Online Energy Generation Scheduling for Microgrids with Intermittent Energy Sources and Co-Generation

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1 Online Energy Generation Sheduling for Mirogrids with Intermittent Energy Soures and Co-Generation Lian Lu Information Engineering Dept. he Chinese Univ. of Hong Kong Jinlong u Information Engineering Dept. he Chinese Univ. of Hong Kong Minghua Chen Information Engineering Dept. he Chinese Univ. of Hong Kong ABSRAC Mirogrids represent an emerging paradigm of future eletri power systems that an utilize both distributed and entralized generations. wo reent trends in mirogrids are the integration of loal renewable energy soures (suh as wind farms) and the use of o-generation (i.e., to supply both eletriity and heat). However, these trends also bring unpreedented hallenges to the design of intelligent ontrol strategies for mirogrids. raditional generation sheduling paradigms rely on perfet predition of future eletriity supply and demand. hey are no longer appliable to mirogrids with unpreditable renewable energy supply and with o-generation (that needs to onsider both eletriity and heat demand). In this paper, we study online algorithms for the mirogrid generation sheduling problem with intermittent renewable energy soures and o-generation, with the goal of maximizing the ost-savings with loal generation. Based on the insights from the struture of the offline optimal solution, we propose a lass of ompetitive online algorithms, alled CHASE (Competitive Heuristi Algorithm for Sheduling Energy-generation), that trak the offline optimal in an online fashion. Under typial settings, we show that CHASE ahieves the best ompetitive ratio among all deterministi online algorithms, and the ratio is no larger than a small onstant 3. We also extend our algorithms to intelligently leverage on limited predition of the future, suh as near-term demand or wind foreast. By extensive empirial evaluations using real-world traes, we show that our proposed algorithms an ahieve near offline-optimal performane. In a representative senario, CHASE leads to around Chi-Kin Chau Masdar Institute of Siene and ehnology Xiaojun Lin Shool of Eletrial and Computer Engineering Purdue Univ. 2% ost redution with no future look-ahead, and the ost redution inreases with the future look-ahead window. Categories and Subjet Desriptors C.4 [PERFORMANCE OF SYSEMS]: Modeling tehniques; Design studies; F.1.2 [Modes of Computation]: Online omputation; I.2.8 [Problem Solving, Control Methods, and Searh]: Sheduling General erms Algorithms, Performane Keywords Mirogrids; Online Algorithm; Energy Generation Sheduling; Combined Heat and Power Generation 1. INRODUCION Mirogrid is a distributed eletri power system that an autonomously o-ordinate loal generations and demands in a dynami manner [23]. Illustrated in Fig. 1, modern mirogrids often onsist of distributed renewable energy generations (e.g., wind farms) and o-generation tehnology (e.g., supplying both eletriity and heat loally). Mirogrids an operate in either grid-onneted mode or islanded mode. here have been worldwide deployments of pilot mirogrids, suh as the US, Japan, Greee and Germany [7]. Mirogrid Households he first two authors are in alphabetial order. Eletriity Eletriity Wind Energy Eletriity Grid Heating Permission to make digital or hard opies of all or part of this work for personal or lassroom use is granted without fee provided that opies are not made or distributed for profit or ommerial advantage and that opies bear this notie and the full itation on the first page. o opy otherwise, to republish, to post on servers or to redistribute to lists, requires prior speifi permission and/or a fee. SIGMERICS 13, June 17-21, 213, Pittsburgh, PA, USA. Copyright 213 ACM /13/6...$15.. Gas Natural Gas Loal Heating Combined Heat and Power Figure 1: An illustration of a typial mirogrid. Mirogrids are more robust and ost-effetive than traditional approah of entralized grids. hey represent an

2 emerging paradigm of future eletri power systems [3] that address the following two ritial hallenges. Power Reliability. Providing reliable and quality power is ritial both soially and eonomially. In the US alone, while the eletri power system is 99.97% reliable, eah year the eonomi loss due to power outages is at least $15 billion [33]. However, enhaning power reliability aross a large-sale power grid is very hallenging [12]. With loal generation, mirogrids an supply energy loally as needed, effetively alleviating the negative effets of power outages. Integration with Renewable Energy. he growing environmental awareness and government diretives lead to the inreasing penetration of renewable energy. For example, the US aims at 2% wind energy penetration by 23 to de-arbonize the power system. Denmark targets at 5% wind generation by 225. However, inorporating a signifiant portion of intermittent renewable energy poses great hallenges to grid stability, whih requires a new thinking of how the grid should operate [4]. In traditional entralized grids, the atual loations of onventional energy generation, renewable energy generation (e.g., wind farms), and energy onsumption are usually distant from eah other. hus, the need to oordinate onventional energy generation and onsumption based on the instantaneous variations of renewable energy generation leads to hallenging stability problems. In ontrast, in mirogrids renewable energy is generated and onsumed in the loal distributed network. hus, the unertainty of renewable energy is absorbed loally, minimizing its negative impat on the stability of the entral transmission networks. Furthermore, mirogrids bring signifiant eonomi benefits, espeially with the augmentation of ombined heat and power (CHP) generation tehnology. In traditional grids, a substantial amount of residual energy after eletriity generation is often wasted. In ontrast, in mirogrids this residual energy an be used to supply heat domestially. By simultaneously satisfying eletriity and heat demand using CHP generators, mirogrids an often be muh more eonomial than using external eletriity supply and separate heat supply [18]. However, to realize the maximum benefits of mirogrids, intelligent sheduling of both loal generation and demand must be established. Dynami demand sheduling in response to supply ondition, also alled demand response [33, 11], is one of the useful approahes. But, demand response alone may be insuffiient to ompensate the highly volatile flutuations of wind generation. Hene, intelligent generation sheduling, whih orhestrates both loal and external generations to satisfy the time-varying energy demand, is indispensable for the viability of mirogrids. Suh generationside sheduling must simultaneously meet two goals. (1) o maintain grid stability, the aggregate supply from CHP generation, renewable energy generation, the entralized grid, and a separate heating system must meet the aggregate eletriity and heat demand. (We do not onsider the option of using energy storage in the paper, e.g., to harge at lowprie periods and to disharge at high-prie periods. his is beause for the typial size of mirogrids, e.g., a ollege ampus, energy storage systems with omparable sizes are very expensive and not widely available.) (2) It is highly desirable that the mirogrid an oordinate loal generation and external energy prourement to minimize the overall ost of meeting the energy demand. We note that a related generation sheduling problem has been extensively studied for the traditional grids, involving both Unit Commitment [34] and Eonomi Dispath [14], whih we will review in Se. 6 as related work. In a typial power plant, the generators are often subjet to several operational onstraints. For example, steam turbines have a slow ramp-up speed. In order to perform generation sheduling, the utility ompany usually needs to foreast the demand first. Based on this foreast, the utility ompany then solves an offline problem to shedule different types of generation soures in order to minimize ost subjet to the operational onstraints. Unfortunately, this lassial strategy does not work well for the mirogrids due to the following unique hallenges introdued by the renewal energy soures and o-generation. he first hallenge is that mirogrids powered by intermittent renewable energy generations will fae a signifiant unertainty in energy supply. Beause of its smaller sale, abrupt hanges in loal weather ondition may have a dramati impat that annot be amortized as in the wider national sale. In Fig. 2a, we examine one-week traes of eletriity demand for a ollege in San Franiso [1] and power output of a nearby wind station [4]. We observe that although the eletriity demand has a relative regular pattern for predition, the net eletriity demand inherits a large degree of variability from the wind generation, asting a hallenge for aurate predition. MWh Eletriity Wind Net t /hour (a) Ele. demand, wind gen. herm Heat t /hour (b) Heat demand Figure 2: Eletriity demand, heat demand and wind generation in a week. In (a), the net demand is omputed by subtrating the wind generation from the eletriity demand. Seondly, o-generation brings a new dimension of unertainty in sheduling deisions. Observed from Fig. 2b, the heat demand exhibits a different stohasti pattern that adds diffiulty to the predition of overall energy demand. Due to the above additional variability, traditional energy generation sheduling based on offline optimization assuming aurate predition of future supplies and demands annot be applied to the mirogrid senarios. On the other hand, there are also new opportunities. In mirogrids there are usually only 1-2 types of small reiproate generators from tens of kilowatts to several megawatts. hese generators are typially gas or diesel powered and an be fired up with large ramping-up/down level in the order of minutes. For example, a diesel-based engine an be powered up in 1-5 minutes and has a maximum ramp up/down rate of 4% of its apaity per minute [41]. he fast responding nature of these loal generators opens up opportunities to inrease the frequeny of generator on/off sheduling that substantially hanges the design spae for energy generation sheduling. Beause of these unique hallenges and opportunities, it remains an open question of how to design effetive strategies for sheduling energy generation for mirogrids. 1.1 Our Contributions In this paper, we formulate a general problem of energy generation sheduling for mirogrids. Sine both the future

3 demands and future renewable energy generation are diffiult to predit, we use ompetitive analysis and study online algorithms that an perform provably well under arbitrarily time-varying (and even adversarial) future trajetories of demand and renewable energy generation. owards this end, we design a lass of simple and effetive strategies for energy generation sheduling named CHASE (in short for Competitive Heuristi Algorithm for Sheduling Energy-generation). Compared to traditional predition-based and offline optimization approahes, our online solution has the following salient benefits. First, CHASE gives an absolute performane guarantee without the knowledge of supply and demand behaviors. his minimizes the impat of inaurate modeling and the need for expensive data gathering, and hene improves robustness in mirogrid operations. Seond, CHASE works without any assumption on gas/eletriity pries and poliy regulations. his provides the grid operators flexibility for operations and poliy design without affeting the energy generation strategies for mirogrids. We summarize the key ontributions as follows: 1. In Se , we devise an offline optimal algorithm for a generi formulation of the energy generation sheduling problem that models most mirogrid senarios with intermittent energy soures and fast-responding gas- /diesel-based CHP generators. Note that the offline problem is hallenging by itself beause it is a mixed integer problem and the objetive funtion values aross different slots are orrelated via the startup ost. We first reveal an elegant struture of the single-generator problem and exploit it to onstrut the optimal offline solution. he strutural insights are further generalized in Se. 3.3 to the ase with N homogeneous generators. he optimal offline solution employs a simple load-dispathing strategy where eah generator separately solves a partial sheduling problem. 2. In Ses , we build upon the strutural insights from the offline solution to design CHASE, adeterministi online algorithm for sheduling energy generations in mirogrids. We name our algorithm CHASE beause it traks the offline optimal solution in an online fashion. We show that CHASE ahieves a ompetitive ratio of min (3 2α, 1/α) 3. In other words, no matter how the demand, renewable energy generation and grid prie vary, the ost of CHASE without any future information is guaranteed to be no greater than min (3 2α, 1/α) times the offline optimal assuming omplete future information. Here the onstant α =( o + m/l)/(p max + η g) (, 1] aptures the maximum prie disrepany between using loal generation and external soures to supply energy. We also prove that the above ompetitive ratio is the best possible for any deterministi online algorithm. 3. he above ompetitive ratio is attained without any future information of demand and supply. In Se. 3.2, we then extend CHASE to intelligently leverage limited look-ahead information, suh as near-term demand or wind foreast, to further improve its performane. In partiular, CHASE ahieves an improved ompetitive ratio of min (3 2 g(α, ω), 1/α) when it an look into a future window of size ω. Here, the funtion g(α, ω) [α, 1] aptures the benefit of looking-ahead and monotonially inreases from α to1asω inreases. Hene, the larger the look-ahead window, the better the performane. In Se. 4, we also extend CHASE to the ase where generators are governed by several additional operational onstraints (e.g., ramping up/down rates and minimum on/off periods), and derive an upper bound for the orresponding ompetitive ratio. 4. In Se. 5, by extensive evaluations using real-world traes, we show that our algorithm CHASE an ahieve satisfatory empirial performane and is robust to look-ahead error. In partiular, a small look-ahead window is suffiient to ahieve near offline-optimal performane. Our offline (resp., online) algorithm ahieves a ost redution of 22% (resp., 17%) with CHP tehnology. he ost redution is omputed in omparison with the baseline ost ahieved by using only the wind generation, the entral grid, and a separate heating system. he substantial ost redutions show the eonomi benefit of mirogrids in addition to its potential in improving energy reliability. Furthermore, interestingly, deploying a partial loal generation apaity that provides 5% of the peak loal demands an ahieve 9% of the ost redution. his provides strong motivation for mirogrids to deploy at least a partial loal generation apability to save osts. Due to spae limitations, all proofs are in our tehnial report [27]. 2. PROBLEM FORMULAION Notation Definition he total number of intervals (unit: min) N he total number of loal generators β he startup ost of loal generator ($) m he sunk ost per interval of running loal generator ($) o he inremental operational ost per interval of running loal generator to output an additional unit of power ($/Watt) g he prie per unit of heat obtained externally using natural gas ($/Watt) on he minimum on-time of generator, one it is turned on off he minimum off-time of generator, one it is turned off R up he maximum ramping-up rate (Watt/min) R dw he maximum ramping-down rate (Watt/min) L he maximum power output of generator (Watt) η he heat reovery effiieny of o-generation a(t) he net power demand (Watt) h(t) he spae heating demand (Watt) p(t) he spot prie per unit of power obtained from the eletriity grid (P min p(t) P max) ($/Watt) σ(t) he joint input at time t: σ(t) (a(t),h(t),p(t)) y n(t) he on/off status of the i-th loal generator (on as 1 and off as ), 1 n N u n(t) he power output level when the i-th generator is on (Watt), 1 n N s(t) he heat level obtained externally by natural gas (Watt) v(t) he power level obtained from eletriity grid (Watt) Note: we use bold symbols to denote vetors, e.g., a (a(t)) t=1. Brakets indiate the units. able 1: Key notations. We onsider a typial senario where a mirogrid orhestrates different energy generation soures to minimize ost

4 for satisfying both loal eletriity and heat demands simultaneously, while meeting operational onstraints of eletri power system. We will formulate a mirogrid ost minimization problem (MCMP) that inorporates intermittent energy demands, time-varying eletriity pries, loal generation apabilities and o-generation. We define the notations in able 1. We also define the aronyms for our problems and algorithms in able 2. Aronym Meaning MCMP Mirogrid Cost Minimization Problem fmcmp MCMP for fast-responding generators fmcmp s fmcmp with single fast-responding generator SP A simplified version of fmcmp s CHASE s he baseline version of CHASE for fmcmp s CHASE s+ CHASE for fmcmp s CHASE lk(ω) s he baseline version of CHASE for fmcmp s with look-ahead CHASE lk(ω) s+ CHASE for fmcmp s with look-ahead CHASE lk(ω) CHASE for fmcmp with look-ahead CHASE lk(ω) gen CHASE for MCMP with look-ahead able 2: Aronyms for problems and algorithms. 2.1 Model Intermittent Energy Demands: We onsider arbitrary renewable energy supply (e.g., wind farms). Let the net demand (i.e., the residual eletriity demand not balaned by wind generation) at time t be a(t). Note that we do not rely on any speifi stohasti model of a(t). External Power from Eletriity Grid: he mirogrid an obtain external eletriity supply from the entral grid for unbalaned eletriity demand in an on-demand manner. We let the spot prie at time t from eletriity grid be p(t). We assume that P min p(t) P max. Again, we do not rely on any speifi stohasti model on p(t). Loal Generators: he mirogrid has N units of homogeneous loal generators, eah having an maximum power outputapaity L. Based on a ommon generator model [21], we denote β as the startup ost of turning on a generator. Startup ost β typially involves the heating up ost (in order to produe high pressure gas or steam to drive the engine) and the time-amortized additional maintenane osts resulted from eah startup (e.g., fatigue and possible permanent damage resulted by stresses during startups) 1. We denote m as the sunk ost of maintaining a generator in its ative state per unit time, and o as the operational ost per unit time for an ative generator to output an additional unit of energy. Furthermore, a more realisti model of generators onsiders advaned operational onstraints: 1. Minimum On/Off Periods: If one generator has been ommitted (resp., unommitted) at time t, itmustremain ommitted (resp., unommitted) until time t + on (resp., t + off ). 2. Ramping-up/down Rates: he inremental power out- 1 It is ommonly understood that power generators inur startup osts and hene the generator on/off sheduling problem is inherently a dynami programming problem. However, the detailed data of generator startup osts are often not revealed to the publi. Aording to [13] and the referenes therein, startup osts of gas generators vary from several hundreds to thousands of US dollars. Startup osts at suh level are omparable to running generators at their full apaities for several hours. put in two onseutive time intervals is limited by the ramping-up and ramping-down onstraints. Most mirogrids today employ generators powered by gas turbines or diesel engines. hese generators are fast-responding in the sense that they an be powered up in several minutes, and have small minimum on/off periods as well as large ramping-up/down rates. Meanwhile, there are also generators based on steam engine, and are slow-responding with non-negligible on, off, and small ramping-up/down rates. Co-generation and Heat Demand: he loal CHP generators an simultaneously generate eletriity and useful heat. Let the heat reovery effiieny for o-generation be η, i.e., for eah unit of eletriity generated, η unit of useful heat an be supplied for free. Alternatively, without o-generation, heating an be generated separately using external natural gas, whih osts g per unit time. hus, η g is the saving due to using o-generation to supply heat, provided that there is suffiient heat demand. We assume o η g. In other words, it is heaper to generate heat by natural gas than purely by generators (if not onsidering the benefit of o-generation). Note that a system with no o-generation an be viewed as a speial ase of our model by setting η =. Let the heat demand at time t be h(t). o keep the problem interesting, we assume that o+ m < L P max + η g. his assumption ensures that the minimum o-generation energy ost is heaper than the maximum external energy prie. If this was not the ase, it would have been optimal to always obtain power and heat externally and separately. 2.2 Problem Definition We divide a finite time horizon into disrete time slots, eah is assumed to have a unit length without loss of generality. he mirogrid operational ost in [1,] isgivenby Cost(y, u, v, s) { t=1 p(t) v(t)+ g s(t)+ (1) [ o u n(t)+ m y n(t)+β[y n(t) y n(t 1)] +]}, N n=1 whih inludes the ost of grid eletriity, the ost of the external gas, and the operating and swithing ost of loal CHP generators in the entire horizon [1,]. hroughout this paper, we set the initial ondition y n() =, 1 n N. We formally define the MCMP as a mixed integer programming problem, given eletriity demand a,heatdemand h, and grid eletriity prie p as time-varying inputs: min Cost(y, u, v, s) (2a) y,u,v,s s.t. u n(t) L y n(t), (2b) N n=1un(t)+v(t) a(t), (2) η N n=1un(t)+s(t) h(t), (2d) u n(t) u n(t 1) R up, (2e) u n(t 1) u n(t) R dw, (2f) y n(τ) 1 {yn(t)>yn(t 1)},t+1 τ t+ on-1, (2g) y n(τ) 1-1 {yn(t)<yn(t 1)},t+1 τ t+ off -1,(2h) var y n(t) {, 1},u n(t),v(t),s(t) R +,n [1,N],t [1,], where 1 { } is the indiator funtion and R + represents the set of non-negative numbers. he onstraints are similar to those in the power system literature and apture the operational onstraints of generators. Speifially, onstraint

5 (2b) aptures the onstraint of maximal output of the loal generator. Constraints (2)-(2d) ensure that the demands of eletriity and heat an be satisfied, respetively. Constraints (2e)-(2f) apture the onstraints of maximum ramping-up/down rates. Constraints (2g)-(2h) apture the minimum on/off period onstraints (note that they an also be expressed in linear but hard-to-interpret forms). 3. FAS-RESPONDING GENERAOR CASE his setion onsiders the fast-responding generator senario. Most CHP generators employed in mirogrids are based on gas or diesel. hese generators an be fired up in several minutes and have high ramping-up/down rates. hus at the timesale of energy generation (usually tens of minutes), they an be onsidered as having no minimum on/off periods and ramping-up/down rate onstraints. hat is, on =, off =,R up =, R dw =. Weremarkthat this model aptures most mirogrid senarios today. We will extend the algorithm developed for this responsive generator senario to the general generator senario in Se. 4. o proeed, we first study a simple ase where there is one unit of generator. We then extend the results to N units of homogenous generators in Se Single Generator Case We first study a basi problem that onsiders a single generator. hus, we an drop the subsript n (the index of the generator) when there is no soure of onfusion: fmcmp s : min Cost(y, u, v, s) (3a) y,u,v,s s.t. u(t) L y(t), (3b) u(t)+v(t) a(t), (3) η u(t)+s(t) h(t), (3d) var y(t) {, 1},u(t),v(t),s(t) R +,t [1,]. Note that even this simpler problem is hallenging to solve. First, even to obtain an offline solution (assuming omplete knowledge of future information), we must solve a mixed integer optimization problem. Further, the objetive funtion values aross different slots are orrelated via the startup ost β[y(t) y(t 1)] +, and thus annot be deomposed. Finally, to obtain an online solution we do not even know the future. Remark: Readers familiar with online server sheduling in data enters [25, 28] may see some similarity between our problem and those in [25, 28], i.e., all are dealing with the sheduling diffiulty introdued by the swithing ost. Despite suh similarity, however, the inherent strutures of these problems are signifiantly different. First, there is only one ategory of demand (i.e., workload to be satisfied by the servers) in online server sheduling problems. In ontrast, there are two ategories of demands (i.e., eletriity and heat demands) in our problem. Further, beause of o-generation, they an not be onsidered separately. Seond, there is only one ategory of supply (i.e., server servie apability) in online server sheduling problem, and thus the demand must be satisfied by this single supply. However, in our problem, there are three different supplies, inluding loal generation, eletriity grid power and external heat supply. herefore, the design spae in our problem is larger and it requires us to orhestrate three different supplies, instead of single supply, to satisfy the demands. Next, we introdue the following lemma to simplify the struture of the problem. Note that if (y(t)) t=1 are given, the startup ost is determined. hus, the problem in (3a)- (3d) redues to a linear programming and an be solved independently in eah time slot. Lemma 1. Given (y(t)) t=1 and the input (σ(t)) t=1, the solutions (u(t),v(t),s(t)) t=1 that minimize Cost(y, u, v, s) are given by:, { if p(t)+η g o u(t)= min h(t) },,a(t),l y(t) if p(t) < η o <p(t)+η g { } min a(t),l y(t), if o p(t) (4) and v(t) =[a(t) u(t)] +, s(t) =[h(t) η u(t)] +. (5) We note in eah time slot t, the above u(t), v(t) and s(t) are omputed using only y(t) and σ(t) inthesametimeslot. he result of Lemma 1 an be interpreted as follows. If the grid prie is very high (i.e., higher than o), then it is always more eonomial to use loal generation as muh as possible, without even onsidering heating. However, if the grid prie is between o and o η g, loal eletriity generation alone is not eonomial. Rather, it is the benefit of supplying heat through o-generation that makes loal generation more eonomial. Hene, the amount of loal generation must onsider the heat demand h(t). Finally, when the grid prie is very low (i.e., lower than o η g), it is always more ost-effetive not to use loal generation. As a onsequene of Lemma 1, the problem fmcmp s an be simplified to the following problem SP, wherewe only need to onsider the deision of turning on (y(t) =1) or off (y(t) = ) the generator. SP :min Cost(y) y var y(t) {, 1},t [1,], where Cost(y) t=1 (ψ ( σ(t),y(t) ) ) + β [y(t) y(t 1)] +, ψ ( σ(t),y(t) ) ou(t)+p(t)v(t)+ gs(t)+ my(t) and (u(t),v(t),s(t)) are defined aording to Lemma Offline Optimal Solution We first study the offline setting, where the input (σ(t)) t=1 is given ahead of time. We will reveal an elegant struture of the optimal solution. hen, in Setion we will exploit this struture to design an effiient online algorithm. he problem SP an be solved by the lassial dynami programming approah. We refer interested readers to our tehnial report [27] for details. However, the solution provided by dynami programming does not seem to bring signifiant insights for developing online algorithms. herefore, in what follows we study the offline optimal solution from another angle, whih diretly reveals its struture. Define ( ) ( ) δ(t) ψ σ(t), ψ σ(t), 1. (6) δ(t) an be interpreted as the one-slot ost differene between using or not using loal generation. Intuitively, if

6 δ(t) > (resp. δ(t) < ), it will be desirable to turn on (resp. off) the generator. However, due to the startup ost, we should not turn on and off the generator too frequently. Instead, we should evaluate whether the umulative gain or loss in the future an offset the startup ost. his intuition motivates us to define the following umulative ost differene Δ(t). We set the initial value as Δ() = β and define Δ(t) indutively: { } Δ(t) min, max{ β, Δ(t 1) + δ(t)}. (7) Note that Δ(t) is only within the range [ β,]. Otherwise, the minimum ap ( β) and maximum ap () will apply to retain Δ(t) within[ β,]. An important feature of Δ(t) useful later in online algorithm design is that it an be omputed given the past and urrent input σ(τ), 1 τ t. Next, we onstrut ritial segments aording to Δ(t), and then lassify segments by types. Eah type of segments aptures similar episodes of demands. As shown later in heorem 1, it suffies to solve the ost minimization problem over every segment and ombine their solutions to obtain an offline optimal solution for the overall problem SP. Definition 1. We divide all time intervals in [1,]into disjoint parts alled ritial segments: [1, 1 ], [ 1 +1, 2 ], [ 2 +1, 3 ],..., [ k +1,] he ritial segments are haraterized by a set of ritial points: 1 <2 <... < k. We define eah ritial point i along with an auxiliary point i, suh that the pair ( i, i ) satisfies the following onditions: (Boundary): Either ( Δ(i )=andδ( i )= β) or ( Δ(i )= β and Δ( i )=). (Interior): β <Δ(τ) < for all i <τ< i. In other words, eah pair of (i, i ) orresponds to an interval where (t) goesfrom-β to or to -β, without reahing the two extreme values inside the interval. For example, (1, 1 )and( 2, 2 ) in Fig. 3 are two suh pairs, while the orresponding ritial segments are (1,2 )and(2,3 ). It is straightforward to see that all (i, i ) are uniquely defined, thus ritial segments are well-defined. See Fig. 3 for an example. One the time horizon [1,] is divided into ritial segments, we an now haraterize the optimal solution. Definition 2. We lassify the type of a ritial segment by: type-start (also all type-): [1, 1 ] type-1: [ i +1, i+1], if Δ( i )= β and Δ( i+1) = type-2: [ i +1, i+1], if Δ( i )=andδ( i+1) = β type-end (also all type-3): [ k +1,] We define the ost with regard to a segment i by: Cost sg-i (y) i+1 t= i +1 ψ ( σ(t),y(t) ) + i+1 +1 t= i +1 β [y(t) y(t 1)] + Δ(t) β Arrivals of demands type-start type-1 type-2 type-1 type-end y OFA y CHASES y lk( ω ) CHASE s ω Figure 3: An example of Δ(t), y OFA,y CHASEs and y lk(ω). CHASE s In the top two rows, we have a(t) {, 1}, h(t) {,η}. he prie p(t) is hosen as a onstant in ( o η g, o). In the next row, we ompute Δ(t) aording to a(t) andh(t). For ease of exposition, in this example we set the parameters so that Δ(t) inreases if and only if a(t) = 1 and h(t) = η. he solutions y OFA,y CHASEs and y lk(ω) at the bottom rows are obtained CHASE s aordingly to (8), Algorithms 1 and 3, respetively. and define a subproblem for ritial segment i by: SP sg-i (yi, l yi r ):mincost sg-i (y) s.t. y(i )=yi,y( l i+1 +1)=yi r, var y(t) {, 1},t [i +1,i+1]. Note that due to the startup ost aross segment boundaries, in general Cost(y) Cost sg-i (y). In other words, we should not expet that putting together the solutions to eah segment will lead to an overall optimal offline solution. However, the following lemma shows an important struture property that one optimal solution of SP sg i (yi,y l i r ) is independent of boundary onditions (yi,y l i r ) although the optimal value depends on boundary onditions. Lemma 2. (y OFA(t)) i+1 t= i +1 in (8) is an optimal solution for SP sg-i (yi, l yi r ), despite any boundary onditions (yi, l yi r ). his lemma an be intuitively explained by Fig. 3. In type-1 ritial segment, Δ(t) has an inrement of β, whihmeans that setting y(t) = 1 over the entire segment provides at least a benefit of β, ompared to keeping y(t) =. Suh benefit ompensates the possible startup ost β if the boundary onditions are not aligned with y(t) = 1. herefore, regardless of the boundary onditions, we should set y(t) =1on type-1 ritial segment. Other types of ritial segments an be explained similarly. We then use this lemma to show the following main result on the struture of the offline optimal solution. heorem 1. An optimal solution for SP is given by {, if t [i +1, y OFA(t) i+1] is type-start/-2/-end, 1, if t [i +1,i+1] istype-1. (8) heorem 1 an be interpreted as follows. Consider for example a type-1 ritial segment in Fig. 3 that starts from ω ω ω

7 1. Sine Δ(t) inreases from β after 1, it implies that δ(t) >, and thus we are interested in turning on the generator. he diffiulty, however, is that immediately after 1 we do not know whether the future gain by turning on the generator will offset the startup ost. On the other hand, one Δ(t) reahes, it means that the umulative gain in the interval [1, 1 ] will be no less than the startup ost. Hene, we an safely turn on the generator at 1. Similarly, for eah type-2 segment we an turn off the generator at the beginning of the segment. (We note that our offline solution turns on/off the generator at the beginning of eah segment beause all future information is assumed to be known.) he optimal solution is easy to ompute. More importantly, the insights help us design the online algorithms Our Proposed Online Algorithm CHASE Denote an online algorithm for SP by A. We define the ompetitive ratio of A by: CR(A) max σ Cost(y A) Cost(y OFA) Reall the struture of optimal solution y OFA: one the proess is entering type-1 (resp., type-2) ritial segment, we should set y(t) =1(resp.,y(t) = ). However, the diffiulty lies in determining the beginnings of type-1 and type-2 ritial segments without future information. Fortunately, as illustrated in Fig. 3, it is ertain that the proess is in a type-1 ritial segment when Δ(t) reahes for the first time after hitting β. his observation motivates us to use the algorithm CHASE s, whih is given in Algorithm 1. If β <Δ(t) <, CHASE s maintains y(t) =y(t 1) (sine we do not know whether a new segment has started yet.) However, when Δ = (resp. Δ(t) = β), we know for sure that we are inside a new type-1 (resp. type-2) segment. Hene, CHASE s sets y(t) =1(resp. y(t) = ). Intuitively, the behavior of CHASE s is to trak the offline optimal in an online manner: we hange the deision only after we are ertain that the offline optimal deision is hanged. Algorithm 1 CHASE s[t, σ(t),y(t 1)] 1: find Δ(t) 2: if Δ(t) = β then 3: y(t) 4: else if Δ(t) =then 5: y(t) 1 6: else 7: y(t) y(t 1) 8: end if 9: set u(t), v(t), and s(t) aording to (4) and (5) 1: return (y(t),u(t),v(t),s(t)) Even though CHASE s is a simple algorithm, it has a strong performane guarantee, as given by the following theorem. heorem 2. he ompetitive ratio of CHASE s satisfies CR(CHASE s) 3 2α <3, (1) where α ( o + m/l)/(p max + η g) (, 1] (11) aptures the maximum prie disrepany between using loal generation and external soures to supply energy. Remark: (i) he intuition that CHASE s is ompetitive an be explained by studying its worst ase input shown (9) Δ(t) β y OFA y CHASES y lk( ω ) CHASE s type-1 ω type-2 Figure 4: he worst ase input of CHASE s, and the orresponding y CHASEs, y lk(ω) and the offline optimal solution y CHASE OFA. s in Fig. 4. he demands and pries are hosen in a way suh that in interval [,1 ]Δ(t) inreases from β to, andininterval[ 1,2 ]Δ(t) dereases from to β. We see that in the worst ase, y CHASEs never mathes y OFA.But even in this worst ase, CHASE s pays only 2β more than the offline solution y OFA on [,2 ], while y OFA pays at least a startup ost β at time. Hene, the ratio of the online ost over the offline ost annot be too bad. (ii) heorem 2 says that CHASE s is more ompetitive when α is large than it is small. his an be explained intuitively as follows. Large α implies small eonomi advantage of using loal generation over external soures to supply energy. Consequently, the offline solution tends to use loal generation less. It turns out CHASE s will also use less loal generation 2 and is ompetitive to offline solution. Meanwhile, when α is small, CHASE s starts to use loal generation. However, using loal generation inurs high risk sine we have to pay the startup ost to turn on the generator without knowing whether there are suffiient demands to serve in the future. Laking future knowledge leads to a large performane disrepany between CHASE s and the offline optimal solution, making CHASE s less ompetitive. he result in heorem 2 is strong in the sense that CR(CHASE s) is always upper-bounded by a small onstant 3, regardless of system parameters. his is ontrast to large parameterdependent ompetitive ratios that one an ahieve by using generi approah, e.g., the metrial task system framework [8], to design online algorithms. Furthermore, we show that CHASE s ahieves lose to the best possible ompetitive ratio for deterministi algorithms as follow. heorem 3. Let ɛ> be the slot length under the disretetime setting we onsider in this paper. he ompetitive ratio for any deterministi online algorithm A for SP is lower bounded by CR(A) min(3 2α o(ɛ), 1/α), (12) where o(ɛ) vanishestozeroasɛ goes to zero and the disretetime setting approahes the ontinuous-time setting. Note that there is still a gap between the ompetitive ratios in (1) and (12). he differene is due to the term 1/α =(P max + η g)/( o + m/l). his term an be interpreted as the ompetitive ratio of a naive strategy that always uses external power supply and separate heat supply. Intuitively, if this 1/α term is smaller than 3 2α, we 2 CHASE s will turn on the loal generator when Δ(t) inreases to. he larger the α is, the slower Δ(t) inreases, and the less likely CHASE s will use the loal generator. ω

8 should simply use this naive strategy. his observation motivates us to develop an improved version of CHASE s, alled CHASE s+, whih is presented in Algorithm 2. Corollary 1 shows that CHASE s+ loses the above gap and ahieves the asymptoti optimal ompetitive ratio. Note that whether or not the 1/α term is smaller an be ompletely determined by the system parameters. Algorithm 2 CHASE s+[t, σ(t),y(t 1)] 1: if 1/α 3 2α then 2: y(t), u(t), v(t) a(t), s(t) h(t) 3: return (y(t),u(t),v(t),s(t)) 4: else 5: return CHASE s[t, σ(t),y(t 1)] 6: end if Corollary 1. CHASE s+ ahieves the asymptoti optimal ompetitive ratio of any deterministi online algorithm, as CR(CHASE s+) min(3 2α, 1/α). (13) Remark: At the beginning of Se. 3.1, we have disussed the strutural differenes of online server sheduling problems [25, 28] and ours. In what follows, we summarize the solution differenes among these problems. Note that we share similar intuitions with [28], both make swithing deisions when the penalty ost equals the swithing ost. he signifiant differene, however, is when to reset the penalty ounting. In [28], the penalty ounting is reset when the demand arrives. In ontrast, in our solution, we need to reset the penalty ounting only when Δ(t), given in the nontrivial form in (7), touhes or β. his partiular way of resetting penalty ounting is ritial for establishing the optimality of our proposed solution. Meanwhile, to ompare with [25], the approah in [25] does not expliitly ount the penalty. Furthermore, the online server sheduling problem in [25] is formulated as a onvex problem, while our problem is a mixed integer problem. hus, there is no known method to apply the approah in [25] to our problem. 3.2 Look-ahead Setting We onsider the setting where the online algorithm an predit a small window ω of the immediate future. Note that ω = returns to the ase treated in Setion 3.1.2, when there is no future information at all. Consider again a type-1 segment [1,2 ] in Fig. 3. Reall that, when there is no future information, the CHASE s algorithm will wait until 1, i.e., when Δ(t) reahes, to be ertain that the offline solution must turn on the generator. Hene, the CHASE s algorithm will not turn on the generator until this time. Now assume that the online algorithm has the information about the immediate future in a time window of length ω. Bythe time 1 w, the online algorithm has already known that Δ(t) will reah at time 1. Hene, the online algorithm an safely turn on the generator at time 1 w. Asaresult, the orresponding loss of performane ompared to the offline optimal solution is also redued. Speifially, even for the worst-ase input in Fig. 4, there will be some overlap (of length ω) between y CHASEs and y OFA in eah segment. Hene, the ompetitive ratio should also improve with future information. his idea leads to the online algorithm CHASE lk(ω) s, whih is presented in Algorithm 3. We an show the following improved ompetitive ratio when limited future information is available. Algorithm 3 CHASEs lk(ω) [t, (σ(τ)) t+w τ=t,y(t 1)] 1: find (Δ(τ)) t+w τ=t 2: set τ min { τ = t,..., t + w Δ(τ) =or = β } 3: if Δ(τ )= β then 4: y(t) 5: else if Δ(τ )=then 6: y(t) 1 7: else 8: y(t) y(t 1) 9: end if 1: set u(t), v(t), and s(t) aording to (4) and (5) 11: return (y(t),u(t),v(t),s(t)) heorem 4. he ompetitive ratio of CHASE lk(ω) s ( ) CR CHASE lk(ω) s satisfies 3 2 g (α, ω), (14) where ω is the look-ahead window size, α (, 1] is defined in (11), and (1 α) g(α, ω) =α + 1+β (L o + m/(1 α)) / (ω(l o +. m) m) (15) aptures the benefit of looking-ahead and monotonially inreases from α to 1 as ω inreases. In partiular, CR(CHASE lk() s )=CR(CHASE s). We replae CHASE s by CHASEs lk(ω) in CHASE s+ and obtain an improved algorithm for the look-ahead setting, named CHASE lk(ω) s+. Fig. 5 shows the ompetitive ratio of CHASElk(ω) s+ as a funtion of α and ω. ) ( CHASE lk(w) s+ CR α ω 2 Figure 5: he ompetitive ratio of CHASE lk(ω) s+ as a funtion of α and ω. 3.3 Multiple Generator Case Now we onsider the general ase with N units of homogeneous generators, eah having an maximum power apaity L, startupostβ, sunk ost m and per unit operational ost o. We define a generalized version of problem: fmcmp : min Cost(y,u, v, s) y,u,v,s s.t. Constraints (2b), (2), and (2d) var y n(t) {, 1},u n(t),v(t),s(t) R +, Next, we will onstrut both offline and online solutions to fmcmp in a divide-and-onquer fashion. We will first partition the demands into sub-demands for eah generator, and then optimize the loal generation separately for eah subdemand. Note that the key is to orretly partition the demand so that the ombined solution is still optimal. Our strategy below essentially slies the demand (as a funtion of t) into multiple layers from the bottom up (see Fig. 6). Eah layer has at most L units of eletriity demand and

9 heorem 6. he ompetitive ratio of CHASE lk(ω) satisfies CR(CHASE lk(ω) ) min(3 2 g(α, ω), 1/α), (18) where α (, 1] is defined in (11) and g(α, ω) [α, 1] is defined in (15). (a) An example of (a ly n ). (b) An example of (h ly n ). Figure 6: An example of (a ly n )and(h ly n ). In this example, N =2. We obtain 3 layers of eletriity and heat demands, respetively. η L units of heat demand. he intuition here is that the layers at the bottom exhibit the least frequent variations of demand. Hene, by assigning eah of the layers at the bottom to a dediated generator, these generators will inur the least amount of swithing, whih helps to redue the startup ost. More speifially, given (a(t),h(t)), we slie them into N + 1layers: a ly-1 (t) =min{l, a(t)}, h ly-1 (t) =min{η L, h(t)} (16a) a ly-n (t) =min{l, a(t)- n 1 r=1 aly-r (t)},n [2,N] (16b) h ly-n (t) =min{η L, h(t)- n 1 r=1 hly-r (t)},n [2,N] (16) a top (t) =min{l, a(t) N r=1 aly-r (t)} (16d) h top (t) =min{η L, h(t) N r=1 hly-r (t)} (16e) It is easy to see that eletriity demand satisfies a ly-n (t) L and heat demand satisfies h ly-n (t) η L. hus,eahlayer of sub-demand an be served by a single loal generator if needed. Note that (a top,h top ) an only be satisfied from external supplies, beause they exeed the apaity of loal generation. Based on this deomposition of demand, we then deompose the fmcmp problem into N sub-problems fmcmps ly-n (1 n N), eah of whih is an fmcmp s problem with input (a ly-n,h ly-n,p). We then apply the offline and online algorithms developed earlier to solve eah sub-problem fmcmps ly-n (1 n N) separately. By ombining the solutions to these sub-problems, we obtain offline and online solutions to fmcmp. For the offline solution, the following theorem states that suh a divide-and-onquer approah results in no optimality loss. heorem 5. Suppose (y n,u n,v n,s n) is an optimal offline solution for eah fmcmps ly-n (1 n N). hen ((yn,u n) N n=1,v,s ) defined as follows is an optimal offline solution for fmcmp: yn(t) =y n(t), v (t) =a top (t)+ N n=1 vn(t) u n(t) =u n(t), s (t) =h top (t)+ N n=1 sn(t) (17) For the online solution, we also apply suh a divide-andonquer approah by using (i) a entral demand dispathing module that slies and dispathes demands to individual generators aording to (16a)-(16e), and (ii) an online generation sheduling module sitting on eah generator n (1 n N) independently solving their own fmcmp ly-n s sub-problem using the online algorithm CHASE lk(ω) s+. he overall online algorithm, named CHASE lk(ω),issimple to implement without the need to oordinate the ontrol among multiple loal generators. Sine the offline (resp. online) ost of fmcmp is the sum of the offline (resp. online) osts of fmcmps ly-n (1 n N), it is not diffiult to establish the ompetitive ratio of CHASE lk(ω) as follows. 4. SLOW-RESPONDING GENERAOR CASE We next onsider the slow-responding generator ase, with the generators having non-negligible onstraints on the minimum on/off periods and the ramp-up/down speeds. For this slow-responding version of MCMP, its offline optimal solution is harder to haraterize than fmcmp due to the additional hallenges introdued by the ross-slot onstraints (2e)-(2h). In the slow-responding setting, loal generators annot be turned on and off immediately when demand hanges. Rather, if a generator is turned on (resp., off) at time t, it must remain on for at least on (resp., off ) time. Further, the hanges of u n(t) u n(t 1) must be bounded by R up and R down. A simple heuristi is to first ompute solutions based on CHASE lk(ω), and then modify the solutions to respet the above onstraints. We name this heuristi CHASE lk(ω) gen and present it in Algorithm 4. For simpliity, Algorithm 4 is a single-generator version, whih an be easily extended to the multiple-generator senario by following the divide-andonquer approah elaborated in Se Algorithm 4 CHASE lk(ω) gen [t, (σ(τ)) t+ω τ=1,y(t 1)] 1: ( y ) s(t),u s(t),v s(t),s s(t) CHASE lk(ω) [ ( ( )) t+w s t, σ τ τ=1,y( t-1 )] 2: if y(τ 1 ) 1 1 {ys(t)>y(t 1)}, τ 1 [max(1,t off ),t 1] and y(τ 2 ) 1 {ys(t)<y(t 1)}, τ 2 [max(1,t on),t 1] then 3: y(t) y s(t) 4: else 5: y(t) y(t 1) 6: end if 7: if u s(t) >u(t 1) then 8: u(t) u(t 1) + min ( R up,u s(t) u(t 1) ) 9: else 1: u(t) u(t 1) min ( R dw,u(t 1) u ) s(t) 11: end if 12: v(t) [ a(t) u(t) ] + 13: s(t) [ h(t) η u(t) ] + 14: return ( y(t),u(t),v(t),s(t) ) We now explain Algorithm 4 and its ompetitive ratio. At eah time slot t, we obtain the solution of CHASE lk(ω) s,inluding y s(t),u s(t),v s(t),s s(t), as a referene solution (Line 1). hen in Line 2-6, we modify the referene solution s y s(t) to our atual solution y(t), to respet the onstraints of minimum on/off periods. More speifially, we follow the referene solution s y s(t) (i.e., y(t) =y s(t)) if and only if it respets the minimum on/off periods onstraints (Line 2-3). Otherwise, we let our atual solution s y(t) equal our previous slot s solution (y(t) = y(t 1)) (Line 4-5). Similarly, we modify the referene solution s u s(t) to our atual solution s u(t), to respet the onstraints on ramp-up/down speeds (Line 7-11). At last, in our atual solution, we use (v(t),s(t)) to ompensate the supply and satisfy the demands (Line 12-13). In summary, our atual solution is designed to be aligned with the referene solution as muh as possible. ratio of CHASE lk(ω) gen We derive an upper bound on the ompetitive as follows.

10 heorem 7. he ompetitive ratio of CHASE lk(ω) gen is upper bounded by (3 2g(α, ω)) max ( ) r 1,r 2, where g(α, ω) is defined in (15) and {( ) Pmax + g η r 1 =1 + max max {, ( )} L R up L + m o max {, ( )} L R dw, m β + m on and r 2 = + L( P max + g η ) ( on + off ). β β We note that when on = off =,R up = R dw =, the above upper bound mathes that of CHASE lk(ω) in heorem 6 (speifially the first term inside the min funtion). 5. EMPIRICAL EVALUAIONS We evaluate the performane of our algorithms based on evaluations using real-world traes. Our objetives are threefold: (i) evaluating the potential benefits of CHP and the ability of our algorithms to unleash suh potential, (ii) orroborating the empirial performane of our online algorithms under various realisti settings, and (iii) understanding how muh loal generation to invest to ahieve substantial eonomi benefit. 5.1 Parameters and Settings Demand rae: We obtain the demand traes from California Commerial End-Use Survey (CEUS) [1]. We fous on a ollege in San Franiso, whih onsumes about 154 GWh eletriity and therms gas per year. he traes ontain hourly eletriity and heat demands of the ollege for year 22. he heat demands for a typial week in summer and spring are shown in Fig. 7. hey display regular daily patterns in peak and off-peak hours, and typial weekday and weekend variations. Wind Power rae: We obtain the wind power traes from [4]. We employ power output data for the typial weeks in summer and spring with a resolution of 1 hour of an offshore wind farm right outside San Franiso with an installed apaity of 12MW. he net eletriity demand, whih is omputed by subtrating the wind generation from eletriity demand is shown in Fig. 7. he highly flutuating and unpreditable nature of wind generation makes it diffiult for the onventional predition-based energy generation sheduling solutions to work effetively. Eletriity and Natural Gas Pries: he eletriity and natural gas prie data are from PG&E [5] and are shown in able 3. Besides, the grid eletriity pries for a typial week in summer and winter are shown in Fig. 7. Both the eletriity demand and the prie show strong diurnal properties: in the daytime, the demand and prie are relatively high; at nights, both are low. his suggests the feasibility of reduing the mirogrid operating ost by generating heaper energy loally to serve the demand during the daytime when both the demand and eletriity prie are high. Generator Model: We adopt generators with speifiations the same as the one in [6]. he full output of a single generator is L = 3MW. he inremental ost per unit time to generate an additional unit of energy o is set to be.51/kw h, whih is alulated aording to the natural gas prie and the generator effiieny. We set the heat reovery effiieny of o-generation η to be 1.8 aording to } [6]. We also set the unit-time generator running ost to be m = 11$/h, whih inludes the amortized apital ost and maintenane ost aording to a similar setting from [36]. We set the startup ost β equivalent to running the generator at its full apaity for about 5 hrs at its own operating ost whih gives β = 14$. In addition, we assume for eah generator on = off =3h and R up = R dw =1MW/h, unless mentioned otherwise. For eletriity demand trae we use, the peak demand is 3MW. hus, we assume there are 1 suh CHP generators so as to fully satisfy the demand. Loal Heating System: We assume an on-demand heating system with apaity suffiiently large to satisfy all the heat demand by itself and without on-off ost or ramp limit. he effiieny of a heating system is set to.8 aording to [2], and onsequently we an ompute the unit heat generation ost to be g =.179$/KW h. Cost Benhmark: We use the ost inurred by using only external eletriity, heating and wind energy (without CHP generators) as a benhmark. We evaluate the ost redution due to our algorithms. Comparisons of Algorithms: We ompare three algorithms in our simulations. (1) our online algorithm CHASE; (2) the Reeding Horizon Control (RHC) algorithm; and (3) the OFFLINE optimal algorithm we introdue in Se. 4. RHC is a heuristi algorithm ommonly used in the ontrol literature [22]. In RHC, an estimate of the near future (e.g., in a window of length w) is used to ompute a tentative ontrol trajetory that minimizes the ost over this timewindow. However, only the first step of this trajetory is implemented. In the next time slot, the window of future estimates shifts forward by 1 slot. hen, another ontrol trajetory is omputed based on the new future information, and again only the first step is implemented. his proess then ontinues. We note that beause at eah step RHC does not onsider any adversarial future dynamis beyond the time-window w, there is no guarantee that RHC is ompetitive. For the OFFLINE algorithm, the inputs are system parameters (suh as β, m and on), eletriity demand, heat demand, wind power output, gas prie, and grid eletriity prie. For online algorithms CHASE and RHC, the input is the same as the OFFLINE exept that at time t, only the demands, wind power output, and pries in the past and the look-ahead window (i.e., [1,t+ w]) are available. he output for all three algorithms is the total ost inurred during the time horizon [1,]. MWh $/KWh 4 24 Eletriity Net Heat Grid prie.1 5 t/hour1 15 (a) Summer herm MWh $/KWh 4 Eletriity Net 24 Heat Grid prie.1 5 t/hour1 15 (b) Winter Figure 7: Eletriity net demand and heat demand for a typial week in summer and winter. he net demand is omputed by subtrating the wind generation from the eletriity demand. he net eletriity demand and the heat demand need to be satisfied by using the loal CHP generators, the eletriity grid, and the heating system. 5.2 Potential Benefits of CHP Purpose: he experiments in this subsetion aim to answer two questions. First, what is the potential savings with mirogrids? Note that eletriity, heat demand, wind sta- herm

11 Eletriity Summer (May-Ot.) Winter (Nov.-Apr.) $/kwh $/kwh On-peak.232 N/A Mid-peak Off-peak Natural Gas.419$/therm.486$/therm able 3: PG&E ommerial tariffs and natural gas tariffs. In the table, summer on-peak, mid-peak, and off-peak hours are weekday 12-18, weekday 8-12, and the remaining hours, respetively. Winter mid-peak and off-peak hours are weekday 8-22 and the remaining hours, respetively. he gas prie is an average; monthly pries vary slightly aording to PG&E. tion output as well as energy prie all exhibit seasonal patterns. As we an see from Figs. 7a and 7b, during summer (similarly autumn) the eletriity prie is high, while during winter (similarly spring) the heat demand is high. It is then interesting to evaluate under what settings and inputs the savings will be higher. Seond, what is the differene in ost-savings with and without the o-generation apability? In partiular, we ondut two sets of experiments to evaluate the ost redutions of various algorithms. Both experiments have the same default settings, exept that the first set of experiments (referred to as CHP) assumes the CHP tehnology in the generators is enabled, and the seond set of experiments (referred to as NOCHP) assumes the CHP tehnology is not available, in whih ase the heat demand must be satisfied solely by the heating system. In all experiments, the look-ahead window size is set to be w =3 hours aording to power system operation and wind generation foreast pratie [3]. he ost redutions of different algorithms are shown in Fig. 8a and 8b. he vertial axis is the ost redution as ompared to the ost benhmark presented in Se Observations: First, the whole-year ost redutions obtained by OFFLINE are 21.8% and 11.3% for CHP and NOCHP senarios, respetively. his justifies the eonomi potential of using loal generation, espeially when CHP tehnology is enabled. hen, looking at the seasonal performane of OF- FLINE, weobservethatoffline ahieves muh more ost savings during summer and autumn than during spring and winter. his is beause the eletriity prie during summer and autumn is very high, thus we an benefit muh more from using the relatively-heaper loal generation as ompared to using grid energy only. Moreover, OFFLINE ahieves muh more ost savings when CHP is enabled than when it is not during spring and winter. his is beause, during spring and winter, the eletriity prie is relatively low and the heat demand is high. Hene, just using loal generation to supply eletriity is not eonomial. Rather, loal generation beomes more eonomial only if it an be used to supply both eletriity and heat together (i.e., with CHP tehnology). Seond, CHASE performs onsistently lose to OFFLINE aross inputs from different seasons, even though the different settings have very different harateristis of demand and supply. In ontrast, the performane of RHC depends heavily on the input harateristis. For example, RHC ahieves some ost redution during summer and autumn when CHP is enabled, but ahieves ost redution in all the other ases. Ramifiations: In summary, our experiments suggest that exploiting loal generation an save more ost when the eletriity prie is high, and CHP tehnology is more 2 %Cost Redution4 RHC CHASE OFFLINE Spring Summer Autumn Winter Whole Year 2 %Cost Redution4 RHC CHASE OFFLINE Spring Summer Autumn Winter Whole Year (a) Loal generators with CHP(b) Loal generators without CHP Figure 8: Cost redutions for different seasons and the whole year. %Cost Redution w /hour Figure 9: Cost redution as a funtion of look ahead window size ω. %Cost Redution %Generation Capaity Figure 1: Cost redution as a funtion of loal generation apaity. ritial for ost redution when heat demand is high. Regardless of the problem setting, it is important to adopt an intelligent online algorithm (like CHASE) to shedule energy generation, in order to realize the full benefit of mirogrids. 5.3 Benefits of Looking-Ahead Purpose: WeomparetheperformanesofCHASE to RHC and OFFLINE for different sizes of the look-ahead window and show the results in Fig. 9. he vertial axis is the ost redution as ompared to the ost benhmark in Se. 5.1 and the horizontal axis is the size of lookahead window, whih varies from to 2 hours. Observations: We observe that the performane of our online algorithm CHASE is already lose to OFFLINE even when no or little look-ahead information is available (e.g., w =, 1, and 2). In ontrast, RHC performs poorly when the look-ahead window is small. When w is large, both CHASE and RHC perform very well and their performane are lose to OFFLINE when the look-ahead window w is larger than 15 hours. An interesting observation is that it is more important to perform intelligent energy generation sheduling when there is no or little look-ahead information available. When there are abundant look-ahead information available, both CHASE and RHC ahieve good performane and it is less ritial to arry out sophistiated algorithm design. In Fig. 11a and 11b, we separately evaluate the benefit of looking-ahead under the fast-responding and slowresponding senarios. We evaluate the empirial ompetitive ratio between the ost of CHASE and OFFLINE, and ompare it with the theoretial ompetitive ratio aording to our analytial results. In the fast-responding senario (Fig. 11a), for eah generator there are no minimum on/off period and ramping-up/down onstraints. Namely, on =, off =, R up =, R dw =. In the slowresponding senario (Fig. 11b), we set on = off =3h and R up = R dw =1MW/h. Inbothexperiments,weobserve that the theoretial ratio dereases rapidly as look-ahead window size inreases. Further, the empirial ratio is already lose to one even when there is no look-ahead information. 5.4 Impats of Look-ahead Error Purpose: Previous experiments show that our algorithms have better performane if a larger time-window of aurate look-ahead input information is available. he input infor-

12 Ratio Empirial ratio heoretial ratio w /hour (a) Fast-responding senario Ratio Empirial ratio heoretial ratio upper bound w /hour (b) Slow-responding senario Figure 11: heoretial and empirial ratios for CHASE, as funtions of look-ahead window size ω. Note that the theoretial ompetitive ratios (or their bounds) measure the worst-ase performane and are often muh larger than the empirial ratios observed in pratie. %Cost Redution 4 2 w=3 h w=1 h %standard deviation (a) Wind power foreast error %Cost Redution 4 2 w=3 h w=1 h %standard deviation (b) Heat demand foreast error Figure 12: Cost redution as a funtion of the size predition error (measured by the standard deviation of the predition error as a perentage of (a)installed apaity and (b)peak heat demand). mation in the look-ahead window inludes the wind station power output, the eletriity and heat demand, and the entral grid eletriity prie. In pratie, these look-ahead information an be obtained by applying sophistiated predition tehniques based on the historial data. However, there are always predition errors. For example, while the day-ahead eletriity demand an be predited within 2-3% range, the wind power predition in the next hours usually omes with an error range of 2-5% [2]. herefore, it is important to evaluate the performane of the algorithms in the presene of predition error. Observations: o ahieve this goal, we evaluate CHASE with look-ahead window size of 1 and 3 hours. Aording to [2], the hour-level wind-power predition-error in terms of the perentage of the total installed apaity usually follows Gaussian distribution. hus, in the look-ahead window, a zero-mean Gaussian predition error is added to the amount of wind power in eah time-slot. We vary the standard deviation of the Gaussian predition error from to 12% of the total installed apaity. Similarly, a zero-mean Gaussian predition error is added to the heat demand, and its standard deviation also varies from to 12% of the peak demand. We note that in pratie, predition errors are often in the range of 2-5% for 3-hour predition [2]. hus, by using a standard deviation up to 12%, we are essentially stress-testing our proposed algorithms. We average 2 runs for eah algorithm and show the results in Figs. 12a and 12b. As we an see, both CHASE and RHC are fairly robust to the predition error and both are more sensitive to the wind-power predition error than to the heat-demand predition error. Besides, the impat of the predition error is relatively small when the look-ahead window size is small, whih mathes with our intuition. 5.5 Impats of System Parameters Purpose: Mirogrids may employ different types of loal generators with diverse operational onstraints (suh ramping up/down limits and minimum on/off times) and heat reovery effiienies. It is then important to understand the impat on ost redution due to these parameters. In this experiment, we study the ost redution provided by our of- %Cost Redution % Rup/L (Rup = Rdw) on/hour (on = off) (a) ost redu. vs. R up and R dw (b) ost redu. vs. on and off %Cost Redution Summer η () ost redution vs. η %Cost Redution %Cost Redution Winter η (d) ost redution vs. η Figure 13: Cost redution as funtions of generator parameters. fline and online algorithms under different settings of R up, R dw, on, off and η. Observations: Fig. 13a and 13b show the impat of ramp limit and minimum on/off time, respetively, on the performane of the algorithms. Note that for simpliity we always set R up = R dw and on = off. AsweanseeinFig. 13a, with R up and R dw of about 4% of the maximum apaity, CHASE obtains nearly all of the ost redution benefits, ompared with RHC whih needs 7% of the maximum apaity. Meanwhile, it an be seen from Fig. 13b that on and off do not have muh impat on the performane. his suggests that it is more valuable to invest in generators with fast ramping up/down apability than those with small minimum on/off periods. From Fig. 13 and 13d, we observe that generators with large η save muh more ost during the winter beause of the high heat demand. his suggests that in areas with large heat demand, suh as Alaska and Washington, the heat reovery effiieny ratio is a ritial parameter when investing CHP generators. 5.6 How Muh Loal Generation is Enough hus far, we assumed that the mirogrid had the ability to supply all energy demand from loal power generation in every time-slot. In pratie, loal generators an be quite expensive. Hene, an important question is how muh investment should a mirogrid operator makes (in terms of the installed loal generator apaity) in order to obtain the maximum ost benefit. More speifially, we vary the number of CHP generators from 1 to 1 and plot the orresponding ost redutions of algorithms in Fig. 1. Interestingly, our results show that provisioning loal generation to produe 6% of the peak demand is suffiient to obtain nearly all of the ost redution benefits. Further, with just 5% loal generation apaity we an ahieve about 9% of the maximum ost redution. he intuitive reason is that most of the time demands are signifiantly lower than their peaks. 6. RELAED WORK Energy generation sheduling is a lassial problem in power systems and involves two aspets, namely Unit Commitment (UC) and Eonomi Dispathing (ED). UC optimizes the startup and shutdown shedule of power generations to meet the foreasted demand over a short period, whereas ED alloates the system demand and spinning reserve apaity among operating units at eah speifi hour

13 of operation without onsidering startup and shutdown of power generators. For large power systems, UC involves sheduling of a large number giganti power plants of several hundred if not thousands of megawatts with heterogeneous operating onstraints and logistis behind eah ation [34]. he problem is very hallenging to solve and has been shown to be NP-Complete in general 3 [16]. Sophistiated approahes proposed in the literature for solving UC inlude mixed integer programming [1], dynami programming [35], and stohasti programming [37]. here have also been investigations on UC with high renewable energy penetration [38], based on overprovisioning approah. After UC determines the on/off status of generators, ED omputes their output levels by solving a nonlinear optimization problem using various heuristis without altering the on/off status of generators [14]. here is also reent interest in involving CHP generators in ED to satisfy both eletriity and heat demand simultaneously [17]. See omprehensive surveys on UC in [34] and on ED in [14]. However, these studies assume the demand and energy supply (or their distributions) in the entire time horizon are known aprior. As suh, the shemes are not readily appliable to mirogrid senarios where aurate predition of small-sale demand and wind power generation is diffiult to obtain due to limited management resoures and their unpreditable nature [42]. Several reent works have started to study energy generation strategies for mirogrids. For example, the authors in [18] develop a linear programming based ost minimization approah for UC in mirogrids. [19] onsiders the fuel onsumption rate minimization in mirogrids and advoates to build IC infrastruture in mirogrids. [24, 26] disuss the energy sheduling problems in data enters, whose models are similar with ours. he differene between these works and ours is that they assume the demand and energy supply are given beforehand, and ours does not rely on input predition. Online optimization and algorithm design is an established approah in optimizing the performane of various omputer systems with minimum knowledge of inputs [8, 32]. Reently, it has found new appliations in data enters [15, 9, 39, 25, 28, 29]. o the best of our knowledge, our work is the first to study the ompetitive online algorithms for energy generation in mirogrids with intermittent energy soures and o-generation. he authors in [31] apply online onvex optimization framework [43] to design ED algorithms for mirogrids. However, the ED problem does not take into aount the startup ost. In ontrast, our work jointly onsider UC and ED in mirogrids with o-generation. Furthermore, the two works adopt different frameworks and provide online algorithms with different types of performane guarantee. 7. CONCLUSION In this paper, we study online algorithms for the mirogrid generation sheduling problem with intermittent renewable energy soures and o-generation, with the goal of maximizing the ost-savings with loal generation. Based on 3 We note that fmcmp in (3a)-(3d) is an instane of UC, and that UC is NP-hard in general does not imply that the instane fmcmp is also NP-hard. insights from the struture of the offline optimal solution, we propose a lass of ompetitive online algorithms, alled CHASE that trak the offline optimal in an online fashion. Under typial settings, we show that CHASE ahieves the best ompetitive ratio of all deterministi online algorithms, and the ratio is no larger than a small onstant 3. We also extend our algorithms to intelligently leverage on limited predition of the future, suh as near-term demand or wind foreast. By extensive empirial evaluations using real-world traes, we show that our proposed algorithms an ahieve near offline-optimal performane. here are a number of interesting diretions for future work. First, energy storage systems (e.g., large-apaity battery) have been proposed as an alternate approah to redue energy generation ost (during peak hours) and to integrate renewable energy soures. It would be interesting to study whether our proposed mirogrid ontrol strategies an be ombined with energy storage systems to further redue generation ost. However, urrent energy storage systems an be very expensive. Hene, it is ritial to study whether the ombined ontrol strategy an redue suffiient ost with limited amount of energy storage. Seond, it remains an open issue whether CHASE an ahieve the best ompetitive ratios in general ases (e.g., in the slow-responding ase). 8. ACKNOWLEDGMENS he work desribed in this paper was partially supported by China National 973 projets (No. 212CB31594 and 213CB3367), several grants from the University Grants Committee of the Hong Kong Speial Administrative Region, China (Area of Exellene Projet No. AoE/E-2/8 and General Researh Fund Projet No and 41111), two gift grants from Mirosoft and Ciso, and Masdar Institute- MI Collaborative Researh Projet No. 11CAMA1. Xiaojun Lin would like to thank the Institute of Network Coding at he Chinese University of Hong Kong for the support of his sabbatial visit, during whih some parts of the work were done. 9. REFERENCES [1] California ommerial end-use survey. Internet: [2] Green energy. Internet: [3] he irish meteorologial servie online. Internet: [4] National renewable energy laboratory. Internet: [5] Paifi gas and eletri ompany. Internet: [6] eogen. Internet: [7] M. Barnes, J. Kondoh, H. Asano, J. Oyarzabal, G. Ventakaramanan, R. Lasseter, N. Hatziargyriou, and. Green. 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