Queueing analysis of service deferrals for load management in power systems

Transcription

1 Qeeing analysis of service deferrals for load management in power systems Andrés Ferragt and Fernando Paganini Universidad ORT Urgay Abstract With the advent of renewable sorces and Smart- Grid deployments, it is increasingly common to control demands in order to redce power consmption variability and ths the need for reglation, with load aggregators now exploiting the deferability of some power loads to smooth the consmption profile. In this paper, we analyze the impact of service deferrals and schedling on power consmption variability sing tools from qeeing theory. We consider a generic model for a load aggregator that receive job reqests, involving a certain amont of energy to be provided and a deadline. We analyze different schedling policies and examine the impact of service deferrals, qantifying the tradeoff between variance redction and attained deadlines. I. INTRODUCTION One of the basic fnctions of a power system is to maintain instantaneos balance between demand and spply, de to the limited availability of storage. Since the first-order effect of any imbalance is a deviation of AC freqency from its nominal vale, the term freqency reglation is normally sed to describe the real-time actions by the system operator to correct sch imbalance. Historically, freqency reglation has been implemented by adjsting the amont of power injected by fast ramping generators (hydro or gas trbines) in real time into the power grid [. This procedre comes from the traditional rationale of an exogenos ncertain demand, and spply nder control of the operator. However, with the advent of renewable energy sorces, spply is now also becoming npredictable, which may reqire the installation of extra reglation capacities [2. On the other hand, Smart-Grid deployments open the possibility of controlling power demand in real time by signaling sers. While sch demand response cold inclde modifying mean consmption levels, in the context of short-term reglation it is more meaningfl to focs on merely deferring some consmption (heating, AC, EV battery charging, etc.) to contribte to power balance. Sch a reglation service on the demand side cold be provided by a load aggregator (e.g., [3, [4) making dispatch decisions on behalf of a large enogh nmber of individal loads. Some recent references exploring this potential for thermostatically controlled loads are [5 [7. More generically, [8, [9 analyze a collection of loads characterized by arrival times, deadlines, and power and energy reqirements. In [8 different schedling methods are stdied from a This work was partially spported by AFOSR nder grant FAA and ANII-Urgay nder grant FSE ferragt@ort.ed.y. nmerical perspective, comparing classical approaches from processor schedling (earliest deadline first, least laxity first [) with a model predictive control proposal. In [9 the athors attempt to characterize the aggregate flexibility provided by sch load arrival profile in deterministic terms, invoking an eqivalent electricity storage. From a control perspective, recently [ propose an architectre which enables the approximation of an aggregation of loads by a linear time invariant model to enable tracking of an external reference signal. Or previos work [2 prses a similar objective with flid models of the load arrival/service process, similar to those of large-scale qeeing systems. Also in [2 is a stdy, from a flid aggregate perspective, of a basic tradeoff for stochastically arriving loads: deferring service can redce consmed power variability (and ths helps with reglation), bt also impacts qality expectations of sers in terms of meeting deadlines. This tradeoff is the central focs of the present paper, this time sing more detailed stochastic qeeing models, and considering a wider variety of service disciplines. The problem formlation is laid ot in Section II, setting the main assmptions and parameters. In Section III we assme a fixed level of service deferral, and stdy the impact of this choice on both the consmption variance and the probability of missing deadlines. In particlar we analyze an eqal sharing policy with tools of M/G/ qees, and provide approximate reslts for the LLF policy. In Section IV we consider an alternate sitation in which deadlines are strictly enforced, and the remaining laxity is administered to redce consmption variance. We se tools of point-process theory and Markov processes to characterize the variability for two relevant policies. Conclsions are given in Section V. II. QUEUEING MODEL FOR FIXED DEFERRED SERVICE We now formlate the problem and notation by means of an initial qeeing model, which will be later extended in Section IV. Consider a load aggregator entity which receives random energy reqests, arriving as a Poisson process of intensity. The nominal power of each reqest is for simplicity taken to be a common parameter p, bt the amont of energy Q k of reqest k is random. Its nominal service time is ths given by k = Q k p. See also [3 for qeeing analysis of aggregation, withot deferability.

2 Individal load reqests may also have a deadline, before which they shold have received fll service. We model this by assming that reqest k has an initial laxity l k, which is the amont of spare time or slackness it has on arrival. In other words, each load has a service deadline of d k = k +l k time nits after its arrival. If not served after l k time nits, laxity expires and service mst be provided at fll power to meet the deadline. If remaining laxity becomes negative, the job will miss its deadline even at fll power. Or initial focs is on the case where both k and l k are independent exponential random variables, with E[ k = /µ and E[l k = /γ. Nevertheless other cases will be considered below and we will indicate when the reslts remain valid for more general distribtions. The load profile is ths characterized by the parameters, µ and γ, and the nominal power of each load p. The mean aggregate power needed for a given load profile is given by: p = E[Q k = p E[ k = p µ, () and this qantity is independent of any decision on load deferral or schedling. In particlar, given the expected energy reqirements this amont of power wold be prchased in advance by the load aggregator. At any given time t, the load aggregator has a qee of n(t) jobs present in the system, all of which may receive service or be deferred. To model service deferrals, we consider a fixed service level (,, which represents the fraction of nominal power sed by the system. Under this policy, the power consmed by the load aggregate at any time t is: p(t) = p n(t). Setting = represents no service deferral, all loads are served at fll power pon arrival. Choosing < means that either fewer loads are served at any given time, or that they are served with less power and consming laxity, leading to deadline misses. Once the service level is chosen, the system has also the choice of which loads to serve, i.e. the detailed schedling mechanism. If we assme that all loads are eventally served (before or after deadline), then the average consmed power shold match the average power of the load profile given by () and therefore in steady state: p = p µ = E[p(t) = E[p n(t), meaning that for a fixed service level the expected nmber of loads in the system is: n = µ. (2) We now state the two main objectives of the load aggregator in mathematical terms. The main prpose of the aggregation is to smooth the consmed power profile, minimizing deviations from the mean power p. Let s denote by δp(t) = p(t) p this real-time deviation. Then or first objective is to redce the steady state variance of p(t), E[(δp) 2, a measre of the reglation cost. The second objective is to serve as many load reqests as possible before their deadlines. To qantify this aspect we introdce the deferability factor of the load profile: := E[l k E[ k = µ γ. (3) Let T k denote the time load k spends in the system, and T its mean. Applying Little s law, we have from eqation (2): n = T = T = µ = E[ k. This imposes a first constraint on : in order to meet deadlines on average we shold have T < E[ k + l k i.e. > E[ k E[ k + E[l k = +. (4) Here η := + is the minimm service level that loads shold receive to meet their deadlines, in terms of the deferability factor. As deferability increases, η and the system gains in flexibility, meaning that loads arrive with larger slack time. Of corse, to evalate system performance, other metrics shold also be inclded, mainly the missed deadline probability α := P (T k > k + l k ). Using this characterization of the load aggregator as a qeeing system, we now explore the aforementioned tradeoffs between consmed power variability and missed deadlines, in terms of the service level and the schedling policies implemented by the system. III. FIXED SERVICE DEFERRAL IMPACT ON POWER CONSUMPTION VARIABILITY AND MISSED DEADLINES Or definition of the qeeing system states that the total otpt power for service level is given by p(t) = p n(t). We now analyze two simple schedling policies operating nder this constraint: first we consider an eqal sharing policy, where every load is served at individal rate p. While this is not always possible in the context of energy distribtion, it serves as a baseline to compare other policies. Also, it can be approximately implemented by serving a sbset of the crrent jobs taken at random from the poplation, with probability. This schedling was analyzed throgh a flid model by the athors in [2. The second case is the least-laxity-first (LLF) policy, introdced by [ in the context of schedling and discssed in [6, [7 in the context of power systems. Here the idea is to serve first the loads with least spare time remaining, p to the aggregate power determined by the service level. A. Eqal sharing policy Under the eqal sharing policy, every load present in the system is served at the redced rate p, ths taking longer to finish. The time in the system is given by: T k = Q k p = k

3 Since all reqests are served in parallel, the system behaves as an infinite server qee, with arrival rate and average service time E[T k = /(µ). In particlar, the nmber of loads in steady state satisfies: ( ) n(t) Poisson. µ We remark here that, since the steady state poplation in an M/G/ qee is insensitive to the detailed characteristics of the job size distribtion, the above reslt holds for general distribtion of k. We conclde that, in steady state, n = E[n(t) = µ, p = E[p(t) = E[p n(t) = p µ consistently with or previos analysis. Or analysis can however now go frther, compting the variance of consmed power E[(δp) 2 = E[p 2 2 (n(t) n) 2 = p 2 2 Var(n(t)) = p p, (5) where we have invoked the variance of the Poisson distribtion. A more normalized way of expressing variability is the coefficient of variation cv 2 (p) defined by cv 2 (p) = Var(p) p 2, which can be readily compted from (), (5) to yield: cv 2 (p) = p. (6) p So we find that variability redces linearly with the service level. Also, as aggregation grows large in the sense that p/p is large, the system redces its variability in consmed power. We now trn or attention to deadline misses. The probability of missing the deadline is: α = P (T k > k + l k ) = P ( k = P > l k ( k > k + l k ) ). (7) The previos eqation can be reinterpreted as follows: when a job arrives with service time k and initial laxity l k and is served at rate, after a time dt the remaining service time will be = k dt. Since its deadline is k + l k dt time nits ahead, the remaining laxity after a time dt will be: l = k + l k dt ( k dt) = l k ( )dt. This means that for service level, laxity is consmed at rate, and (7) simply states that laxity is consmed before service. A depiction of the eqal sharing policy in the service-laxity space is given in Fig. : all loads present in the system consme service and laxity in certain fixed proportions, therefore points move following the same vector. Under the exponential assmptions, α can be readily calclated by observing that k exp(µ) and l k l Fig.. Eqal sharing schedling for = /3. α ds α dl η Fig. 2. Missed deadline probability as a fnction of for = 2. exp(( )γ). Using the minimm of two exponential random variables we get: α = γ( ) µ + γ( ) = ( ) + ( ). (8) Deadline misses are decreasing in as expected. In particlar, for = η = /( + ), α = /2. Analogos calclations can be performed for any joint distribtion in (, l). For comparison prposes we compte also the probability for deterministic service time k µ which yields: ( α ds = P µ > l ) k = e. For deterministic laxity l k γ the corresponding expression is: ( ) α dl k = P > = e. γ( ) The three cases are depicted in Fig. 2 for a deferability parameter of = 2. As we can see, deadline misses are rather high even for moderate vales of > η. We trn to an alternate policy that seeks to minimize deadline misses.

4 l 8 θ Remaining laxity Not in service In service Remaining service Fig. 3. LLF schedling for = / Not in service In service B. Least-laxity-first The previos policy is agnostic to whether the deadlines are abot to expire. Assming global information and central control, it wold be better to schedle service taking into accont the remaining laxity of the loads. The least-laxityfirst (LLF) policy is defined in the following way: sort the crrent jobs by increasing laxity, and serve the first k(t) = n(t) at nominal power. 2 The remaining jobs will consme laxity ntil they are schedled. The LLF policy was introdced in the context of processor time schedling [, and has been thoroghly analyzed in the case of singleserver qees (cf. [4 for a recent treatment). The difference here is that we are dealing with an infinite server system. A depiction of the LLF policy in service-laxity space is given in Fig. 3. It is convenient to define the frontier process n(t) θ(t) := sp l : {lk l} < n(t). (9) k= Then θ(t) represents the maximm laxity of the loads crrently in service. Loads with laxity greater than θ(t) only consme laxity and are not served. We now analyze the steady state occpation and power otpt of the LLF policy for exponentially distribted service times. We begin by the following: Proposition : Under the LLF policy and exponential service times, the total occpation of the system n(t) evolves as in the eqal sharing policy. Proof: De to the LLF definition, at any time t there are n(t) loads in service. De to the memoryless property of the exponential distribtion, the service process of the system for occpation state n(t) and service level corresponds to n(t) exponential servers in parallel. Therefore, the total poplation evolves as a birth-death process with birth rate and death rate µn, i.e. as in the M/M/ of the eqal sharing policy. In particlar we conclde that in steady state, n Poisson (/(µ)), the average system occpation is again 2 More precisely, n(t) are served at fll power. The remaining power shold be allocated proportionally to the difference between n(t) and k(t) bt this ronding error is negligible in a large scale system. Remaining laxity Remaining service Fig. 4. Remaining service and laxities for LLF when > η (above) and < η (below), with /µ = 5. n = /(µ) and the otpt variance is again given by (6), i.e. it is linear in. This was observed empirically in [2. We now analyze the behavior of the system when the scale is large ( ). It was also observed in [2 that nder the LLF policy deadline misses show a sharp decline when the service level satisfies > η. To nderstand this, we plot in Fig. 4 two simlation experiments for a system with /µ = 5 and = 2 (η = /3). In the first case =.5 > η and the frontier process θ(t) finds a positive eqilibrim θ. Loads arriving with laxity greater that θ consme laxity down to level θ and then they are served. In the second case, with =.2 < η, loads expire their laxities before being served when they reach an eqilibrim vale of θ <. By applying Little s law, we can characterize the eqilibrim vale θ throgh a fixed-point analysis. We do so in the case where laxity is exponentially distribted. Recall that, from Proposition, the average nmber of clients in the system is n = /(µ). Therefore, the average time in the system by Little s law is T = /(µ). If θ > we can compte the average time as: T = µ = E[l k θ l k > θ P (l n > θ ) + E[ k. The first term simply states that loads arriving with laxity greater that θ shold wait to consme their laxity p to level θ before being served. If l k exp(γ) the above eqation becomes: µ = γ e γθ + µ,

5 Fraction of missed deadlines η Fig. 5. Empirical missed deadline probability as a fnction of for LLF schedling with = 2 (η = /3) and /µ = 5. or eqivalently: θ = [ γ log. () Note that () yields a positive soltion provided >, i.e. > + = η. Therefore, provided > η, the frontier converges to a positive eqilibrim and deadlines are attained. If θ <, the average time mst be compted as: T = µ = (E[l k θ ) + µ, which follows from the fact that loads mst wait an extra time θ before receiving service. Solving for θ we get: θ = γ µ. () Note that provided < η, () gives a negative soltion, as expected. Therefore, in steady state, the frontier converges to a negative eqilibrim and all deadlines are missed. While the above discssion is valid in the large scale limit (where the process θ(t) becomes constant), the approximation is indeed good for moderate vales of, as depicted in Fig. 5. The main conclsion of this analysis is that, for large scale systems sing LLF, the service level can be redced almost p to η (thereby redcing variance), withot great impact on deadline misses. Of corse, this comes at the cost of having a complex schedling policy: the load aggregator shold possess detailed information of the remaining laxity of each job reqest in order to perform schedling. In the next Section we analyze a different class of policies that cope with deadlines in a way more amenable to decentralization. IV. POLICIES THAT STRONGLY ENFORCE DEADLINES In the previos section the focs was on schedling policies that redce the global service level by a fixed amont, and we analyzed the impact of this choice in the amont of variability in consmed power as well as on deadline misses. We now trn or attention to a different class of schedling policies: in these, we impose that no deadlines are missed, and analyze the impact on steady state variance. A. Exact schedling The first policy in this family is called exact schedling. Here, service reqests arrive as a Poisson process of intensity, each load bringing a service time k and initial laxity l k given by a joint density f(, l) in the positive orthant. Given k and l k, the service level for load k is chosen as: k = k, k + l k and therefore the time spent by load k in the system is: T k = k k = k + l k. Namely, each reqest is served at exactly the amont of power needed to meet its deadline. A depiction of the policy is given in the first graph of Fig. 6. Note that this policy is very easy to decentralize provided that loads can tne their service level, since job reqests are already aware of their energy reqests and deadline. We wold like to compte the steady state variance of consmed power E[(δp) 2 for this system. The main challenge here is that the service level is determined by the job reqest characteristics. In order to express the steady state characteristics it is better to perform the following change of variables: z = + l, = + l. Here z represents the amont of time spent in the system and [, the service level. For given z and we can recover the original variables as = z, l = ( )z. The Jacobian is given by J (,l) (z, ) = z. The joint distribtion of (z k, k ) can be readily compted as: g(z, ) = zf(z, ( )z). (2) Note that in these new variables, reqests are served in parallel at rate along the variable z, while is fixed throghot service, as depicted in the second graph of Fig. 6. The state of the system at time t is expressed throgh the following conting measre [5 in R + [, : n(t) Φ t = δ (zi, i), (3) i= where z i represents the remaining service time of job i and i its service level. With the above definitions, Φ t is a measre-valed Markov process with simple dynamics: crrent jobs are translated to the left at rate while new jobs arrive as a Poisson process with marks z k, k distribted as g(z, ) given by (2). Since all jobs are served in parallel at rate the system behaves as an M/G/ qee. We have the following Theorem, derived from the steady state characteristics of the M/G/ qee [5, [6: Theorem : Under the exact schedling policy with reqests distribted as f(, l), the distribtion of Φ t in steady

6 l Fig. 6. Exact schedling depicted in service-laxity space and in the timeservice level coordinates. state is a Poisson point process on R + [, with mean measre density: h(z, ) = z g(w, )dw. (4) As a conseqence of this reslt, average characteristics of the steady-state process can be compted by sitable integrals with respect to the above density; we refer to [7 for these derivations. In particlar: The mean nmber of loads present in the system is n = E[n(t) = h(z, )dzd. (5) The mean aggregate power of the loads is given by E[p(t) = p E [ i i = p h(z, )dzd. (6) The variance of aggregate power is given by Var[p(t) = p 2 Var [ i i = p 2 z 2 h(z, )dzd. (7) To gain some intition, it is worth specializing the above reslts to the case of exponential reqests. In that case we have: f(, l) = µγe µ γl,, l >, and in conseqence: g(z, ) = µγze (µ+γ( ))z Define ν := µ + γ( ), then the steady state density is given by compting the integral in (4) to yield: [ νz + h(z, ) = µγ e νz. ν 2 With this density we can carry ot the calclations indicated in (5-7). In particlar, it is easily checked that h(z, )dz = 2µγ ν 3, so the mean nmber of cstomers is given by 2µγ n = (µ + γ( )) 3 d = µγ γ 2 µ γ µ ν 3 dν = µγ ( µ γ µ 2 ) γ ( 2 = µ + ). (8) γ Eqation (8) simply states that the average nmber of cstomers in the system is the arrival rate, times the expected service time E[ k +l k = µ + γ, consistent with the fact that the system behaves as an M/G/ qee. It can also be readily verified that E[p(t) = p E[ i i = p /µ = p, as expected. While service level in this system is load dependent, an effective service level can be compted as point of comparison, dividing the mean power consmed by the mean nominal power of the loads present: eff = E[p(t) p E[n(t) = /µ ( ) = γ µ + µ + γ = η, γ i.e. the exact schedling policy works at a service level comparable to taking = η in the preceding policies. More importantly, sing (7) and integration by parts we can compte the steady state power variance to be [ E[(δp) 2 = p 2 µγ (µ γ)µ 2 2 (µ γ) 2 µ 2 + (µ γ) 3 log ( µ γ ). (9) A more amenable measre is again the coefficient of variation cv 2 (p) = E[(δp) 2 / p 2 which can be derived from (9) and expressed in terms of the deferability factor: cv 2 (p) = p p [ 2 ( ) log( ) ( ) 3. (2) We shall se the above expression to compare the performance of the different policies at the end of the Section. B. Laxity expiring policy Finally, let s consider the following simple policy: Apply a fixed service level ũ (, only to loads with positive remaining laxity 3. If laxity of load k expires, serve the load at fll power. A depiction of the trajectories of this policy is given in Fig. 7. The main advantage of this policy is that it is very easy to decentralize. The system operator fixes a service level ũ 3 This is not an overall service level, hence the new notation ũ.

7 l Fig. 7. Laxity expiring schedling for = /3. for those loads that still have laxity, and distribtes this as a common signal. When a given load reaches the point where it cannot be deferred any longer, it starts consming power at maximm rate. Under the exponential job/laxity size assmption, the above qee has a very simple model. Let n(t) denote the nmber of jobs with positive laxity and m(t) those whose laxity has expired. Then (n(t), m(t)) is a continos time Markov chain with state space N 2 and the following transition rates: (n, m) (n +, m) : (n, m) (n, m) : µũn (2) (n, m) (n, m + ) : γ( ũ)n (n, m) (n, m ) : µm m γ( ũ)n µũn The above Markov chain has a prodct form soltion: Proposition 2: The eqilibrim distribtion of the Markov process defined by (2) is given by: π(n, m) = e ρn ρm ρn n, n, m N (22) n! m! i.e. in steady-state n and m behave as independent Poisson random variables with parameters: ρ n = µm ρ m m µũ + γ( ũ), ρ m = n γ( ũ) ρ n µ These average vales can be rewritten in terms of the following: ν := µũ + γ( ũ), γ( ũ) α := µũ + γ( ũ) With the above definition, /ν is the average time before either the laxity or the service of a given job ends, and α is the probability that the laxity expires before the job ends, and ths the job starts service in the second qee. Note that α has the same form as the missed deadline probability for the eqal sharing policy in the previos section. We can then rewrite: ρ n = ν, ρ m = α µ Noting that the otpt power p is p ũ for the first n loads and p for the remaining m loads, we can compte: ( p = E[p(t) = E[p (nũ + m) = p ũ ν + α ) µ [ũµ + να = p = p µ ν µ, }{{} = as expected. We can also qantify the deviations from eqilibrim in steady-state as: E[(δp) 2 = Var[p (n(t)ũ + m(t)) = p 2 (ũ 2 Var(n(t)) + Var(m(t))) = p 2 ( ρn ũ 2 ) + ρ m [ = p 2 µũ( ũ). µ ν where we have sed that n and m are independent random variables in steady state de to the prodct form distribtion. We can see that E[(δp) 2 p 2 µ and in fact this is achieved for ũ = or ũ =. The case ũ = corresponds to not giving any service ntil laxity expires, effectively delaying arrival for all jobs to the second qee and losing control on deferability. The case ũ = corresponds to serving the loads at fll power pon arrival, so in steady state m and the system behaves as in the eqal sharing policy with =. Again, it is better to express the variability in normalized nits, by compting the coefficient of variation as: cv 2 (p) = E[(δp)2 p 2 = p [ ũ( ũ) (23) p ũ + ( ũ) The above expression is minimized at: ũ = +, and the minimal vale of cv 2 (p) for this policy is: cv 2 (p) [ ũ=ũ = p p ( +. / ) 2 Note that both the optimal vale of ũ as well as the ratio between optimal and maximal variance do not depend on the arrival rate, and only on the ratio between the parameters µ and γ for the load service and laxity times. The effective service level in the optimal vale can be compted as: p eff = p E[n(t) + m(t) ũ=ũ =

8 cv 2 (p) Expiring Exact ES, LLF Fig. 8. Normalized coefficient of variation for the different schedling policies. It trns ot that eff > +, consistent with the fact that some of the loads are given fll power and ths losing control on their deferral. As a final comparison, we smmarize below the variability measres for the for schedling policies discssed in the paper. For Eqal Sharing and LLF we se the coefficient of variation for the case = η; this is the minimm vale for which LLF can comply with deadlines with high probability, as shown before 4. For the laxity expiring policy we se the optimal vale discssed above. The expressions are: cv 2 (p) ES = cv 2 (p) LLF = p p +, (24a) cv 2 (p) ExS = p [ p 2 ( ) log( ) ( ) 3, (24b) [ cv 2 (p) ExpL = p p ( + / ) 2. (24c) The above expressions are plotted for comparison in Fig. 8. As deferability increases ( grows), the best policy is LLF, at the cost of having a detailed information of the crrent system state. The laxity expiring policy is conseqently the worst since it only controls a fraction of the loads, with the exact schedling achieving intermediate reslts. V. CONCLUSIONS In this paper, we analyzed how deferring service of power loads can be sed to redce power consmption variability, an important problem in Smart-grid deployments aimed at redcing freqency reglation needs. We derived a qeeing model for a load aggregator entity that manages service reqests, characterized by service times and deadlines. We analyzed different qeeing policies with different degrees of complexity and attention to deadlines. For these policies, we compted the coefficient of variation of power consmption in terms of the deferability characteristics of the load profile, as well as the probability of missed deadlines, therefore qantifying the tradeoff between variance redction and qality of service. Several lines of ftre work remain open. In the case of the least-laxity-first schedling, a more detailed model for general service times and laxities is in order, as well as the transient behavior of in-service/not-in-service loads and the frontier process. In more general terms, an economic analysis of how sers may be incentivized to cooperate by adjsting the service level and declare their tre deadlines is an interesting line to prse. ACKNOWLEDGEMENTS The athors wold like to thank Adam Wierman for many hors of insightfl discssions and Federico Bliman for his help with the analysis of simlations. REFERENCES [ New York Independent System Operator, Ancillary services manal, operations/docments/ Manals and Gides/Manals/Operations/ancserv.pdf. [2 H. Holttinen and et al., Design and operation of power systems with large amonts of wind power, IEA wind, Tech. Rep., 29. [3 EnergyPool, [4 GoodEnergy, [5 S. Koch, J. Mathie, and D. Callaway, Modeling and Control of Aggregated Heterogeneos Thermostatically Controlled Loads for Ancillary Services, in Proc. of the 7th Power Systems Comptation Conference, 2. [6 H. Hao, B. M. Sanandaji, K. Poolla, and T. L. Vincent, A Generalized Battery Model of a Collection of Thermostatically Controlled Loads for Providing Ancillary Service, in Proc. of the 5st Allerton conference on Commnication, Control and Compting, 23. [7, Freqency Reglation from Flexible Loads: Potential, Economics, and Implementation, in Proc. of the American Control Conference (ACC), 24. [8 A. Sbramanian, M. Garcia, D. Callaway, K. Poolla, and P. Varaiya, Real-Time Schedling of Distribted Resorces, IEEE Transactions on Smart Grid, vol. 4, pp , 23. [9 A. Nayyar, J. Taylor, A. Sbramanian, K. Poolla, and P. Varaiya, Aggregate Flexibility of a Collection of Loads, in Proc. of the 52nd IEEE Conference on Decision and Control, 23. [ J. Hong, X. Tan, and D. Towsley, A performance analysis of minimm laxity and earliest deadline schedling in a real-time system, IEEE Trans. on Compters, vol. 38, pp , 989. [ S. Meyn, P. Barooah, A. Bšić, Y. Chen, and J. Ehren, Ancillary service to the grid sing intelligent deferrable loads, IEEE Trans. on Atomatic Control, to appear. [2 F. Bliman, A. Ferragt, and F. Paganini, Controlling Aggregates of Deferrable Loads for Power System Reglation, in Proc. of the 25 American Control Conference, Chicago, IL, Jne, 25. [3 G. D. Bella, L. Giarrè, M. Ippolito, A. Jean-Marie, G. Neglia, and I. Tinnirello, Modeling Energy Demand Aggregators for Residential Consmers, in Proc. of the 52nd IEEE Conference on Decision and Control, 23. [4 H. C. Gromoll and L. Krk, Heavy traffic limit for a processor sharing qee with soft deadlines, Annals of Applied Probability, vol. 7, no. 3, pp. 49, 27. [5 P. Robert, Stochastic networks and qees. Springer, 23. [6 S. Zachary, A note on insensitivity in stochastic networks, Jornal of Applied Probability, vol. 44, pp , 27. [7 F. Baccelli and B. Błaszczyszyn, Stochastic Geometry and Wireless Networks, Volme I - Theory. Now Pblishers, For Eqal Sharing the fraction of missed deadlines wold be high.