Distributed Coordination of Deferrable Loads: A Real-time Market with Self-fulfilling Forecasts
Hazem A. Abdelghany, Simon H. Tindemans, Mathijs M. de Weerdt, Han la Poutré
DDistributed Coordination of Deferrable Loads: A Real-time Market withSelf-fulfilling Forecasts (cid:73)
Hazem A. Abdelghany a,d , Simon H. Tindemans a , Mathijs M. de Weerdt b , Han la Poutr´e a,c a Department of Electrical Sustainable Energy, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University ofTechnology, The Netherlands. b Department of Software Technology, Faculty of Electrical Engineering, Mathematics and Computer Science, Delft University of Technology,The Netherlands. c Centrum Wiskunde & Informatica (CWI), Amsterdam, The Netherlands. d Electrical and Control Engineering Department, Arab Academy for Science, Technology and Maritime Transport, Cairo, Egypt.
Abstract
Increased uptake of variable renewable generation and further electrification of energy demand necessitate efficientcoordination of flexible demand resources to make most efficient use of power system assets. Flexible electrical loads aretypically small, numerous, heterogeneous and owned by self-interested agents. Considering the multi-temporal nature offlexibility and the uncertainty involved, scheduling them is a complex task. This paper proposes a forecast-mediatedreal-time market-based control approach (F-MBC) for cost minimizing coordination of uninterruptible time-shiftable(i.e. deferrable) loads. F-MBC is scalable, privacy preserving, and usable by device agents with small computationalpower. Moreover, F-MBC is proven to overcome the challenge of mutually conflicting decisions from equivalent devices.Simulations in a simplified but challenging case study show that F-MBC produces near-optimal behaviour over multipletime-steps.
Keywords:
Market-based Control, Markov Decision Process, Flexibility, Demand Response, Distributed EnergyResources.
1. Introduction
Power systems have seen an increasing penetration ofdistributed energy resources (DERs), such as distributedgenerators, flexible demand, and small-scale renewable gen-eration. This trend has significant impacts on the network,leading to congestion, reduced network utilization, andeven instability or system inoperability at the distributionlevel [1]. Consequently, the transition to future powersystems requires either a great deal of investment in gridreinforcement, or efficient use of flexibility from DERsthrough coordination.Optimal coordination among DERs is a complex multi-dimensional problem, especially in settings with small,numerous, heterogeneous DERs owned by self-interestedagents. The complexity is further amplified by inter-temporal constraints introduced by shifting energy con-sumption and uncertainties in DER usage patterns andrenewable-based generation. A suitable coordination ap-proach for such a setting is required to be simple and usableby agents with small computational power [2], scalable for (cid:73)
This project has received funding from the European Union’s Hori-zon 2020 research and innovation programme under Marie Sklodowska-Curie grant agreement No. 675318 (INCITE). ∗ Corresponding author
Email address: [email protected] (Hazem A.Abdelghany) settings with numerous DERs, and privacy preserving sincethe DERs being considered are owned by self-interestedagents.This problem has been considered in a number of set-tings, including electric vehicle charging [3], deferrableloads such as washing machines, dish washers, and ther-mostatically controlled loads. Most control techniques forflexible demand are based either on centralized coordina-tion, top-down control, or price response [4, 5]. Centralizedand top-down approaches (e.g. [6]) are not suitable whenconsidering privacy, autonomy, and scalability constraints,whereas completely decentralized approaches relying onone-way communication (e.g. price response [7, 8]) have un-certain realized system response. A comprehensive reviewof advantages and disadvantages of control approaches canbe found in [9].
A natural fit for the problem of coordinating self-interestedDERs is transactive control, which refers to control ap-proaches which perform coordination and control tasksby using economic incentive signaling to exchange infor-mation about generation, consumption, constraints, andresponsiveness of assets over dynamic, real-time forecastingperiods [10]. Market-based control (MBC) describes a classof transactive control algorithms that take the form of amediated market [11]. In an attempt to find a middle way
Preprint submitted to Sustainable Energy, Grids and Networks. June 16, 2020 a r X i v : . [ ee ss . S Y ] J un etween the aforementioned approaches, this paradigmprovides simultaneously a degree of privacy, autonomy,certainty, and openness compared to the aforementionedapproaches [5, 9]. However, when used for coordinationamong numerous DERs, over multiple time-steps and tak-ing into account uncertainty, MBC approaches rapidly growin complexity, limiting their scalability and practical fea-sibility. For example, multi-settlement markets, such asin [12, 13] require complex bid formulation algorithms,which is especially hard for devices with small compu-tational power [2]. Accounting for uncertainty similarlyincreases complexity, as is evident in the hierarchical MBCapproach in [14]. In [15, 16], iterative approaches for co-ordination were proposed. An iterative approach basedon Mean-field games was proposed in [17]. However, [18]indicates iterative approaches are not suitable for real-time operations due to uncertain convergence time anddependence on initial conditions. The same logic appliesfor negotiation approaches such as in [19]. On the otherhand, approaches based on the assumption of cooperativeagents [20, 21, 22] are not suitable for the settings withself-interested agents. In this paper, we use the term “Real-time market-basedcontrol (RTMBC)” to describe a simple and scalable formof MBC. In RTMBC, DERs are represented by autonomousagents participating in a spot power market. The marketis cleared for the upcoming time-step (i.e. in real time)by means of a double auction. The use of decentralizeddecision making and a centralized one-shot market clearingsimplifies the whole process. Device level constraints andobjectives are taken into account in the process of bid/offerformulation. An example of such approach can be seenin [9].Despite these beneficial properties, in practice RTMBCoften leads to poor performance over multiple time-stepsdue to uncertainty, inter-temporal constraints of uninter-ruptible devices, and mutually-conflicting decisions thatarise from decentralization and the self-interested behaviourof agents [23, 24, 18]. For example, in [25] the effect of suchbehaviour is shown to lead to exhaustion of flexibility inthe system. An approach for coordination among thermo-statically controlled loads was presented in [26]. This wasfurther studied in [23] where it was found prone to loadsynchronization and power oscillations. Agents submittingsimilar bids (i.e. Bulk switching), and clustering at lowerprice periods are phenomena that occur when optimal de-cisions from the agents’ perspective conflict and lead tosub-optimal outcomes both at the agent level and systemlevel. This is most apparent in case of identical devicesgiven the same information. Therefore, identical devicespose a challenge to many coordination approaches.
In this paper, we aim at solving the problem of schedul-ing a set of uninterruptible deferrable loads over multiple time-steps to minimize generation cost taking into accountuncertainty. We will refer to this as the “optimal coordi-nation problem”. To achieve this, we propose the forecast-mediated market-based control approach (F-MBC). F-MBCrelies on decentralized bid formulation and centralized one-shot market clearing to coordinate among these devices.The proposed approach is scalable and preserves end-userprivacy and autonomy. It relies on probabilistic price fore-casts obtained by a facilitator that accounts for uncertaintyin renewable-based generation and DER usage patterns.Moreover, we design a low-complexity Markov decision pro-cess(MDP) based optimal bidding algorithm for deferrableloads, which is usable by a device with limited computa-tional power (e.g. embedded systems) to formulate a bidthat minimizes its own expected cost given probabilisticprice forecasts. We show that the combination of prob-abilistic reference prices, optimal bidding, and real-timemarket clearing solves the problem of mutually-conflictingdecisions among identical devices; that is, two identicaldevice agents with different deadlines will never have thesame bid. This is shown mathematically in Section 2. Ad-ditionally, we design a tie-breaking mechanism to assist inmarket clearing when several agents are indifferent betweendifferent actions at the market-clearing price. Moreover, weprove approximate consistency of the approach by bound-ing the deviation from the optimal solution that occursif the forecast correctly identifies an optimal feasible solu-tion. We show by simulation that the proposed F-MBCapproach achieves near-optimal system level performanceover multiple time-steps (i.e. minimizes overall generationcost) in Section 3.
2. Methodology
Consider a setting of uninterruptible deferrable loads,with deadlines set by their respective owners. This re-sembles a collection of devices such as irrigation pumps,greenhouse lighting, or home appliances such as washingmachines, dryers, etc. [27, 28]. We assume that each of thedeferrable loads acts in its economic best interest, minimiz-ing its consumption cost subject to device level constraints(e.g. deadline, uninterruptibility).The challenge is to design a scheme that fully or approxi-mately solves the optimal coordination problem, schedulingthe flexible demand over multiple discrete time-steps withthe objective of minimizing the overall generation cost. Itis important to note that the global cost minimization isequivalent to social welfare maximization since the totalenergy demand (and, therefore, the utility) is fixed. There-fore, for the remainder of the paper we will just use theterm “optimal coordination”.To achieve this, we rely on the idea of “self-fulfillingforecasts”. As illustrated in Figure 1, F-MBC comprisesthree types of autonomous agents; A facilitator, an auction-eer, and a device agent per flexible device. The facilitatoris a central entity which, in general, does not have ac-cess to private information (e.g. deadlines, cycle durations)2nd cannot directly control the devices. This is a sensibleassumption in settings where DERs are small, numerousand owned by self-interested agents. Such an approachis similar to the vision of layered decentralized optimiza-tion architecture in [29]. The facilitator utilizes aggregatehistorical information, forecasts, behaviour patterns, andsystem models to estimate an “offline optimal” solution tothe optimal coordination problem. Some examples of tech-niques to solve such a problem can be found in [30, 31, 32].The resulting estimated schedule is probabilistic and resultsin a probabilistic reference price for each time-step (in theform of a probability distribution), thus taking into accountuncertainty. Throughout, we assume that the price of en-ergy paid by devices equals the marginal cost of generationat the relevant time step. The probabilistic reference pricesare then communicated to the flexible demand agents whichuse this information for bid formulation. Device agentsformulate their respective bids in a self-interested manner(i.e. minimizing the expected cost incurred by the agent).A device agent takes into account local deadline and un-interruptibility constraints in addition to the probabilisticreference prices provided by the facilitator. Bids are thensubmitted to a central auctioneer in the form of a demandfunction. Finally, an allocation is made through a one-shotdouble auction and an additional tie breaking mechanism.The facilitator updates the “estimate” for the future tak-ing into account the market outcome which results in anupdated probabilistic reference price signal. The wholeprocess is repeated for every time-step.It is noteworthy here that aggregation of bids can bedone centrally or through hierarchical aggregation of bidfunctions. This means that the complexity of aggregatingbids is linear, at worst, or logarithmic, at best, when thesystem is organized as a binary tree. This, combined withdecentralized bid optimization, one-shot market clearingand the non-iterative nature of the approach makes it scal-able and simple to implement even in scenarios where agentshave small computational power. Moreover, the outcomeof this process is a near-optimal system-level behaviourover multiple time-steps. The resulting coordination ap-proximates the “offline optimal coordination” estimated apriori, so the probabilistic reference prices can be consid-ered “self-fulfilling”.
Consider a scheduling horizon consisting of the set ofdiscrete time steps T = { , . . . , T } with fixed intervals∆ t . The subscript t will be used to refer both to the in-stant t as well as the interval that immediately follows,depending on the context. The system comprises a set A ofuninterruptible deferrable devices owned by self-interestedconsumers. Each device is represented by an agent a de-fined by a deadline, duration and a power consumptionpattern d a , D a , { P a , . . . , P aD a − } respectively. The systemalso has inflexible demand, and flexible generation with anon-decreasing marginal cost m t ( P ) (which may includezero-cost renewable generation). An optimal coordination · · · Probabilisticreference pricesDevice agentbids Marketoutcome Local controlactions Aggregate historical dataWeather forecastLoad forecastAggregate behaviour patternsSystem models Updatedhistorical data Figure 1: Schematic overview of the proposed F-MBC approach denotes the allocation of flexible devices over the schedulinghorizon, such that the overall cost of generation to meetthe aggregate demand P T is minimized: P ∗ T = arg min P T T (cid:88) i =1 ∆ t (cid:90) P t m t ( P (cid:48) )d P (cid:48) . (1)This is subject to system-level constraints (i.e. supply/demandmatching, flexible generation limits), and agent-level con-straints (i.e. deadlines, uninterruptibility). At this point, we describe how an agent may computeand optimize its bid given the probabilistic reference pricessupplied by the facilitator. We represent these prices, hav-ing the form of time-dependent probability distributions,by independent random variables X t with bounded ex-pectation E ( X t ) = ¯ x t < ∞ . Each device agent aims atminimizing its expected cost out of self-interest. For that,we develop a MDP model for optimal bidding which consistsof a state space, action space and a set of rewards/costs.We show that the MDP-based bidding algorithm minimizesthe expected cost for the device (i.e. optimal in expec-tation) given the available information (i.e. probabilisticprice reference) and the assumption that a single deviceis a price taker. For an uninterruptible deferrable device,the action space only consists of two actions on , off . Thestate consists of a possible realization of the price, and thestatus ( s at ) of the device, where s at = 0 for a device thathas not started yet (i.e. waiting ), s at = { , . . . , D a − } fora device that has started (i.e. running ), and s at = D for adevice that has run for D time-steps (i.e. finished ).If the uninterruptible device a switches from the waiting to the running state at time t with a market clearing price x t , its expected total running cost is a combination of thecost of starting at t with a price of x t , and the sum of the3xpected costs for the remainder of the device’s cycle, C s,at ( x t ) = x t · P a · ∆ t + D a − (cid:88) i =1 ¯ x t + i · P ai · ∆ t (2)The agent aims to minimize its running cost. It does so by,at each time step, submitting a bid function b at ( x ), definedby a threshold price ˆ x at . The definition of an optimal bidfunction is given below. Theorem 1.
For a sequence of independent referenceprices X t with bounded expectation, agent a minimizesits expected running cost by submitting the threshold-basedbid function b at ( x ) , where b at ( x ) = (cid:40) P a x ≤ ˆ x at x > ˆ x at , (3) P a = (cid:40) P as at if s at < D a otherwise (4)ˆ x at = −∞ if s at = D a ∞ if s at = 1 , . . . , D a − z at if s at = 0 , (5) z at = ∞ t ≥ d a − D aC ∗ at +1 − (cid:80) Da − i =1 ¯ x t + i · P ai · ∆ tP a · ∆ t t < d a − D a , (6) and C ∗ at is the optimal expected cost at t , which is recur-sively defined in reverse order for t ≤ d a − D a by C ∗ ad a − D a = D a − (cid:88) i =0 ¯ x d a − D a + i · P ai · ∆ t, (7) C ∗ at =Pr( X t > ˆ x at ) · C ∗ at +1 + Pr( X t ≤ ˆ x at ) · E [ C s,at ( X t ) | X t ≤ ˆ x at ] . (8) Proof.
In order for b at ( x ) to be optimal, the optimal actionfor an agent must be on if the clearing price x t is smallerthan or equal to the threshold bid ˆ x at , and off if it is largerthan the threshold bid. First, if s at = D (i.e. finished ),the only feasible, thus optimal, action is off regardlessof the price ( b at ( x ) = 0, i.e. ˆ x at = −∞ ). Similarly, if s at = 1 , . . . D a − running ), and has not completed itstask, the only feasible, thus optimal, action is on regardlessof the price ( b at ( x ) = P ai , i.e. ˆ x at = ∞ ), where i is therelevant time period in the devices program. Finally, a waiting device has different optimal actions based on thefollowing logic. • At time-step t = d a − D a , a waiting device a mustswitch to the running state to meet the deadline, sothe optimal action is on irrespective of the clearingprice (i.e. ˆ x at = ∞ ). The expected cost associatedwith starting immediately is therefore also optimal: C ∗ ad a − D a = E (cid:2) C s,ad a − D a ( X d a − D a ) (cid:3) , resulting in (7). • At time-steps t < d a − D a , if a has not startedyet, the action on is optimal when the expected costfor switching on is less than the expected cost forwaiting and acting optimally at t + 1, that is, if C s,at ( X t ) < C ∗ a t +1 . Conversely, if C s,at ( X t ) > C ∗ a t +1 ,only off is optimal. Therefore, the threshold z at for t < d a − D a in (6) is derived from the equality C s,at (ˆ x at ) = C ∗ a t +1 . (9)When the equality holds, agent a is indifferent be-tween starting and waiting.Given the existence of optimal threshold bids ˆ x at and (7),the optimal expected cost (8) for t < d a − D a follows bybackwards induction.In the following, we consider how different deadlinesimpact the bids of otherwise identical agents. Identicaldevices pose a challenge due to the increased possibility forsynchronised and conflicting decisions [23, 24, 18, 25, 26].We argue that F-MBC provides a natural way to resolvesuch conflicts.In the proofs, we shall assume that at any time, theforecast price has a non-zero probability to exceed thelargest finite threshold price: Pr( X t > ˆ x at ) > , ∀ a, ∀ t : t Agents { , . . . , n } are rapid-starting, identi-cal and deadline-ordered if their power requirement and ser-vice duration are identical and they start consuming immedi-ately ( ∀ a, i : D a ≡ D, P ai ≡ P i , P a (cid:54) = 0 ), but their deadlinessatisfy d < d < . . . < d n . They are weakly deadline-ordered if their deadlines satisfy d ≤ d ≤ . . . ≤ d n . Lemma 2. A collection of n rapid-starting, identical,deadline-ordered devices that is in the waiting state attime t , operating under the optimal MDP policy, will bidwith a strictly decreasing sequence of threshold prices: ˆ x t > ˆ x t > . . . > ˆ x nt . A weakly deadline-ordered collection willbid with a non-increasing sequence of threshold prices: ˆ x t ≥ ˆ x t ≥ . . . ≥ ˆ x nt .Proof. Contained in Appendix A. Theorem 3. A collection of n rapid-starting, identical,deadline-ordered devices, operating under the optimal MDPpolicy, will start (and complete) in order of their deadlines. If this condition does not hold for a given pair { a, t } , agent a concludes that it is always optimal to start at time t (or at an earliertime), i.e. for all possible realisations of the random clearing price X t . This effectively adjusts the deadline d a → ˜ d a = t + D a , thusremoving the differentiation in threshold bids among affected devices.We note that this is desirable behaviour if the forecaster correctlyidentified the range of X t , but may cause problems if this range wasunderestimated, hence including a non-vanishing tail probability inthe forecast is recommended. roof. Prior to the first auction, all agents are in the waiting state. In the auction, agents with a thresholdbid exceeding (and sometimes including) the clearing pricetransition to the running state. Lemma 2 guarantees thatthese are agents with the earliest deadlines. This processis repeated for subsequent auctions with devices that havenot started yet. The market is cleared via a one-shot double auctionfor each time-step. We assume that generation truthfullyreveals its marginal cost function. The aggregate offerfunction accounts for flexible generation and inflexible gen-eration in the upcoming time-step in the form of a marginalcost function. Device agents submit their bids only for theupcoming time-step. The aggregate bid function includesinflexible demand and the bids submitted by flexible de-mand. The market is cleared at time-step t at the price x t at which supply meets demand. Then, the market-clearingprice is communicated to device agents which determinetheir local control actions based on their earlier submittedbids.Although Lemma 2 ensures differentiation of bids amongdevices with different deadlines, equal bids may be submit-ted, for example if identical devices have identical deadlines.A tie situation occurs when the market clears at the pricebid by multiple agents, x t = ˆ x at = ˆ x bt = . . . . The aggregatebid/offer functions for such a case are shown in Figure 2.A large step in the aggregate bid can cause difficulties inmarket clearing (i.e. bulk switching). To address this issue,we introduce a tie breaking mechanism among such agents.The tie breaking mechanism determines which of thetied agents can start at the current time-step and whichwill wait for a later time-step. Each agent submits a ran-dom number ρ a along with its bid. When the auctioneerdetects a tie situation, it determines a value ρ ∗ so thatonly bids with ρ a ≤ ρ ∗ will be accepted. ρ ∗ is chosen suchthat demand most closely approximates the supply at theclearing price x t .Due to the discrete nature of the loads, an exact matchmay not be found. In such a case, the bid of the marginaldevice a is accepted with probability γP a , where γ is thedifference between the supply at x t and the demand withoutthe marginal device. Agents will be charged the marketclearing price x t while generation should supply at a slightlyhigher (lower) set-point, and is paid accordingly. Thisresults in a budget imbalance that vanishes in expectation(i.e. averages to zero in the long term). This is illustratedin Figure 2.We note that the random tie breaking mechanism doesnot affect optimality or fairness as it is only used to breakties among agents with bids that are equal to the marketclearing price. Agents are indifferent between starting andwaiting at their bid price, so those who are not allocatedwill wait for a later time-step and eventually incur the sameexpected cost as those which were allocated. Therefore,they have no incentive to game the tie-breaking mechanism. Supply x t Allocateddemand Non-allocateddemand x t cut-o ff γ AllocatedNon-allocatedAllocated with probability γ Price x P o w e r ( k W ) Higherset-pointLowerset-point Price x Figure 2: A tie situation; devices that are indifferent between startingor waiting at x t are allocated randomly. Also, because ρ a is generated locally, tie breaking can beimplemented using a broadcast of ρ ∗ . The alternative,where ρ a is determined by the auctioneer, would require atargeted message to each device. According to the previously stated definition of theoptimal coordination problem, our objective is to steerthe cluster of flexible devices towards an optimal system-level behaviour (i.e. total generation cost minimization).To guarantee a stable optimum, it is necessary that theoptimal coordination corresponds to a Nash equilibrium.This guarantees that it is in the best interest of the deviceagents not to deviate from such behaviour. Therefore, weshow that the global cost minimizing solution indeed cor-responds to a Nash equilibrium. To analyse the potentialfor F-MBC to achieve optimal system-level behaviour, wefirst consider the schedule achieved by a clairvoyant opti-mizer with complete information. We give conditions underwhich this schedule corresponds to the outcome of a Nashequilibrium, i.e. agents cannot benefit by deviating fromthe starting time-step allocated by the central optimizer.These results indicate that the central F-MBC facilitatorshould aim to estimate the prices that correspond to such asystem optimal allocation, so that devices are incentivisedto realise the reference prices.We consider a cost-optimal allocation of flexible devices,characterized by an aggregate load profile P ∗ t and a startingtime t a for each flexible device a , summarized as S =( { P ∗ t } t =1:T , { t a } a =1: A ). Without loss of generality, in thefollowing we take the perspective of an arbitrary deferrabledevice agent a that has a duration D and uninterruptibleconsumption pattern { P a , . . . , P aD − } , which is scheduledto start at t = t a under the cost-optimal allocation. Noassumptions are made about the properties of other flexibleloads. Let P ¬ at be the cost-optimal load pattern P ∗ t minusthe consumption of device a starting at t a , and m t ( P ) bethe monotone increasing function in P which representsthe marginal cost of a unit of generation at generation level5 and time t . The cost to the system of running device a at time t is K at = ∆ t D − (cid:88) i =0 (cid:90) P ¬ at + i + P ai P ¬ at + i m t + i ( P )d P. (10)The fact that the starting time t a is optimal with respectto overall system cost, implies that K at a ≤ K at , ∀ t ∈ T . (11)Switching from the system perspective to that of an in-dividual, we assume that the a pays a price equal to themarginal cost of energy. The total price paid by agent a starting at t isΠ at = ∆ t D − (cid:88) i =0 m t + i ( P ¬ at + i + P ai ) P ai . (12)The allocation S is a Nash equilibrium if for each agent a ,Π at a ≤ Π at , ∀ t ∈ T . (13)In the following, we identify conditions where global cost-optimality (11) implies the Nash equilibrium condition (13).We first consider a (restrictive) special case in which theimplication holds exactly; we then consider a weaker set ofconditions that results in Theorem 5 and Corollary 6 withmuch broader applicability. Theorem 4. If m t ( P ) is an affine function with constantslope d m t ( P ) / d P = c, ∀ t , then S is a Nash equilibrium.Proof. Evaluating the integral in (10) using the affine struc-ture of m t ( P ) yields K at = Π at − c ∆ t D − (cid:88) i =0 ( P ai ) . (14)Because the last term does not depend on t or P ∗ , (11)implies (13) and S is a NE.Note that m t ( P ) does not need to be strictly affine withslope c for all P , but only for those marginal power levelsthat are accessible by flexible devices. This is the case inthe example in Section 3. Definition 2. The allocation S is a δ -relaxed Nash equi-librium if the condition (13) is replaced by the weaker con-dition Π at a ≤ (1 + δ )Π at , ∀ t ∈ T . (15)The δ -relaxed Nash equilibrium is effectively a Nashequilibrium for devices that are insensitive to relative pricedifferentials of size δ . Clearly, it converges to a regularNash equilibrium in the limit δ ↓ 0. We note that this isclosely related to the concept of an ε -equilibrium [33]. Theorem 5. If there exists an ε < so that, m t ( P ¬ at + P ai ) − m t ( P ¬ at ) ≤ εm t ( P ¬ at + P ai ) , ∀ t ∈ T , ∀ i ∈ { , . . . , D − } , (16) then S is a δ -relaxed Nash equilibrium with δ = ε/ (1 − ε ) Proof. From definitions (10), (12), (16) and the fact that m t ( P ) is non-decreasing, it follows that(1 − ε )Π at ≤ K at ≤ Π at , ∀ t ∈ T . (17)By chaining the first inequality (for t = t a ) with (11) andthe second inequality, we obtain(1 − ε )Π at a ≤ Π at , ∀ t ∈ T . (18)Substitution of δ = ε/ (1 − ε ) and comparison with (15)completes the proof. Corollary 6. In the limit where agents are price takers(individually), S is a Nash equilibrium.Proof. When agents are price takers their influence on m t is negligibly small; this implies that Theorem 5 applies withthe limit ε ↓ 0. Therefore, δ ↓ collectively influencing prices significantly. In this section we quantify the consistency of the pro-posed F-MBC approach. Ideally, if the facilitator is able tosupply the agents with reference prices that are realisableand near-optimal, the agents should respond with bids thatresult in start times consistent with that profile. If this isthe case, the (near-)Nash Equilibrium that is encoded inthe reference prices would become self-fulfilling . In orderto quantify this property, we investigate deviations fromthe optimal coordination solution in the limit where thereference prices correspond to such a solution. We do so forthe special case of a collection of rapid-starting, identicaldevices, and a single time step t . In the following, super-scripts a are dropped for identical quantities (e.g. durations D ). Near-optimality of price forecasts is represented byprice forecasts X t (cid:48) = m t (cid:48) ( P ∗ t (cid:48) ) + ∆ t (cid:48) for t (cid:48) > t , where P ∗ t : T represents a feasible cost-optimal consumption scheduleand the magnitude of ∆ t : T is strictly bounded from below. Lemma 7. Consider a collection of rapid starting devicesoperating under the F-MBC framework, and a cost-optimalallocation S , characterised by a starting time t a ≥ t for eachdevice a , and an aggregate load P ∗ t : T (including inflexibleload). f devices receive near-optimal reference prices X t (cid:48) = m t (cid:48) ( P ∗ t (cid:48) ) + ∆ t (cid:48) , where E [∆ t (cid:48) ] = 0 and Pr(∆ t (cid:48) > − η ) = 1 for η > for all t (cid:48) ∈ { t, . . . , T } , then the difference betweenthe clearing price x t and the reference price x ∗ t = m t ( P ∗ t ) is bounded by − (cid:80) D − i =0 P i [∆ m t,i + η ] P ≤ x t − x ∗ t ≤ max t (cid:48) ∈{ t +1 ,...,T } (cid:80) D − i =0 P i ∆ m t (cid:48) ,i P (19) with ∆ m t,i = m t ( P ∗ t ) − m t ( P ∗ t − P i ) (20) Proof. Contained in Appendix B.Let n ∗ t be the number of devices starting at t underthe optimal allocation S , and n t the number of devicesstarting using the F-MBC dispatch method. Under theadditional assumption of smoothness of the marginal cost ofgeneration, it is possible to derive bounds for the difference n t − n ∗ t , as follows. Theorem 8. Assume that Lemma 7 holds, and that m t ( P ) can be approximated around P ∗ t by m t ( P ) = m t ( P ∗ t ) + c t [ P − P ∗ t ] . (21) Then − (cid:38) (cid:80) D − i =0 (cid:2) c t + i P i + ηP i (cid:3) c t P (cid:39) ≤ n t − n ∗ t ≤ max t (cid:48) ∈{ t +1 ,...,T } (cid:38) (cid:80) D − i =0 c t (cid:48) + i P i c t P (cid:39) . (22) Proof. The optimal clearing price x ∗ t is associated withthe desired number of starting devices n ∗ t . Linearity ofthe marginal cost function and rounding up/down to thenearest integer results in (cid:22) x t − x ∗ t c t P (cid:23) ≤ n t − n ∗ t ≤ (cid:24) x t − x ∗ t c t P (cid:25) (23)Combining with (19) and making use of∆ m t,i = c t P i (24)results in (22). Corollary 9. In the special case where m t ( P ) is an affinefunction with constant slope c t = c , devices consume aconstant amount of power ( P i = P ) and in the limit ofvanishing uncertainty ( η ↓ ), we have − D − ≤ n t − n ∗ t ≤ D. (25) These results show that the F-MBC dispatch convergesto the optimal dispatch within hard limits. These limits donot depend on the total number of devices, so the relativeperformance increases with the number of devices.Moreover, the analysis above considers only a singletime step t . If the number of devices n t starting at t exceeds n ∗ t , this results in higher prices for subsequent time steps,thus reducing the number of devices that start at t + 1 until t + D − 1. Conversely, if the number of device starts islower than scheduled, this will incentivise additional startsin subsequent time steps. Although not quantified here,this self-regulating effect further reinforces the convergenceto the reference solution. 3. Experimental analysis Using simulations, we illustrate two features of theproposed F-MBC approach. First, we show that F-MBCperformance is near-optimal over multiple time-steps whenprice uncertainty is negligible (consistency). Second, weanalyze the robustness of the solution to varying amountsof uncertainty in price forecasts in order to qualify the needfor accurate estimation of reference prices. For this case study, we consider a system with identicaldeferrable loads. This represents a particularly challengingscenario, due to a high probability of ties occurring and alumpiness of loads that does not permit full ‘valley filling’of the solution. A full day (24 hours, starting at 21:00)was simulated with market clearing at 5 min time steps. Afixed horizon at 20:55 the next day was used for forecastingand bid formation. The system included 1200 deferrableloads, with a duration of 1 h and fixed consumption of2 kW each. Deadlines were distributed in two clusters of600 devices, normally distributed with a standard deviationof 1 hour around 7:00 in the early morning and 17:00 in theearly evening, and rounded to the nearest 5 min time-step.Inflexible demand was modelled using load data from [34]aggregated and scaled to a peak of 350 kW. Wind genera-tion with a peak of 500 kW was generated using [35] anda simple wind turbine model that approximates the per-formance of a 100 kW wind turbine [36], scaled to 500 kW.We assume that wind power generation is free and curtail-able. The simulation input data can be seen in Figure 3.Simulations were performed in Matlab.Flexible generation was represented by a time-independent,linearly increasing marginal cost function m ( P g ) = P g k , P g ≥ , (26)where P g is the power generated by the flexible generatorfleet and k = 500 kW min, with arbitrary units for cur-rency. With this choice, the total cost of generation (windand flexible generation), has an affine marginal cost, pro-vided that P g > 0. Therefore, it follows from Theorem 47 W i nd P o w e r ( k W ) I n f l e x i b l e l o a d ( k W ) Wind powerInflexible load21:00 24:00 3:00 6:00 9:00 12:00 15:00 18:00 20:55 Time (Hr) N u m b e r o f d e v i ce d ea d li n e s Figure 3: Simulation input data. Top: Wind power generation andinflexible load profile. Bottom: Distribution of device deadlines acrossthe 5 minute time intervals. that the device schedule from a clairvoyant optimizer withcomplete information corresponds to the outcome of a Nashequilibrium. To establish the potential of F-MBC as a coordinationmechanism via simulations, we first identify the theoreti-cal optimal coordination that can be obtained only by aclairvoyant optimizer with complete control. Accordingly,we obtain optimal reference prices that reflect an optimalallocation of demand using perfect foresight. Due to ourselection of identical, fixed consumption devices, this canbe done by solving the mixed-integer quadratic program(MIQP) that finds the optimal number of devices to startat each time-step σ t , and optimal flexible power generationfor each time-step P g t such that the total generation costover multiple time-steps is minimized:minimize P g t ,σ t ,o t (cid:88) t ∈ T · ( P g t ) k · ∆ t, (27)subject to, ∀ t ∈ T , P g t ≥ , (28) P g t + P r t ≥ o t · P a + P l t , (29) t (cid:88) i =1 σ i ≥ φ d ( t + D ) , t ≤ T − D, (30) t (cid:88) i =1 σ i = φ d (T) , T − D + 1 ≤ t ≤ T (31) σ t ≥ , (32) o t = (cid:40)(cid:80) tj =1 σ j t ≤ Dσ t + ( o t − − σ t − D ) t > D (33) where at time-step t , φ d ( t ) is the number of devices withdeadlines before or at t , P l t is power consumption by inflex-ible load, P r t is the power from renewable sources, o t is thenumber of devices running at t . The objective function in(27) is the integral of the marginal cost (26). Generatorlimits and supply/demand matching constraints are repre-sented by (28) and (29), respectively. It is assumed thatrenewable generation is curtailed when a generation surplusoccurs. The number of device start-ups to any time-step t must be at least equal to the number of devices whichhave a deadline before or at t + D , and for the last timeperiods it should be exactly the total number of deviceswith a deadline before T . This is represented by (30)-(31).Device uninterruptibility is ensured by (32)-(33). Com-bined, (30)-(33) guarantee that devices will not miss theirrespective deadlines. By solving the MIQP, a cost-optimalsystem load profile is obtained, which corresponds to a setof reference prices x ∗ t ∀ t ∈ T . While the reference solutionhere does not account for specific allocation for each agent,one realization of the reference schedule can be achieved bygiving priority to devices according to their proximity totheir respective deadlines, with ties being broken randomly,and assuming that devices do not switch off until their cy-cle (duration) is complete. The optimization was repeatedafter each market clearing to account for deviations fromthe previous reference solution, to effectively generate an“up-to-date” forecast at each time step. As previously established, probabilistic reference pricesare required. In reality, the facilitator would provide prob-abilistic reference prices that depend on actual forecastsand information used in generating the reference. Instead,for simulation purposes, probabilistic price forecasts weregenerated by adding noise to the deterministic referenceprices x ∗ t as follows. It was assumed that uncertainties areexogenous and independent for each time-step, and fore-cast prices at each time-step are log-normally distributed,with a standard deviation that increases with time. Thestandard deviation of the price X t as forecast at t (cid:48) ≤ t isparametrised by the day-ahead uncertainty ν h as SD t = x ∗ t · ν h · ( t − t (cid:48) )24 h . (34)Moreover, forecasting errors were simulated by adjust-ing the mean of the log-normal forecasts: the expectedprices ¯ x t were sampled from the log-normal distributionwith mean x ∗ t and standard deviation SD t . The values¯ x t , SD t ∀ t ∈ T were communicated to agents to be usedfor bid formulation. Figure 4 shows simulation results obtained with negligi-ble uncertainty ( ν h = 10 − ), demonstrating that F-MBCachieves near-optimal performance over multiple time-steps.8t can be seen from top panel that the device schedule ob-tained by F-MBC closely resembles the schedule obtainedfrom the MIQP reference solution. The difference in to-tal generation cost in this case is 0 . 08 % compared to thereference solution.Moreover, the centre panel shows the approximate ‘val-ley filling’ behaviour of the solution, especially comparedto the system without flexible demand (dotted line). Wenote that perfect flattening of flexible power generation isnot feasible due to the extended run time (1 hour, i.e. 12time steps) of loads. In the bottom panel, costs incurredby devices are plotted against their starting times. Theactual costs obtained using F-MBC are very close to thenearly identical costs obtained using MIQP.We compare the performance of F-MBC to three al-ternative coordination techniques in Figure 5, depictingthe same information as the lower two panels of Figure 4.Lack of coordination is represented by the “latest start” ap-proach where devices start at the latest time-step possiblewithout missing their respective deadlines. The “NaiveMBC” approach implements naive agents that submita bid between the minimum expected price x at,min andmaximum expected price x at,max that occur before theirlatest start time d a − D − 1. The bid placed is ˆ x at = x at,min + t ( x at,max − x at,min ) / ( d a − D a − expected prices and places an optimal bid using backward induction.This approach performs sub-optimally and yields a totalcost error of 9 . ν h from10 − to 1 (i.e. 100%). Figure 6 shows the results of20 independent simulation runs for each value of ν h . Thetop panel shows the distribution of realised cost of flexi-ble generation, compared with the reference solution. Itdemonstrates near-optimal performance even for significantuncertainties in forecast prices.The middle panel compares the individual paymentsmade by device agents (1200 × 20 for each value of ν h )against the payments under the reference schedule. Forforecast uncertainties up to 10%, these are approximatelyzero-mean, so that devices are on average as well off us-ing the F-MBC coordination scheme as under the Nashequilibrium.Finally, the bottom panel depicts the distribution ofregret that device agents have as a result of F-MBC (i.e.the difference between the actual price paid and the lowestpossible price in retrospect). The positive values indicatesmall deviations from a Nash equilibrium. However, thecomputed regret can only be used to generate cost savingsindividually: collectively, devices would quickly equaliseprice savings, as is evidenced by the small system costdeviations in the top panel.Collectively, these results demonstrate that when sup- Time (Hr) C o s t o f s t a r ti ng ( a . u . ) Cost of starting (F-MBC)Cost of starting (MIQP)21:00 24:00 3:00 6:00 9:00 12:00 15:00 18:00 20:55-500-300-1000100300500 P o w e r ( k W ) Inflexible net loadTotal net load (MIQP)Total net load (F-MBC)21:00 24:00 3:00 6:00 9:00 12:00 15:00 18:00 20:5505001000 C u m u l a ti v e nu m b e r o f d e v i ce s t a r t s MIQP scheduleF-MBC schedule Figure 4: Simulation output. Top: Cumulative device starts obtainedby F-MBC compared to the MIQP reference solution. Middle: Loadminus inflexible generation (i.e. power supplied by flexible generation)for three scenarios: without flexible loads, MIQP reference solutionand F-MBC solution. Bottom: The realized total cost paid by devicesplotted as a function of their starting times. plied with nearly optimal reference prices, F-MBC is ableto approximate the optimal schedule, but small differencesremain due to the ‘lumpiness’ of load, in line with theapproximate consistency results in Subsection 2.5. How-ever, these differences are small when averaged over manyruns, and are expected to reduce further as the system sizeincreases. 4. Discussion & Conclusion In this paper we considered a setting of uninterruptibledeferrable devices with deadlines set by their respectiveowners. By relying on decentralized decision making andcentralized forecasting and market clearing, the F-MBCapproach provides a simple and scalable means of DERcoordination. In terms of communication infrastructure,the proposed mechanism can be implemented using onlygathering of bids and broadcasts of prices and tie-breakingcut-off values from the auctioneer to all devices, signif-icantly reducing implementation complexity. Moreover,since the information in these broadcasts only concernspublic information, F-MBC preserves end user privacy andautonomy. The bidding algorithm was shown to automati-9 P o w e r ( k W ) Latest startF-MBCMIQP Inflexible net loadPoint forecast MBCNaive MBC21:00 24:00 3:00 6:00 9:00 12:00 15:00 18:00 20:55 Time (Hr) C o s t o f s t a r ti ng ( a . u . ) MIQP Point forecastNaive MBC Latest start F-MBC Figure 5: Comparison of different coordination techniques. Top:Total net load. Bottom: starting cost against starting time per device cally resolve mutually-conflicting decisions between deviceswith different deadlines and a tie breaking procedure wasproposed to resolve conflicts between indifferent devices. Itwas shown by simulation that near-optimal performancecan be attained by a clairvoyant facilitator, establishingthe consistency of the approach. Moreover, an analysis ofthe sensitivity to price forecast uncertainty demonstratesthe robustness of the approach. It was able to achievegood system-level and device-level performance across anextended horizon, making use of simple agent logic andsingle-period market clearing.Prices obtained using F-MBC are determined in real-time, thus exposing users to price uncertainty. The resultssuggest that the resulting cost fluctuations even out in thelong run, so that users are not worse off - especially incomparison with less-optimal schemes. If such exposure isnevertheless undesirable, an alternative is to use F-MBCwith a virtual currency, only for coordination and control.A different payment scheme (e.g. fixed subscription, averageprice, etc. ) can be operated in parallel.This paper has introduced the F-MBC concept andestablished its desirable properties in a limited set of ap-plications, thus laying the groundwork for various gener-alizations. As a proof of concept, we use uninterruptibledeferrable loads. However, relevant extensions for futurework are the inclusion of heterogeneous sets of deferrable C h a ng e i n a g e n t p a y m e n t w . r . t . r e f e r e n ce ( % ) Maximum standard deviation (%) A g e n t r e g r e t ( % o f ac t u a l c o s t ) -2 -1 I n c r ea s e i n s y s t e m c o s t ( % ) Figure 6: Sensitivity to uncertainty. Top: Increase in system cost withrespect to the optimal solution. Middle: Distribution of change inagent payments compared to optimal solution. 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(A.1)Using the definition p = Pr( X t > ˆ x t ), the optimal expectedcost (8) for agent 2 is rewritten as C ∗ d − D = p (cid:104) C ∗ d − D +1 − C s, d − D (ˆ x d − D ) (cid:105) + p ˆ x d − D · P · ∆ t + (1 − p ) E [ X d − D | X d − D ≤ ˆ x d − D ] · P · ∆ t + D − (cid:88) i =1 ¯ x d − D + i · P i · ∆ t. (A.2)The first term vanishes as a result of the threshold pricedefinition, so the inequality C ∗ d − D > C ∗ d − D to be provencan be simplified to E [ X d − D ] >p ˆ x d − D + (1 − p ) E [ X d − D | X d − D ≤ ˆ x d − D ] . (A.3)Considering E [ X d − D ] as the weighted sum of two con-ditional expectations (above and below ˆ x d − D ), this cansimplified to p (cid:0) E [ X d − D | X d − D > ˆ x d − D ] − ˆ x d − D (cid:1) > . (A.4)This is positive under the assumption that p > x t ). Therefore (A.3) holds and C ∗ d − D > C ∗ d − D . Lemma 11. For any two rapid-starting, deadline-orderedagents in the waiting state, and t < d − D , if C ∗ t +1 >C ∗ t +1 , then C ∗ t > C ∗ t .Proof. Analogous to (8), we define C b,at (ˆ x ) the expectedcost incurred by agent a at time t , when it submits a firstbid ˆ x and subsequent optimal bids: C b,at (ˆ x ) = Pr( X t > ˆ x ) · C ∗ at +1 + Pr( X t ≤ ˆ x ) · E [ C s,at ( X t ) | X t ≤ ˆ x ] (A.5)At any time-step t , the expected cost incurred by agent2 for bidding with a price ˆ x t is by definition greater thanor equal its optimal expected cost, C b, t (ˆ x t ) ≥ C b, t (ˆ x t ) = C ∗ t . (A.6)The condition C ∗ t +1 > C ∗ t +1 , combined with (A.6), and theassumption Pr( X t > ˆ x t ) > 0, implies C ∗ t = C b, t (ˆ x t ) > C b, t (ˆ x t ) . (A.7)Combining (A.6), (A.7) yields the desired result C ∗ t >C ∗ t . Proof of Lemma 2 : Consider two identical, deadline-ordered agents in the waiting state. The ordering ofthreshold prices ˆ x t > ˆ x t follows from (6), provided that C ∗ t +1 > C ∗ t +1 . The latter condition is guaranteed byLemma 10 and induction using Lemma 11. Because thisholds for any two agents, it also holds for the entire col-lection. The weakly ordered result follows by consideringagents with equal deadlines. Because such devices areindistinguishable (other aspects were already identical),symmetry requires that their threshold bids are identical. Appendix B. Proof of Lemma 7 Proof. First, if no devices are available to start, then x t = x ∗ t by virtue of P ∗ t being optimal, and 0 lies within thebounds of (19) as required.Next, we prove the lower bound for the clearing price.The case x t < x ∗ t occurs when agents have (unrealistically)low expectations for future costs and therefore submit lowbids. We consider the lowest possible bid placed by adevice that should start at t according to S . Theorem 3guarantees that the lowest bid is placed by the device a with the latest deadline d a (among devices that shouldstart at t ). The bid ˆ x ↓ t of this device is bounded by thelowest possible cost associated with waiting, according tothe inequality: C s,at (ˆ x ↓ t ) ≥ min t (cid:48) ∈{ t +1 ,...,d a − D } ∆ t D − (cid:88) i =0 P i (cid:2) m t (cid:48) + i ( P ∗ t (cid:48) + i ) − η (cid:3) ≥ min t (cid:48) ∈{ t +1 ,...,d a − D } K at (cid:48) − η ∆ t D − (cid:88) i =0 P i , (B.1)where the lower bounds of the price forecasts X t (cid:48) and (10)have been used. Cost-optimality of S and the fact that thedevice should start at t implies K t ≤ K t (cid:48) (11), so that C s,at (ˆ x ↓ t ) ≥ K at − η ∆ t D − (cid:88) i =0 P i , (B.2) ≥ ∆ t D − (cid:88) i =0 P i (cid:2) m t + i ( P ∗ t + i − P i ) − η (cid:3) . (B.3)Expanding C s,at (ˆ x ↓ t ) using (2) and making use of definition(20) results in the inequality P (cid:104) ˆ x ↓ t − x ∗ t (cid:105) ≥ − D − (cid:88) i =0 P i [∆ m t,i + η ] (B.4)If x t < x ∗ t , at least one fewer device has started than wasaccounted for in the allocation S . Therefore, the bid ˆ x ↓ t ofthe marginal device must bound the clearing price x t frombelow, and the lower bound of (19) follows.The upper bound can be derived by considering devicesthat should not start under S , but submit a high bidbecause starting immediately appears to be cheaper than12tarting at their scheduled time t a > t . The magnitude ofsuch bids ˆ x at is bounded by C s,at (ˆ x at ) = C ∗ a t +1 (B.5) ≤ C s,at a ( x t a ) (B.6)= ∆ t D − (cid:88) i =0 P i (cid:2) m t a + i ( P ∗ t a + i ) (cid:3) , (B.7)where we have used (9) and the fact that the optimal ex-pected cost C ∗ a t +1 is always upper-bounded by the expectedcost for one specific feasible time t (cid:48) > t , including thespecial case t (cid:48) = t a . Using the definition of C s,at :∆ tP [ˆ x at − x ∗ t ] ≤ ∆ t D − (cid:88) i =0 P i (cid:2) m t a + i ( P ∗ t a + i ) − m t + i ( P ∗ t + i ) (cid:3) ≤ ∆ t D − (cid:88) i =0 P i (cid:2) m t a + i ( P ∗ t a + i ) (cid:3) − K at ≤ ∆ t D − (cid:88) i =0 P i (cid:2) m t a + i ( P ∗ t a + i ) (cid:3) − K at a , ≤ ∆ t D − (cid:88) i =0 P i ∆ m t a ,i (B.8)where we have used the fact that K at a ≤ K at due to cost-optimality of S . The bound for the highest bid ˆ x ↑ t of adevice that should not run is therefore defined as P (cid:104) ˆ x ↑ t − x ∗ t (cid:105) ≤ max a ∈{ a (cid:48) ∈A : t a >t } D − (cid:88) i =0 P i ∆ m t a ,i (B.9) ≤ max t (cid:48) ∈{ t,...,T } D − (cid:88) i =0 P i ∆ m t (cid:48) ,i (B.10)If x t > x ∗ t , at least one more device started than in theallocation S , so the clearing price x t must be at or belowˆ x ↑ tt