredappropriately. To optimally configure these values, a JSCP isformulated.

B. Problem Formulation

The control objective of the JSCP is to provide the stablepower supply to meet the network load. Meanwhile, powershortage and power spillage should be avoided with minimumESS cost. Therefore, the objective function should capture theoperation costs due to power shortage and power spillage, andthe construction cost of storage.

The cost of purchasing electrical energy from the controllablegeneration unit Cg in dollars is given by

Cg =T∑

t=1

G(t) =T∑

t=1

Δt[Pr(t)− Pout(t)]+p(t) (4)

where [x]+ means maxx, 0 and G(t) is the electricity cost intime slot t. The renewable spillage cost Cs is given by

Cs =T∑

t=1

H(t) =T∑

t=1

Δt[Pout(t)− Pr(t)]+s(t) (5)

where H(t) is the renewable spillage cost in time slot t.The construction cost of the ESS Ce in dollars is given by

Ce = (cbB + ccP c + cdP d) ·TΔt

TL(6)

which gives the weighted construction cost of the ESS for thescheduling horizon of T time slots. cbB is the constructioncost in terms of the size of the ESS and ccP c + cdP d is theconstruction cost in terms of the maximum charging and dis-charging rates. For simplicity, the capacity decay of the ESS isnot considered in this model since during the ESS lifetime, thecapacity decay of ESS is not significant [24].

The objective function f is, then, expressed as

f = Cg + Cs + Ce. (7)

The charging/discharging power of the ESS followsAlgorithm 1. The storage controller aims to equalize Pout(t)and Pr(t) when there exists power discrepancy between thesetwo. Meanwhile, the maximum charging/discharging rates andthe SOC constraints are considered.

The charging power of the ESS, Pc(t) kW, is, thus, given by

Pc(t) = min

P c,B − S(t)

ηΔt,max0, R(t)− Pr(t)

(8)

in which max0, R(t)− Pr(t) gives the amount of chargingdemand in kW in case of power excess, P c is the maximumcharging rate, and B−S(t)

ηΔt is the maximum charging bound toensure that the ESS will not be charged over its upper bound Bduring Slot t.

Similarly, the discharging power of the ESS, Pd(t) kW, is

Pd(t) = min

P c,S(t)β

Δt,max0, Pr(t)−R(t)

(9)

in whichmax0, Pr(t)−R(t)gives the amount of dischargingdemand in kW in case of power shortage, P d is maximum



discharging rate, and S(t)βΔt is the maximum discharging bound

to ensure that the ESS will not be discharged under its lowerbound 0 during slot t.

Since the charging and discharging processes of the ESScannot happen at the same time, we have

Pd(t) · Pc(t) = 0. (10)

The SOC of the ESS is updated as

S(t+ 1) = ηPc(t)Δt− Pd(t)

βΔt+ S(t) (11)

and S(1) denotes its initial SOC. Moreover, the SOC of the ESSis restricted by the size of the ESS, which follows:

B ≥ S(t) ≥ 0. (12)

Based on the aforementioned control objective and con-straints, the optimization for JSCP is formulated as

minimizeB,P c,Pd

(7)

subject to (1), (8)–(12). (13)

By solving (13), the optimal values of P c, P d, and B canbe found. Due to constraints (8)–(10), the problem is a non-linear equality constrained optimization (nonconvex) and, thus,intractable. We need to relax the problem so as to solve it moreefficiently, which will be discussed in Section III-C.

C. Problem Relaxation

Here, JSCP (13) is relaxed as a mixed-integer linear program(MILP) problem. First, we convert constraint (8) into

⎧⎪⎨

⎪⎩

0 ≤ Pc(t) ≤ B−S(t)ηΔt

0 ≤ Pc(t) ≤ a0P c

0 ≤ Pc(t) ≤ R(t).

(14)

(15)

(16)

Similarly, constraint (9) can be converted into⎧⎪⎨

⎪⎩

0 ≤ Pd(t) ≤ S(t)βΔt

0 ≤ Pd(t) ≤ a1P d

0 ≤ Pd(t) ≤ Pr(t).

(17)

(18)

(19)

Note that a0 and a1 need to satisfy

a0 + a1 = 1, where a0, a1 ∈ N . (20)

Equations (14)–(20) are linear constraints with mixed integersand sufficient conditions for (8)–(10). Then, a relaxed MILPproblem is obtained as

minimizeB,P c,Pd,Pc(t),Pd(t),a0,a1

(7)

subject to (1), (11), (12), and (14)–(16).(21)

Theorem 2: Suppose that the electricity price p(t) and theunit spillage cost s(t) are constant. The optimal solutions forP c, P d, B in (21) are these of (13).

Proof: See Appendix B. By Theorem 2, the solution of the original JSCP (13) can be

obtained from (21), which is an MILP.

IV. RELIABILITY ANALYSIS OF RES

In Section III, we have devised POS scheme for the RES.For such an RES, it is important to theoretically analyze theperformance of the system, like system reliability, under theproposed POS scheme. System reliability is an important metricfor the power grid operation, which reflects to what extent the

Fig. 2. Probability density distribution and auto-correlation function of windpower from Markov chain model and real-world data. (a) Probability densitydistribution of wind power. (b) Autocorrelation function of wind power.

power supply can meet the network load. In our model, thesystem reliability is measured by the expected loss of powersupply (LOPSE) [25], which is the expected amount of shortageof the power supply compared to D(t)

LOPSE =1

T

T∑

t=1

max(D(t)− Pout(t)), 0 (22)

where Pout(t) is the power output for RES at time t.In this section, an analytical framework is proposed to analyze

the reliability of the RES with the proposed POS scheme. Basedon the proposed framework, the tradeoff between the size of theESS and the system performance, i.e., system reliability, can beevaluated.

The development of this section is as follows. First, thewind power generation process is modeled as a Markov processinspired by [26] and[27]. Second, the mathematical model forthe network load is provided. Then, we analyze the evolution ofthe SOC of the ESS using MMFQ [28]. Finally, the LOPSE iscomputed theoretically through the framework.

A. Wind Generated Power Modeling

To analyze an RES with stochastic renewable sources, weneed to model the renewable generated power mathematically.In fact, renewable generated power shows stochastic fluctua-tions, day periodic patterns, and seasonal patterns. Thus, it canbe described as a stochastic process. The Markov process forrenewable generated power modeling is a simplified model thatcan capture the stochastic properties of the renewable generationand make it feasible to be analyzed [27]. In this article, we modelthe wind generation process as a continuous time Markov chain(CTMC) [26], [27]. The phase-based Markov process divides thetime-varying wind generated power intoM phases or power lev-els. The transitions among different phases are, then, representedby an generator matrix A = [Aij ], i, j = 0, . . .,M − 1. Thegenerator matrix can be calculated using the method proposedin [27].

To validate the effectiveness of the Markov chain model forrenewable generated power, we conduct a simulation study tocompare the Markov chain model with the real-world data. Thereal data of renewable generated power is obtained from [29] andthe Markov chain model is trained using the method proposedin [27]. According to [27], time series model for wind generationis required to imitate the major statistical properties of the realdata, i.e., the probability density distribution and the autocor-relation function. Based on our simulation results illustratedin Fig. 2, the probability density distribution of the generatedpower from Markov chain model can match well with thereal-world data. The autocorrelation function of the generatedpower modeled by Markov chain can mimic the trend of the



Fig. 3. CTMC model for energy rate.

autocorrelation function of the real-world data. In Fig. 2(b),the periodic fluctuations in the real-world data arise from thedaily periodicity of the wind profile. Note that, due to theintractability of the analytical model, the periodicity of the windgenerated power has not been incorporated in our model. Thiswill not result in much inaccuracy in wind generation modeling,since the periodicity of wind generation is not significant. InSection VI-D2, we further show that although the periodicity isnot incorporated in the wind generation model, the CTMC modelstill shows its accuracy in wind generation modeling since theanalytical results using the CTMC model match well with thesimulation results using the real-world data.

B. Network Load Model

Different from the wind generation process that is highly fluc-tuated and intermittent, the network load profile in a residentialarea reflects a periodic pattern due to the regular electricityconsumption behaviors of residents [30]. To model the dailyperiodicity of the network load profile, the mathematical modelof network load is formulated as

D(t) = L(t− nR) if 0 ≤ t− nR < R, n ∈ N . (23)

According to (23), the network load profile D(t) is a periodicfunction with period R = 1440 min. It is mapped to a daily loadcurve L(t) with domain [0, R).

C. Theoretical Results for System Reliability Using MMFQ

After the wind generation process has been modeled as aCTMC, the RES with the Markovian energy arrival processand the energy departure process following Algorithm 1 can beanalyzed using MMFQ [28]. MMFQ can describe the evolutionof a system with the Markov arrival process and the constant-ratedeparture process. It can provide a framework for the steady-state analysis of such a system. The state of wind generationprocess at time t is represented by Sw(t) and the energy rate ofESS at drift state Sw(t) = s is denoted by d(s, t) = s−D(t),where s = 0, 1, . . .,M − 1. (Here, d(s, t) > 0 means charg-ing and d(s, t) < 0 means discharging). We assume that thecharging/discharging bounds for ESS are sufficiently large, i.e.,d(0) ≥ P c and d(M − 1) ≤ P d, so that the energy rate canalways be satisfied. Note that even if this assumption does nothold, we can still remove the states in CTMC in which theenergy rate cannot be satisfied and modify the generator matrixA accordingly so that the CTMC can model the energy rate of theESS. Transitions among d(s, t) are controlled by the generatormatrix A = [Aij ] as shown in Fig. 3. We define the drift matrixEr as Er = diag[d(0, r), d(1, r), . . ., d(M − 1, r)] = diag[0−L(r), 1− L(r), . . .,M − 1− L(r)], where r ∈ [0, R) and R isthe period of function D(t). To simplify the discussion, it isassumed that β = η = 1. Note that it is easy to extend the modelwhen β = 1 and η = 1 by manipulating the generator matrixA. When the duration of each time slot Δt is small enough,

the discrete processes discussed in Sections II and III can beconsidered as continuous processes and the evolution of the SOCof the ESS S(t), under the proposed POS scheme, is shown as

dS(t)

dt=

⎧⎪⎪⎨

⎪⎪⎩

d(s, t)+ = max0, d(s, t) if S(t) = 0 and Sw(t) = s

d(s, t) if 0 < S(t) < B andSw(t) = s

d(s, t)− = min0, d(s, t) if S(t) = B and Sw(t) = s.

(24)

For an RES, we decide to analyze the system reliability givena certain renewable supply process denoted by the generatormatrix A, the size of the ESS B, and the load profile L(r). Wefirst obtain the limiting probability vector of the SOC of the ESSusing the MMFQ model [31] in order to calculate the systemreliability. Based on [31], Theorems 3 and 4 are introduced tohelp obtain the limiting probability vector of the SOC of the ESS.

Theorem 3: We define F(x, r) = Fr(1, x, r), Fr(2, x, r),. . ., Fr(M,x, r) as the limiting probability vector of the SOCof the ESS under load state r, where 0 ≤ s ≤ M , is the driftstate and x is the SOC. In the finite-storage case with ESS sizeB, F(x, r), which can be modeled as MMFQ, satisfies

F(x, r)A =dF(x, r)

dxEr (25)

with the boundary conditions F (s,B, r) = πs when d(s, r)< 0 and F (s, 0, r) = 0 when d(s, r) > 0, where πr = πr

1,πr2, . . ., π

rs is the limiting probability vector for CTMC.

The above differential equation can be solved using the eigen-value method [32]. The solution of F(x, r) is

F(x, r) =

M−1∑

i=0

αri e

λri xφr

i (26)

where the pair (λri ,φ

ri ) is the eigenvector and eigenvalue for

matrix pair (Er,A) that satisfies φri (λ

riEr −A) = 0. The co-

efficients αri satisfies

M−1∑

i=0

αriφ

rij = 0, j ∈ S+ (27)

M−1∑

i=0

αriφ

rije

λriB = πr

j , j ∈ S− (28)

where S+ and S− denote the state in which d(s, r) > 0 andd(s, r) < 0, respectively.

Theorem 4: The limiting probability vector F(x) =F (1, x), . . ., F (M,x) under the load profile D(t) canbe calculated as

F(x) =

∫ R

r=0

F(x, r). (29)

After the limiting probability vector of the SOC F(x) is ob-tained, we can derive the system reliability measured by LOPSE,which is the expectation for the amount of power shortage giventhe size of the ESS B

LOPSE =

M−1∑

i=0

[

−min

0,

∫ R

r=0

d(i, r)

]

F (i, 0). (30)



V. PRICE-ADAPTIVE POWER SMOOTHING ALGORITHM

The proposed POS scheme in Section III aims to serve the net-work load with the optimal renewable energy utilization. Never-theless, it fails to consider the time-varying electricity price andmay yield high operation costs. With electricity price informa-tion transmitted through IoE in real time, a price-adaptive powersmoothing algorithm is devised based on Lyapunov optimizationtheory. It aims to minimize the operation cost of the RESwithout forecast information on generation, load, and electricityprice. In Section V-C, we provide the deterministic performanceguarantee of the algorithm on operation cost compared to theglobal optimum.

The control objective of the RES is to determine Pc(t) andPd(t) so as to minimize the time-averaged operation cost, whichincludes electricity cost G(t) and spillage cost H(t). Con-sidering the ESS operation constraints, the power schedulingoptimization problem P1 is formulated as

P1 :

fmin= minimizePc(t),Pd(t)

limT→∞

1

T

T−1∑

t=0

E[G(t) +H(t)] (31a)

subject to 0 ≤ Pc(t) ≤ P c (31b)

0 ≤ Pd(t) ≤ P d

(1), (10), (11), and (12). (31c)

One challenge of solving the above optimization problemis that the future renewable generation, electricity price, andnetwork load in P1 are unknown system parameters. Moreover,the constraints on S(t), namely, (11) and (12), make P1 becometime-coupling, which means that the current control decision caninfluence the future control decisions. To solve P1, an approachbased on Lyapunov optimization is proposed, that requires nofuture information about the system parameters.

A. Problem Reformulation

In order to decouple the time-coupling constraints on ESS,we consider a relaxed version of the optimization problem P1,namely, P2, in which all the constraints associated with S(t),i.e., (11) and (12), are relaxed

P2 :

gmin= minimizePc(t),Pd(t)

limT→∞

1

T

T−1∑

t=0

E[G(t) +H(t)] (32a)

subject to limT→∞

1

T

T−1∑

t=0

E[ηPc(t)]− E

[1

βPd(t)

]

=0

(1), (27b), (27c), and (10). (32b)

In this problem, the ESS with finite capacity is relaxed to haveinfinite storage. Hence, we have gmin ≤ fmin. The constraint of(32b) is introduced to guarantee the stability of SOC. Later, itis shown that the Lyapunov optimization approach for solvingP2 can provide tight bounds on S(t) to satisfy (11) and (12).Therefore, the relaxation from P1 to P2 is exact.

B. Lyapunov Optimization Approach

To track the SOC of ESS, we first adopt the concept of virtualqueue in the formulation

X(t) = S(t)− Sth (33)

where S(t) is the SOC of ESS and Sth is introduced so as tosatisfy ESS constraints in Lyapunov optimization. The settingof Sth will be discussed in Theorem 5. We define the Lyapunovfunction with respect to the virtual queue as

L(t) =1

2(X(t))2 =

1

2(S(t)− Sth)

2. (34)

Here, the conditional Lyapunov drift with respect to L(t),which describes the one-slot expected change of the Lyapunovfunction, is given by

Δ(t) = E[L(t+ 1)− L(t)|X(t)]. (35)

Lemma 1: The Lyapunov drift is bounded as follows:

Δ(t) ≤ E[X(t)[ηPc(t)−1

βPd(t)]Δt|X(t)] + T (36)

where T < ∞ is a constant.Proof: See Appendix C.

Based on the Lyapunov optimization framework, (35) is in-troduced to stabilize the virtual queue. To reflect the controlobjective which minimizes the generation cost, we add a functionof the expected generation cost over one slot to (35) and obtainthe following drift-plus-penalty function:

ΔV (t) = Δ(t) + V E[H(t) +G(t)|X(t)]

≤ E[X(t)

[

ηPc(t)−1

βPd(t)

]

Δt

+ V [H(t) +G(t)]|X(t)] + T . (37)

According to Lyapunov optimization theory [33], the decisionis determined to greedily minimize the bound on the drift-plus-penalty function, say, the right-hand side of the inequality(37), for each time slot t. The drift-plus-penalty function iscomposed of two parts, namely, the Lyapunov drift (35) andthe weighted cost function V [G(t) +H(t)]. Intuitively, to min-imize, the Lyapunov drift can stabilize the virtual queue length.The second partV [G(t) +H(t)] is regarded as the penalty term,which describes the weighted operation cost. The weight Vdetermines the tradeoff between minimizing the Lyapunov driftand minimizing the weighted cost function.

The Lyapunov optimization problem P3 is formulated to solveP2, that, in turn, solves the original problem P1P3 :

minimizePc(t),Pd(t)

X(t)

[

ηPc(t)−1

βPd(t)

]

Δt+ V [G(t) +H(t)]

subject to (1), (27b), (27c), and (10). (38)

Problem P3 gives the control decisions Pc(t) and Pd(t) intime slot t, where t ∈ T . The SOC of the ESS S(t) is thenupdated based on (11).

Theorem 5: By choosing the parameters V and Sth as

0 < V ≤(B − Pd

β Δt− P cηΔt)η

pmaxβη + smax(39)

Sth = V pmaxβ +P d

βΔt (40)

where pmax and smax are the maximum values of p(t) and s(t),respectively, the decisions made by the optimization problem(38) are feasible for problem (31). More specifically, the se-quence of SOC S(t) resulting from (38) satisfies

0 ≤ S(t) ≤ B (41)

which satisfies (12).Proof: See Appendix D.



Algorithm 2: LB-PAPS Algorithm.

Input: Real-time data for R(t), network load D(t), SOC ofthe ESS S(t), electricity price p(t), spillage cost s(t).

Output Pc(t), Pd(t).1: for each Time t do2: Compute Pc(t) and Pd(t) based on the Lyapunov

optimization problem P3.3: Update S(t) as (11).4: end for

C. LB-PAPS Algorithm

Based on the above Lyapunov optimization framework, wenow present the LB-PAPS algorithm shown in Algorithm 2 todetermine the charging/discharging power of the ESS.

Theorem 6: The time average operation cost achieved by LB-PAPS algorithm satisfies

limT→∞

1

T

T−1∑

t=0

E[G(t) + H(t)] ≤ fmin +T

V. (42)

Proof: See Appendix E. Theorem 6 gives the performance guarantee of the LB-PAPS

algorithm compared to the global optimum in P1. From (39) and(42), it can be induced that when the size of the ESSB increases,the upper bound of V increases and the optimality gap betweenthe result achieved by the LB-PAPS algorithm and the globaloptimal, namely,

˜TV , decreases. When B goes to infinity, we can

conclude that the time-averaged operation cost achieved by theLB-PAPS algorithm converges to the global optimal fmin.

VI. PERFORMANCE EVALUATION

A. Simulation Setup

We conduct a case study of a small-scale wind power gener-ator, which is located in the north of Los Angeles, CA, USA.A set of real data recorded for the whole 2012 is used in thecase study, with 5-min metering interval [29]. The historicaltime-varying electricity price is obtained from [34]. The size ofESS varies to test its effect on the performance of RES. Themaximum charging and discharging rates of the ESS are set as 8and−8 kW, respectively. The whole operation period of the RESin the simulation is set to one month (30 days), with the durationof each time slot Δt = 5 min. The network load is assumed tobe constant and equals 8 kW in Tests 1 and 2. In Tests 3 and 4,real-world metered load data of the PJM market from 1 January2016 to 30 August 2017 [35] is adopted in the simulation. Themetered load data are normalized with mean of 8 kW to matchwith the wind generation capacity. In Test 5, the network loadsof D(t) with mean of 8 kW and variance of different values areadopted to test the load adaptability of algorithms. Simulationsare conducted for 20 times with different generation and loadprofiles. The results are given with the mean estimates for Tests 1and 2, and the mean estimates and the 95% confidence intervalsfor Tests 3 and 5.

B. Scenarios for Comparison

To evaluate our approaches, we compare six scenarios:(S1) with no ESS (base scenario);(S2) with ESS, using the smoothing control method in [8];

(S3) with ESS, using the POS scheme devised in Section III-A;

(S4) with ESS, using the HB-LAPS algorithm in [17];(S5) with ESS, using the LB-PAPS algorithm in Section V;(S6) the global optimum on operation cost with ESS.Note that S3 and S5 are the algorithms proposed in our article

and are evaluated through the comparisons with S1, S2, S4,and S6.

In S1, no energy management approach is implemented. It isregarded as the base case for performance evaluation. For S2, thecharging or discharging power of the ESS is calculated based onthe dynamic rate limiter proposed in [8]. Basically, S2 keeps therenewable output unchanged if the current output is within thepredefined range of the power fluctuation. Otherwise, it modifiesand smoothens the renewable output by managing the chargingand discharging behaviors of the ESS. In S3, Pc(t) and Pd(t)are computed according to Algorithm 1. It has been proved thatS3 achieves the optimal utilization of renewable energy. Thus,S3 is taken as the benchmark to evaluate the renewable energyutilization. In S5, Sth is set based on (40) and V is set as theright-hand side of (39). In S6, it is assumed that all the operationparameters, including the network load, renewable generationprofile, and electricity prices, are known in advance. Therefore,the global optimum on operation cost can be obtained throughminimizing (43). Note that S6 requires accurate predictions onoperation parameters, that is not practical in real grid operations.S2–S5 are real-time algorithms and do not require predictiveinformation.

C. Performance Metrics

Two performance metrics are considered in the simulationtests: Cavg and URG. Cavg denotes the daily average cost

Cavg =Td

T

T∑

t=1

[G(t) +H(t)] (43)

where Td denotes the time horizon of one day in slots. URG

denotes the renewable energy utilization. It measures the pro-portion of the renewable energy in serving the network load, andis given as

URG =

∑Tt=1 minPout(t), D(t)Δt

∑Tt=1 R(t)Δt

. (44)

URG can also be represented by LOPSE, which is given by (22)and (30)

URG =

⎛

⎝1− LOPSE∑T

t=1 D(t)T

⎞

⎠∑T

t=1 D(t)∑T

t=1 R(t). (45)

An algorithm which results in a lower Cavg and a higher URG

(lower LOPSE) indicates a better performance.In order to evaluate the effect of renewable generation profile

on the ESS configuration, we need to introduce some effectivemeasurements to describe the fluctuation of the renewable gen-eration profile. We adopt two metrics, i.e., standard deviationSTDRG and standard deviation of moving average STDMA,to describe, respectively, the amplitude of fluctuation and theintermittency of renewable generation

STDRG =

√1

T

∑T

t=1

(

R(t)− 1

T

∑T

t=1R(t)

)2

(46)



Fig. 4. Performance improvements under different ESS sizes. (a) Renewableutilization. (b) Reliability.

STDMA

=

√√√√ 1

T−2W

T−W∑

t=1+W

(1

2W + 1

t+W∑

i=t−W

R(i)−1

T

T−W∑

t=1+W

R(t)

)2

(47)

where 2W + 1 is the window size of the moving average and1

2W+1

∑t+Wi=t−WR(i) is the moving average of the renewable

generated power at time slot t. The renewable generated powerwith a larger STDRG has a larger amplitude of fluctuation. Alarger STDMA suffers severer intermittency of the renewablegenerated power, where the occurrences of the supply shortagesbecome more frequent.

D. Simulation Results

Five tests are performed to evaluate the performance of ourmethods and the effect of ESS size on the system performance.In the first test, we examine the effectiveness of our proposedPOS scheme, i.e., S3, for serving the network load and showhow the ESS affects the performance of the POS scheme. In thesecond test, we analyze how the ESS size and the renewablegeneration profile affect the renewable utilization. We also givesome insights in how the ESS should be configured under dif-ferent amplitudes of fluctuations and fluctuation frequencies ofthe renewable generation, so as to satisfy the required renewableutilization. In the third and fourth tests, Scenarios S1, S2, S3,S4, S5, and S6 are compared in terms of operation cost andrenewable utilization, respectively. In the fifth test, we study theload adaptability of S5.

1) Test 1: We focus on smoothing power output of the RESto serve network load and compare S1 and S3 on renewableenergy utilization and reliability under different sizes of ESS.This investigates the capability of the POS scheme to mitigatethe power fluctuations. Fig. 4(a) illustrates the renewable energyutilization with different configurations on the size of the ESSB.It indicates that, as the size of the ESS increases, the renewableenergy utilization achieved by the POS scheme improves butits marginal improvement decreases. It is reasonable because alarger ESS can buffer more renewable energy and, thus, increasethe renewable energy utilization. Fig. 4(b) illustrates the systemreliability, which is reflected by LOPSE calculated in Section IV,under S1 and S3. It reveals that S3 can improve the systemreliability as the ESS size increases but its marginal improve-ment on reliability decreases according to the analytical results.The analytical results shown in Fig. 4(b) match well with thesimulation results shown in Fig. 4(a) based on the relationshipbetween URG and LOPSE given by (45).

Fig. 5. Required ESS size under different renewable generation profiles.(a) Required ESS size under different STDRG. (b) Required ESS size underdifferent STDMA.

Fig. 6. Performance comparison among different algorithms. (a) Performanceon daily average cost. (b) Performance on renewable energy utilization.

Through Test 1, it is shown that S3 can yield both highrenewable energy utilization and reliability while smoothing therenewable generated power. The gain achieved by S3 increaseswith the ESS size.

2) Test 2: In this test, we focus on how the ESS affects therenewable utilization under different amplitudes of fluctuationsof renewable generation and intermittency levels. We fix therenewable utilization of the RES to 90% and find what is therequired ESS size to satisfy this utilization level under differentamplitudes of fluctuations and intermittency levels. In the firstsimulation, we keep the mean of the renewable generated powerunchanged but increase the amplitude of fluctuation throughenlarging the gap between the renewable generated power andthe mean of the renewable generated power. The relationshipbetween STDRG and the size of the ESSB is shown in Fig. 5(a).It reveals that when STDRG is larger than a threshold, therequired ESS size blows up with STDRG to satisfy the requiredrenewable utilization. In the second simulation, we fix STDRG

and the mean of renewable generated power while changing theintermittency level through modifying STDMA, where W is setto 100. The relationship between STDMA and the size of theESS B is shown in Fig. 5(b). It reveals that, given that the meanand the variance of the renewable generated power unchanged,the size of the ESS blows up with STDMA to satisfy the requiredrenewable utilization. It is because when the supply of renewablegenerated power becomes more intermittent, the ESS needs toreserve more energy when the renewable generated power isin excess to compensate for the power shortage with a longerperiod. Therefore, a larger size of the ESS is needed to satisfythe required renewable utilization level.

3) Test 3: We compare scenarios, S1–S6, with time-varyingload to evaluate the achieved daily average cost while the sizeof the ESS varies. The results (both the mean estimate and the95% confidence interval for each configuration) are shown inFig. 6(a). We can see that the cost is the highest in S2 since S2only aims to smooth out the high-frequency fluctuations of thepower output, which does not consider the economical aspectof the RES. The cost in S1 is the second highest comparing toother scenarios with ESS, which indicates that the operation cost



Fig. 7. Performance of the LB-PAPS algorithm with load variations.

can be reduced by employing ESS. Among those scenarios withESS, S5 outperforms S2, S3, and S4. The cost achieved by S5 isa bit higher than the global optimum in S6. According to (42), itcan be proved theoretically that as the operation period and thesize of the ESS go to infinity, the daily average cost achieved byS5 converges to the global optimum in S6.

4) Test 4: Similar to Test 3, we do the comparisons amongS1–S5 in terms of renewable energy utilization. Fig. 6(b) illus-trates the results achieved by these five scenarios, where boththe mean estimate and the 95% confidence interval for each con-figuration are exhibited. Among these five scenarios, S3 yieldsthe optimal utilization. It is because S3 always provides poweroutput to serve the load whenever the renewable generation isavailable. The renewable energy utilization achieved by S4 is alittle lower than S3. The reason why S3 and S4 outperform othersis that both S3 and S4 take renewable energy utilization as theircontrol objectives (S3 takes renewable energy utilization as itsonly control objective and S4 takes renewable energy utilizationas one of its heuristic information). Among those scenarios,which do not take renewable energy utilization as their controlobjectives directly, S5 performs much better than S1 and S2. Thisshows that S5 can achieve good performance on daily averagecost while maintaining relatively high utilization. S2 yields thelowest utilization since it does not aim to satisfy the load demandwhile filtering the high-frequency fluctuations of the renewablegenerated power.

5) Test 5: We focus on the effect of the load variation on ourproposed LB-PAPS algorithm. We evaluate S5 under differentvariances of the network load and the size of the ESS is set to100 kWh. The performance (both the mean estimate and the95% confidence interval for each configuration) are exhibitedin Fig. 7. From Fig. 7, we see that as the variance of the loadincreases, S5 can maintain the mean of the daily average costand the renewable energy utilization at a constant level. It isbecause S5 can adaptively change the power output to counterthe uncertainty of the load. Fig. 7 also reveals that the confidenceintervals on the daily average cost and the renewable energyutilization becomes larger as the variance of the load increases.This indicates that the fluctuation of the load can degrade thedegree of confidence on these two metrics.

VII. CONCLUSION

This article investigates the POS techniques for renewablegeneration in IoE and analyzes the tradeoff between ESS sizeand the performance, i.e., reliability, utilization, or cost, on POS,respectively. By jointly considering the constructional cost ofthe ESS and the operational cost of the RES, an optimiza-tion problem for joint storage configuration is formulated. Theproblem can be relaxed to an MILP under certain assumptions.An analytical framework is proposed to investigate the tradeoff

between ESS size and the system reliability. The simulation re-sults exhibit that the proposed LB-PAPS algorithm outperformsother existing algorithms in terms of operation cost and canmaintain relatively high renewable energy utilization under dif-ferent ESS capacities. We also find that, to satisfy the renewableutilization requirement, the required ESS size blows up withboth the amplitude of fluctuation and the intermittency level ofthe renewable generation. It is noted that the technique proposedin this article applies to POS for the RES, as well as the potentialapplications on other research areas. For instance, the methodsproposed and results obtained in this article can be applicable tothe energy management and distribution in the vehicular energynetwork [36].

APPENDIX

A. Proof of Theorem 1

Due to the maximum charging/discharging rates of the ESSand the SOC constraints, the feasible domain of Pc and Pd

is bounded by 0 ≤ Pc ≤ minMc(t), P c = Ac and 0 ≤ Pd ≤minMd(t), P d = Ad, respectively. Note that the charging anddischarging processes of the ESS cannot happen at the sametime, compared to the POS scheme, other charging schemesmust fall into the following four cases: 1) 0 ≤ Pc ≤ Gc(t) andPd = 0, 2) Gc(t) ≤ Pc ≤ Ac and Pd = 0, 3) 0 ≤ Pd ≤ Gd(t)and Pc = 0, and 4) Gd(t) ≤ Pc ≤ Ad and Pc = 0, where Gc(t)and Gd(t) are defined in Algorithm 1. Define the resulted URG

by using POS algorithm as U ∗RG. We now proof that in all the

above cases, URG cannot be higher than U ∗RG.

For cases 1), if Gc(t) ≤ Ac, satisfying 1) will decrease S(t)by [Gc(t)− Pc(t)]ηΔt and will not affect URG. Since reducingS(t) can only result in the reduction of URG, the obtained URG

cannot be higher thanU ∗RG in this way. IfGc(t) ≥ Ac, satisfying

1) will decrease S(t) by [Ac − Pc(t)]ηΔt and will not affectURG. Since reducing S(t) can only result in the reduction ofURG, the obtained URG cannot be higher than U ∗

RG in this way.For case 2), ifGc(t) ≤ Pc ≤ Ac, satisfying 1) will increaseS(t)by [Pc(t)−Gc(t)]ηΔt and decrease URG by [Pc(t)−Gc(t)]Δt

∑Tt=1 R(t)Δt

.

In this way, even in the optimal case, the increase of [Pc(t)−Gc(t)]ηΔt in S(t) can only increase URG by [Pc(t)−Gc(t)]ηβΔt

∑Tt=1 R(t)Δt

,

which is smaller than [Pc(t)−Gc(t)]Δt∑T

t=1 R(t)Δt. Therefore, the obtained

URG cannot be higher than U ∗RG by satisfying 2).

Similar to cases 1) and 2), it can be concluded that the obtainedURG cannot be higher than U ∗

RG by satisfying 3) and 4). Hence,we can draw the conclusion that the proposed POS scheme canachieve the optimal value of URG.

B. Proof of Theorem 2

Proof: Note that the feasible region of Problem (13) isa subset of the feasible region of Problem (21). ComparingConstraints (8), (9), and (10) in (13) with Constraints (14)–(20) in (21), we categorize the charging/discharging constraintsof the ESS (14)–(20) in Problem (21) into four cases: 1)Pd(t) = 0 and 0 ≤ Pc(t) < max0, R(t)− Pr(t), 2) Pd(t) =0 and max0, R(t)− Pr(t) < Pc(t) ≤ R(t), 3) Pc(t) = 0and 0 ≤ Pd(t) < max0, Pr(t)−R(t), and 4) Pc(t) = 0 andmax0, Pr(t)−R(t) < Pd(t) ≤ Pr(t).



According to (8), (9), and (10), these cases in (14)–(20) canbe mapped to four states of the ESS 1) being undercharged,2) being overcharged, 3) being underdischarged, and 4) beingoverdischarged, respectively. Given that p(t) is fixed, it can beconcluded that f will not decrease by satisfying 2) and 3) sinceCg cannot decrease by reducing the current energy provided andstoring it in the ESS for future use.

To explain, we first consider case 2). Through satisfy-ing 2) instead of letting Pc(t) = max0, R(t)− Pr(t), S(t)is increased by [Pc(t)−max0, R(t)− Pr(t)]ηΔt and theelectricity cost at time t, G(t), is increased by [Pc(t)−max0, R(t)− Pr(t)]p(t)Δt. In this way, the resulting elec-tricity cost Cg is larger than the electricity cost C∗

g when satisfy-ingPc(t) = max0, R(t)− Pr(t). It is because even in the op-timal case, the increase of [Pc(t)−max0, R(t)− Pr(t)]ηΔtin S(t) results in the reduction of Cg by [max0, R(t)−Pr(t) − Pc(t)]ηβp(t)Δt in grid operation, which is smallerthan and cannot compensate for the electricity cost [Pc(t)−max0, R(t)− Pr(t)]p(t)Δt arised by satisfying 2) at timet. Therefore, satisfying 2) cannot reduce f . In case 3), com-paring to letting Pd(t) = max0, Pr(t)−R(t), S(t) is in-creased by [max0, Pr(t)−R(t) − Pd(t)]Δt and the electric-ity cost at time t,G(t), is increased by [max0, Pr(t)−R(t) −Pd(t)]βp(t)Δt. In this way, the resulting electricity cost Cg

is larger than or equal to the electricity cost C∗g when letting

Pd(t) = max0, Pr(t)−R(t). It is because even in the opti-mal case, the increase of [max0, Pr(t)−R(t) − Pd(t)]Δt inS(t) results in the reduction of Cg by [max0, R(t)− Pr(t) −Pc(t)]βp(t)Δt in grid operation, which is equal to the electricitycost arised by satisfying 3) at time t. Therefore, satisfying 3)cannot reduce f .

Similar to the discussion in case 2) and 3), it can be concludedthat f will not be reduced by satisfying 1) and 4).

Based on the above discussions, it is proved that Problem (13)always achieves a value of f not larger than that of Problem (21)for all the four situations, given that P c, P d, and B are fixed.Coupled with the fact that the feasible region of Problem (13) is asubset of the feasible region of Problem (21), it can be concludedthat solving Problem (13) for optimal values of P c, P d, and Bis equivalent to solving Problem (21) for optimal values of P c,P d, and B. Hence, the optimal solutions for P c, P d, and B inProblem (21) are the same as those of Problem (13). C. Proof of Lemma 1

Δ(t) =1

2E(X(t+ 1))2 −X(t)2|X(t)

=1

2

⎧⎪⎪⎨

⎪⎪⎩

E(X(t) + ηPc(t)Δt)2 −X(t)2|X(t)if Pd(t) = 0E(X(t)− 1

βPd(t)Δt)2 −X(t)2|X(t)if Pc(t) = 0

≤ 1

2EX(t)[2(ηPc(t)−

1

βPd(t))]Δt

+1

2max(ηP cΔt)2, (

1

βP dΔt)2|X(t)

= T + EX(t)[ηPc(t)−1

βPd(t)]Δt|X(t) (48)

where T = 12 max(ηP cΔt)2, ( 1βP dΔt)2.

D. Proof of Theorem 4

To proof the left-hand side of (41), we first proof the followinglemma.

Lemma 2: When S(t) < Pd

β Δt, it holds that Pd(t) = 0 ac-cording to (38).

Proof: We prove this Lemma using the proof by contra-diction. First, we assume that Pd(t) > 0 when S(t) < Pd

β Δt.Since the control objective is to minimize (38), it holds that ifPd(t) > 0

X(t)1

β≥ −V pmax

1

β

[

−S(t) + Vpmaxβ +

P d

βΔt

]

≤ Vpmax(using(40))

S(t) ≥ P d

βΔt (49)

(49) violates S(t) < Pd

β Δt. Therefore, it must be that when

S(t) < Pd

β Δt, Pd(t) = 0. According to Lemma 2, it holds that Pd(t) = 0 when S(t) <

Pd

β Δt. WhenS(t) ≥ Pd

β Δt, it holds that 0 ≤ Pd(t) ≤ P d basedon (31c). Therefore, it can be concluded that the left-hand sideof (41) 0 ≤ S(t) holds.

To proof the right-hand side of (41), we introduce the follow-ing lemma.

Lemma 3: When S(t) > B − P cηΔt, it holds that Pc(t) =0 according to (38).

Proof: We prove this lemma using the proof by contradic-tion. First, we assume thatPc(t) > 0whenS(t) > B − P cηΔt.Since the control objective is to minimize (38), it holds that ifPc(t) > 0

X(t)η ≤ V smax

[

S(t)− V pmaxβ − P d

βΔt

]

η ≤ V smax (using (40))

[

S(t)−P d

βΔt

]

η≤(smax+pmaxβη)Bη − (Pd

β η + P cη2)Δt

smax + pmaxβη

S(t) ≤ B − P cηΔt (50)

(50) violates S(t) > B − P cηΔt. Therefore, it must be thatwhen S(t) > B − P cηΔt, Pd(t) = 0.

According to Lemma 3, it holds that Pc(t) = 0 when S(t) >B − P cηΔt. When S(t) ≤ B − P cηΔt, it holds that 0 ≤Pc(t) ≤ P c based on (31b). Therefore, it can be concluded thatthe left-hand side of (41) S(t) ≤ B holds.

Combining Lemmas 2 and 3 as well as the above discussion,we can conclude it holds that 0 ≤ S(t) ≤ B.

E. Proof of Theorem 5

Note that the solution of (38) minimize the drift-plus-penaltyfunction (37) over its feasible region. We have

ΔV (t)

≤E[X(t)

[

ηPc(t)−1

βPd(t)

]

Δt+V [G(t) + H(t)]|X(t)]+T



≤E

[

X(t)[ηPc(t)−1

βPd(t)

]

Δt+V [G(t) + H(t)]|X(t)] + T

= E[X(t)[0]+V [G(t) + H(t)]|X(t)] + T (51)

where the control actions by solving (38) is denoted with thesuperscript and the control actions by solving (32) is denotedwith the superscript . The second inequality is derived due tothe stationary and the randomized policy in the problem (32).The equality holds based on E[ηPc(t)− 1

β Pd(t)|X(t)] = 0.Summing from t = 0 to t = T − 1 on both sides of (51)

T−1∑

t=0

ΔV (t) ≤T−1∑

t=0

E[V [G(t) + H(t)]] + T. (52)

Then, we expand (52) by using (35) and (37) and obtain

EL(T )− L(0)+T−1∑

t=0

E[V [G(t) + H(t)]]

≤T−1∑

t=0

T+E[V [G(t) + H(t)]]. (53)

Divide both sides by V · T and make T → ∞, we have

limT→∞

1

T

T−1∑

t=0

E[G(t) + H(t)]

≤ limT→∞

1

V T

T−1∑

t=0

T+ limT→∞

1

T

T−1∑

t=0

E[G(t) + H(t)]

=T

V+ gmin ≤ T

V+ fmin. (54)

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