Optimization models for consumer flexibility aggregationin smart grids: the ADDRESS approach

Alessandro Agnetis, Gabriella Dellino, Gianluca De Pascale, Giacomo Innocenti, Marco Pranzo, Antonio Vicino

Abstract— This paper addresses the problem of optimalmanagement of consumer flexibility in an electric distributionsystem. Aggregation of a number of consumers clusteredaccording to appropriate criteria, is one of the most promisingapproaches for modifying the daily load profile at nodes of anelectric distribution network. Modifying the daily load profileis recognized as one of the strongest needs both for safe andefficient operation of the network.

The paper proposes an optimization approach allowing theaggregator, i.e., the operator which manages the aggregatedconsumers, to gather flexibility and generate bids for the energymarket, with the aim of maximizing its revenue. It is shownthat this problem can be solved through mixed integer linearprogramming. Numerical simulation results are provided forvalidating the proposed approach.

I. INTRODUCTION

The ever increasing diffusion of renewable energy sourcesin electricity systems has considerably raised the need forresponsive and resilient power delivery systems. Linkingelectricity systems to the most recent methods and tech-nologies in communications and computer control has beenrecognized as the most appropriate path to meet the re-quirements deriving from this new scenario. The vision ofthe ‘smart grids’ as founding pillars of the economy of the21st century has found a solid ground in the very promisingresults achieved by several important research projects andinitiatives developed in recent years ( [1]–[6]). The designof intelligent, self-healing electricity networks allowing forintensive data flow among system components, system op-erators, generators, market traders, retailers and consumersis the primary objective of these and other projects whichare posing important challenges to the information andcommunication research community, including the controlcommunity ( [7]–[12]).

In this context, the ADDRESS European project (ActiveDistribution networks with full integration of Demand anddistributed energy RESourceS) [13], [14] aims at the devel-opment of an ‘Active Demand’ (AD) view to the smart grids,allowing for the active participation of both domestic andsmall commercial consumers in the network management,letting them play an active role in the energy market throughthe newly introduced operator called ‘Aggregator’ [15]. Therationale of the approach is based on different facts which

The authors are with the Dipartimento di Ingegneria dell’Informazione& Centro per lo Studio dei Sistemi Complessi, Universita di Siena, 53100Siena, Italy.Email: agnetis,dellino,pranzo,vicino@ing.unisi.it;luca1975@gmail.com

emerge from an analysis of the present state of the art in themanagement of electricity distribution networks:

• the presence of intermittent generators, typically re-newable energy sources, poses serious problems tothe network operation (power unbalances management,voltage regulation, etc.);

• well assessed investigations on domestic consumptionsstate that at least 10% of the domestic load is ‘flexible’,in the sense that it can be time-shifted, specificallyin the presence of economic incentives as well asenvironmental concerns;

• safe electric system operation requires the continuouspresence of a power ‘reserve’ to be used for facingpossible problems arising in the network.

The basic idea underlying the AD approach is to exploithousehold and small commercial consumption flexibility toreduce as much as possible the generation reserves whichrepresent a significant cost for safe operation of the system,as well as to accomodate unbalances caused by distributedrenewable energy sources. The ADDRESS approach takesinto account different types of equipment and appliancespresent at the consumers’premises, from distributed genera-tion, such as photovoltaic or co-generation plants, to thermaland electrical storage devices and purely passive loads. Themain issues involved in the flexibility evaluation are relatedto the features of devices and their usage and control. In thissense, air conditioning, space and water heating plants showthe highest potential with respect to flexibility. Also, plug-in vehicles, although not explicitly tackled in the ongoingADDRESS project activities, are devices with high potentialfor flexibility procurement.

The main focus of the project is on the Aggregator,which is conceived as an intermediary party between theconsumers and the energy market. This player enrolls con-sumers through signing appropriate contracts with them onthe basis of their typical daily load profile and features likegeographical position, type of devices installed, number ofpeople in the house etc. The aggregator gathers flexibilityfrom its affiliated consumers, asking them to adjust theirdaily load profile according to an optimized schedule. Thisway, he is able to prepare bids for the market in the formof (ancillary) services, consisting in power reduction (orincrease) over specific time intervals. The actuator signalsthe aggregator uses to interface with its consumers areprice-volume signals, through which he offers a reward to

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consumers for their willingness to modify their load profile.According to the setup outlined above, the purpose of this

paper is to propose an optimization framework for solvingthe basic decision problem the aggregator has to solve inhis activity on the energy market. On the basis of forecastsof the energy price, he has to make decisions on whichkind of bids to submit to the market, by evaluating theexpected consumer responses in terms of daily load profilechanges to price-volume signals. We will focus on the day-ahead market and formulate an optimization problem whosesolution allows the aggregator to design an optimal dailybusiness strategy for maximizing his revenue. It is shown thatthe problem can be cast as a mixed integer linear program,whose solution provides an optimal schedule of bids tothe day-ahead market, as well as the related price-volumesignals to be sent to the affiliated consumers to gather therequired energy. In addition, simulation results are illustrated,highlighting advantages and limitations of the approach interms of computational burden.

The paper is structured as follows. Section 2 introducesthe problem formulation and the related key concepts. Sec-tion 3 provides the formulation of the aggregator shortterm business plan as a mixed integer linear programmingproblem, while simulation results are reported in Section 4.Concluding remarks are given in Section 5.

II. PROBLEM FORMULATION

Let us start by briefly recalling the typical mechanismbehind energy markets. Here, there are mainly two kindsof players. On one side, energy “producers” who aims atselling the energy they produce either through contractswith other players or by selling it directly on the wholesalemarket. On the other side, “retailers” who aims at buyingenergy from the producers and selling it to their customers.Both producers and retailers submit bids to the marketspecifying the price, the time and the amount of energythey want to sell or buy. The energy price is fixed bymatching demands with offers and typically changes duringthe day. Finally, the TSO (Transmission System Operators)and DSO (Distribution System Operator), a third categoryof players, verify whether the energy flows are compatiblewith the transmission and distribution networks structure andpossibly curtail some offers to prevent congestion risks andguarantee safe operation of the network. The above activitiesare repeated in two stages, characterized by different timehorizons and known as the day-ahead and the intra-daymarkets.

In such a framework, the consumers are not aware ofthe presence of the energy market since their counterpartis simply the retailer they are related through a contract.However, it is well known that the energy consumptions ofdomestic and small commercial consumers is characterized,at least to some extent, by a certain degree of flexibility,essentially due to:

• “shiftable loads”, whose starting time can be freely seton a specific interval up to a certain extent;

• “adjustable loads”, whose power profile can be modifiedwithin a proper threshold;

• unnecessary consumptions, usually associated towastage or comfort excess.

The consumer flexibility is currently not exploited, andthe aim of the ADDRESS project is to introduce a newmarket player, the Aggregator, able to gather the consumers’flexibility and use it on the market. To this end, eachconsumer is provided with an “Energy Box” which interfacesthe consumer directly with the Aggregator. The EnergyBox is a device able to control and coordinate householdappliances and loads. Each Energy Box receives signals fromthe Aggregator to modify the load profile of the consumersby leveraging on shiftable and flexible loads: as a payoff theconsumer receives a reward. The flexibility request signalsdescribe a request of load reduction (increase) for a giventime period and specify the monetary reward if the consumerfulfills the request.

Therefore, as shown in Fig. 1, the Aggregator task istwofold:

• On the consumer side: he gathers flexibility by suitablyproposing incentives to consumers in order to shapetheir load profile;

• On the energy market side: he sells flexibility into theenergy market and he may provide a balancing serviceto DSO.

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Fig. 1. Relations with market and consumers

To carry out this task the Aggregator groups its affiliatedconsumers into clusters. A cluster includes all the consumerssharing similar characteristics in terms of kind and usage ofthe appliances, number of people at the premises, their habitsand geographical region to which they belong. We assumethat the behavior of the cluster, in the absence of flexibilityrequests, can be represented as a baseline load profile curve,representing the cluster load profile during the day. Besides,the Aggregator is assumed to know how each cluster reactsin response to each admissible price-volume signal sent toit. In other words, the different clusters response functionsto the price-volume signals are known and utilized in orderto optimize the bids presented to the market. In Fig. 3 weshow the baseline load profile of a cluster and the response

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function to a fixed power reduction signal. Observe that,due to the presence of shiftable loads, the modified loadprofile shows a payback effect just after the end of thepower reduction time interval. This means that consumersreducing consumption in one period, tend to recover theirscheduled program in the successive time slots. This behaviorintroduces dynamics in the process which must be accountedfor in the aggregator scheduling optimization.

The Aggregator job is to optimize the available flexibilitythat can be gathered by its consumer portfolio and to presentbids on the market when the energy price is most favorable.Moreover, he has to take into account the possibility of bidscurtailment from DSO, in case his matched market bids willbe deemed unfeasible in the service validation process.

III. THE OPTIMIZATION MODEL

In this section, we show how the problem of preparingbids for the energy market in the different time slots of therelevant time horizon, can be formulated as a mixed inte-ger linear programming problem. Computational complexityissues of the optimization model are discussed.

First, we provide a brief description of the model wedeveloped, without entering into mathematical details. Math-ematical formulation of the model is introduced in subsectionIII-A in which the mathematical optimization model used inthe day-ahead market is presented.

The optimization model involves the following groups ofdecisions (variables):

• A first group of boolean variables represents whetherthe aggregator is able to propose offers to the marketfor the various time slots.

• A second group of variables specifies the amount ofenergy gathered in the various time slots.

• The last group of boolean variables characterizes theparticipation of which cluster to which offer of theaggregator.

Notice that there exist two different time scales in theproblem. A first time scale is used to represent time as seenfrom the market point of view. A different time scale mayrepresent the time slots as seen from the clusters point ofview.

The input data required by the optimization model aremainly given by load profiles of the consumer clusters; i.e.,“baseline” load profiles (in absence of aggregator flexibilityrequests) and load profiles corresponding to each possibleflexibility request of the aggregator. Of course, the clusterflexibility is represented by the difference of these profiles.

In our model we assume that, for each possible flexibilityrequest, the aggregator is able to retrieve the informationabout the overall price to be paid when the cluster accepts itsproposal. The availability of such data allows the optimiza-tion algorithm to handle multi-level price/volume signalsconsisting of multiple volume thresholds and correspondingprice levels. Moreover, the aggregator optimization processcan cope with complex flexibility requests in which, foreach time slot, a different (multi-level) signal is sent to thecluster. Clearly, the computational burden of the algorithm

grows exponentially with the number of admissible flexibilityrequests present in the database.

With reference to the optimization cost function, weassume that the aggregator main objective is to maximize itsrevenue. Hence, the objective function is composed of twoterms: the income from selling the energy on the market, andthe cost to be paid to the consumers for their participationin the service.

The main constraints of the model account for the follow-ing facts:

• Each cluster can receive at most one flexibility requestper day by the aggregator.

• The payback at the level of the load area where theaggregator operates must not exceed a given mini-mum/maximum threshold profile.

• Each single offer to the market must have a minimumand a maximum volume.

• The request to each consumer cluster cannot exceeda maximum threshold related to the reliability of thecluster.

Notice that the aggregator can promptly react to a curtail-ment request coming from the DSO. In fact, if the total load(of the load area) resulting from market clearing turns outtoo large to pass DSO validation in certain critical time slots,the optimization model can be re-run after the constraint onthe maximum load of the load area (in the critical time slots)has been suitably tightened. Possibly, when the model is runagain, flexibility requests in the previous optimal solutionthat are not related to those involved in the curtailment can befixed, if we wish the new flexibility plan to be only partiallydifferent from the previous round.

A. Aggregator Toolbox Optimization Model for the day-ahead market

The indices used in the mathematical formulation of themodel are:

• k: denotes the k-th cluster of consumers; k = 1, . . . ,K,where K is the number of clusters in the load area.

• h: denotes the h-th flexibility request that the aggregatormay send to cluster k; more specifically, it consists of a(possibly multi-level) price/volume signal and the dura-tion of the request. We assume that the aggregator mayformulate a finite number of proposals (correspondingto different combinations of price-volume signals, indifferent time slots), which we denote by H . Therefore,h = 0, 1, . . . ,H , where we set h = 0 for the casein which no request is sent by the aggregator to theconsumers.

• t: denotes the t-th market time slot; t = 1, . . . , TM ,TM being the number of market time slots within theselected time horizon.

• τ denotes the τ -th consumer time slot; τ = 1, . . . , TC ,TC being the number of consumer time slots within theselected time horizon.

To run the optimization model, the aggregator needs thefollowing input data, that he will get from the database:

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• πt: denotes the energy price forecast at market time slott.

• Pkh: gives the overall cost that the aggregator has topay to its consumers if the h-th flexibility request issent to cluster k.

• τ1h: denotes the starting time (expressed in consumertime slots) of the flexibility request h.

• τ2h: denotes the ending time (expressed in consumertime slots) of the flexibility request h.

• fk0(τ): gives the baseline load profile of cluster k. Thediscrete values of the load profile are given for dailyconsumer timeslots τ = 1, . . . , TC . The baseline loadprofile is expressed in kWh.

• fkh(τ): gives the load profile of cluster k when theflexibility request h is activated. The discrete values ofthe modified load profile are given for daily consumertime slots τ = 1, . . . , TC .

• Lmin(τ): is the minimum load compatible with the loadarea for each consumer time slot τ . It is expressed inkWh.

• Lmax(τ): is the maximum load compatible with theload area for each consumer time slot τ . It is expressedin kWh.

• ω is the minimum threshold value for the size of thebid to be made on the energy market.

• Ω is the maximum threshold value for the size of thebid to be made on the energy market.

• ρk: denotes the maximum “risk” for cluster k. It im-poses an upper limit on the request that each clustercan receive from the aggregator. It is measured in kWh.

• σk: is an indicator of cluster k reliability. It ranges from0 to 1, 1 being the maximum reliability.

The variables involved in the model are:

• Yt ∈ 0, 1, t = 1, . . . , TM . This family of booleanvariables is used to represent the offer on the energymarket. If Yt is 1 then the aggregator is able to presentan offer in the market time slot t. Otherwise it assumesvalue of 0.

• Et, t = 1, . . . , TM . This continuous variable representsthe amount of energy that the aggregator is able to offerto the market at time slot t.

• xkh ∈ 0, 1, k = 1, . . . ,K, h = 0, . . . ,H . This familyof boolean variables is used to represent the signals sentfrom the aggregator to its consumers. If xkh = 1, thenthe cluster k receives a flexibility request h from theaggregator. Otherwise, the variable is equal to 0. Recallthat h = 0 denotes the baseline scenario, in which noflexibility is requested to the cluster.

• Rt, t = 1, . . . , TM . This variable represents the differ-ence between the baseline load profile and the actualload profile of the entire load area in the t-th markettime slot. It may assume either positive values (if theclusters reduce their consumption) or negative valueswhen the clusters increase their consumption (note thatthis may happen during the paybacks).

We now introduce the mathematical model in terms of ob-jective function, functional and technical constraints, whichresults in a Mixed Integer Linear Programming (MILP)formulation.

maxTM∑t=1

Etπt −K∑

k=1

H∑h=0

Pkhxkh (1)

subject toH∑

h=0

xkh = 1,∀k (2)

Lmin(τ) ≤K∑

k=1

H∑h=0

xkhfkh(τ) ≤ Lmax(τ),∀τ (3)

ωYt ≤ Et ≤ ΩYt,∀t (4)

(1− σk)H∑

h=0

xkh

∑τ∈[τ1h,τ2h]

(fk0(τ)− fkh(τ)) ≤ ρk,∀k (5)

K∑k=1

(fk0(τ)−H∑

h=0

xkhfkh(τ)) ≥ Rt,∀τ ∈ t,∀t (6)

(1− Yt)(−M) ≤ Rt,∀t (7)

Et ≤ Rt + (1− Yt)(−M),∀t (8)

The objective function of the model – Equation (1) – aimsat maximizing the aggregator’s profits and it is composed oftwo terms. The first term takes into account the gains fromselling the flexibility to the markets, while the second termaccounts for the costs that the aggregator has to pay to itsconsumers.

The functional constraints of the model are as follows.Equation (2) enforces that each cluster k receives exactly

one flexibility request for each day. Recall that if a clusterdoes not receive a flexibility request then xk0 = 1.

Constraint (3) guarantees that in each consumer time slotthe load profile of the whole load area remains boundedwithin two threshold values.

Equation (4) is used to avoid that the aggregator producesoffers that are below a minimum and beyond a maximumvalue. Observe that, due to the presence of variable Yt, ifno offer is presented in the market time slot (i.e., Yt = 0),then the aggregator cannot sell energy to the market (i.e.,Et = 0).

The constraints expressed in Equation (5) are used to limitrisks (i.e., excessive requests to unreliable clusters) for theaggregator business. This is obtained by imposing an upperbound on the energy requested to each cluster and alsoby taking into account the reliability of each cluster. Morespecifically, the higher the cluster reliability, the bigger thebound on the overall flexibility the aggregator can collectfrom cluster k. Notice that this constraint could also be“disabled” (by setting σk = 0 and ρk = ∞), so that it wouldnot be active in the optimization model.

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TABLE ICLUSTER DETAILS

Cluster ID Size Reliability1 900 12 800 13 1250 0.64 1020 0.8

Besides the functional constraints discussed before, weneed to add some technical constraints to guarantee con-sistent solutions.

This family of constraints described in Equation (6) de-fines the variables Rt as the minimum difference betweenbaseline and actual load profiles in the whole load area.

In the last two constraint, we define M as an arbitrarilylarge constant. The Equation (7) is a family of constraintsintroduced to link the two families of decision variables, Yt

and Rt. In fact, Rt must be positive whenever the aggregatoris able to present an offer to the market (i.e., Yt = 1).

The last family of constraints (Equation (8)) guaranteesthat – whenever the aggregator is able to present an offer tothe market (i.e., Yt = 1) – the energy offered to the marketis at most equal to the overall flexibility gathered by theaggregator.

IV. NUMERICAL SIMULATION RESULTS

In this section, numerical results provided by the opti-mization algorithm in a simulated scenario are presentedtogether with a discussion on advantages and limitations ofthe approach.

We perform our experiments assuming that consumersbelong to four clusters, whose characteristics (namely, sizeand reliability) are specified in Table I.

Although the optimization model is flexible, in our testswe assume 24 one hour time slots for the market time scaleand 15 minutes time slots for the consumer time scale.

The energy price forecasts over the 24-hour time horizonhave been derived based on historical data, and are depictedin Fig. 2.

Fig. 2. Energy price forecasts for the day-ahead market

The flexibility requests to be examined by the aggregatoroptimization module are derived as follows. We consider

• two price levels; namely, 0.01e and 0.02e;• a single volume reduction level of 1 kW;• two options for the duration of a flexibility request;

namely, 2 hours (i.e., 8 time slots) and 3 hours (i.e.,12 time slots).

To control the overall computational effort, we do notaccount for all the possible level combinations of the threefactors, limiting the number of flexibility requests for eachcluster to 90 signals. The minimum and maximum size ofthe bid have been fixed to ω = 100 kWh and Ω = 1 MWh,respectively.For each possible flexibility request, the response functionof the cluster is known; an example is given in Fig. 3.

Fig. 3. Baseline load profile and response function of a cluster to aflexibility request

Based on this input data, we solve the optimization prob-lem formulated in Section III-A using CPLEX [16]. Theoptimal solution consists of the following bid made on theenergy market: from 9 to 10, the aggregator offers 187 kWh;from 10 to 11, he offers 1 MWh; from 11 to 12, he offers975.8 kWh and from 12 to 13 he offers 1 MWh. The resultingbid is depicted in Fig. 4, represented by the red bars. Theblue profile shown in this figure represents the total amountof power collected by the aggregator. Notice that in two timeintervals (from 10 to 11 and from 12 to 13) the aggregatorwould be able to offer more than it does, due to a slightlyhigher amount of power collected over the whole markettime slot; however, having fixed a threshold Ω = 1 MWh,the aggregator is not allowed to exceed this value.

The energy collected by the aggregator is due to thefollowing flexibility requests sent to the clusters: clusters 1and 2 receive a reduction request of 1 kW lasting 3 hours,from 9 to 12, with a corresponding reward of 0.02e; clusters3 and 4 receive a reduction request of 1 kW from 11 to 14,with a corresponding reward of 0.02e.

We point out that the aggregator formulates his bids overthe time slots corresponding to higher forecasted energyprices, as it is evident from Fig. 2. The overall aggregator’sprofit for the optimal solution found by the algorithm is equal

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Fig. 4. Aggregator offers to the energy market

Fig. 5. Baseline profile (dotted) of the load area and response profile (solid)to the aggregator’s offer

to 442.79e: 54.37ewill be paid by the aggregator to the con-sumers participating in the active demand, and the remainingamount is the net gain derived from selling the collectedflexibility to the market, according to the formulated bids.Notice that the seemingly low reward received by a singleconsumer corresponds to a significant daily saving on theenergy bill.

Fig. 5 shows the response of the whole load area to theoffer formulated by the aggregator; the dotted line representsthe baseline profile of the load area, obtained by summingup the base profiles of the clusters belonging to that loadarea. This figure highlights the power reduction over thetime intervals of the aggregator’s flexibility request, andthe corresponding payback in the following hours, due toa shift in consumers’ loads. The dashed upper and lowerlines denote the maximum and minimum loads compatiblewith the load area, over the 24-hour time horizon.

This test has been performed using a 3 GHz Intel Core 2Duo processor with 3.25 GB of RAM; the computation timerequired to find an optimal solution was about 3 minutes.Additional experiments with 4000 flexibility requests foreach cluster have been executed, taking a computational timeof approximately 45 minutes. In consideration of the fact thatthe problem is solved off-line (e.g. the day ahead), these com-putational times are practically feasible, i.e., the optimizationmodel can be solved exactly with no need to resort to specificsolution approaches such as dual decomposition or heuristic

techniques.

V. CONCLUSIONS

In this paper an optimization approach has been proposedallowing the aggregator to gather consumer flexibility in anelectricity distribution network, for generating services andbids to be submitted to the energy market. With referenceto the day-ahead market, it is shown how the aggregatorproblem can be formulated and solved through mixed in-teger linear programming. Simulation results are provided,showing the effectiveness and the flexibility features of theproposed approach. Besides the environmental impact, theoptimal solution found suggests considerable daily savingsfor consumers participating in the active demand.Extensions of the presented approach to deal with the intra-day market as well as the management of derivatives in thewholesale energy market are the subject of ongoing work.

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VI. ACKNOWLEDGEMENTS

The research leading to these results has received fundingfrom the European Community’s Seventh Framework Pro-gramme (FP7/2007-2013) under grant agreement no. 207643.

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