Robust Energy-Constrained Frequency Reserves from Aggregations of Commercial Buildings
aa r X i v : . [ c s . S Y ] J un Robust Energy-Constrained Frequency Reservesfrom Aggregations of Commercial Buildings
Evangelos Vrettos ∗ , Frauke Oldewurtel † , and G¨oran Andersson ‡ EEH-Power Systems Laboratory, Swiss Federal Institute of Technology(ETH) Zurich, Switzerland Department of Electrical Engineering & Computer Sciences, Universityof California, Berkeley, USAAugust 25, 2018
Abstract
It has been shown that the heating, ventilation, and air conditioning (HVAC)systems of commercial buildings can offer ancillary services to power sys-tems without loss of comfort. In this paper, we propose a new control frame-work for reliable scheduling and provision of frequency reserves by aggrega-tions of commercial buildings. The framework incorporates energy-constrainedfrequency signals, which are adopted by several transmission system opera-tors for loads and storage devices. We use a hierarchical approach with threelevels: (i) reserve capacities are allocated among buildings (e.g., on a dailybasis) using techniques from robust optimization, (ii) a robust model predic-tive controller optimizes the HVAC system consumption typically every minutes, and (iii) a feedback controller adjusts the consumption to providereserves in real time. We demonstrate how the framework can be used to esti-mate the reserve capacities in simulations with typical Swiss office buildingsand different reserve product characteristics. Our results show that an aggre-gation of approximately buildings suffices to meet the MW minimumbid size of the Swiss reserve market.
In many countries, large amounts of renewable energy sources (RES) are integratedin the power system and they increase the operational uncertainty due to their fluc-tuating nature. In a power system, supply and demand of electric power must be ∗ [email protected] † [email protected] ‡ [email protected] of a building’s fan power can be offered as reserveswithout significant loss of comfort, if the SFC signal is within the frequency band f ∈ [1 / (10 min ) , / (4 sec )] . In [20], a control approach similar to that of [18, 19]was experimentally validated in a real building, but without considering frequen-cies below / (10 min ) to avoid effects on chiller power consumption. The follow-up work [21] included chiller control, which enlarged the frequency band of SFCsignals to / (60 min ) . Ref. [22] investigated SFC by direct control of a heat pump’s(HP) compressor power using a variable speed drive motor in a water-based HVACsystem. 2part from accurate tracking of the SFC signal, a TSO needs guarantees thatthe reserve capacity of commercial buildings will be reliably available. Becausethe buildings are energy-constrained resources reserve scheduling is required, inparticular if the SFC signal is not approximately zero-energy over short periodsbut can be biased towards one direction over long periods of time. A priori re-serve scheduling allows buildings to participate in markets for such reserve prod-ucts without compromising occupants’ comfort. Ref. [23] addressed this issue bydeveloping an MPC-based method to quantify the flexibility of a commercial build-ing and a contractual framework to declare it to the utility. In this paper, we followthis line of research and consider robust reserve scheduling for aggregations ofcommercial buildings.The main contributions of this paper are threefold. First, we propose a newframework to estimate the SFC reserve capacity that can be reliably offered by anaggregation of commercial buildings considering weather conditions, occupancy,electricity prices, reserve payments, and comfort zone. The framework builds on ahierarchical control scheme with three levels, namely reserve scheduling and allo-cation, HVAC control, and reserve provision, and is based on robust optimizationand MPC. The framework actively allocates reserves among aggregation’s build-ings based on their individual characteristics, which is expected to maximize thereserve potential in markets with typical requirements such as constant reservecapacity over a minimum duration and/or equal up- and down-reserve capacities.This is in contrast to [18, 19, 21] that estimated the capacity of a group of build-ings by simply scaling up the estimated capacity of a single building. Second, wepropose new methods to estimate reserve capacities in case of energy-constrainedfrequency signals, a practice adopted by several TSOs for loads or storage devices,e.g., [24]. This is a significant advantage of our method compared to [23]. Arelevant approach is taken in [25], where the reliable AS capacity from a commer-cial building is expressed as a function of the AS signal’s frequency. In contrastto [25], we are interested in the dependence of capacity on the integral of the SFCsignal, i.e., on the signal’s bias directly, which we believe is the major limiting fac-tor when extracting reserves from energy-constrained resources. Third, we deriveupper bounds on the untapped potential for SFC by different building types and fordifferent reserve product characteristics such as duration, symmetry, and energycontent.Although in some cases the SFC signal is approximately zero-energy, this isnot generally true for all power systems. Ideally the bias of the SFC signal wouldbe obtained from generators, e.g., through activation of tertiary control reserves orin a nearly real-time (e.g., 5 minute) market. However, nearly real-time marketsdo not exist in many control areas, and activation of tertiary control reserves is notalways more economical than SFC reserves, because it highly depends on the theavailability of cheap and flexible generators. In addition, with increasing RES pen-etration in power systems, the share of conventional generators in the productionmix will decrease. Despite the improvements in forecasting techniques, the RESforecasts will never be perfect and biased forecast errors will translate into biased3FC signals. With fewer controllable generators in the production mix, other re-sources will have to absorb these biases. For this reason, we believe that accountingfor biases in the SFC signal will be very important in the future, possibly also inpower systems where the SFC signal is today approximately zero-energy. To theauthors’ best knowledge, this is the first time that methods are proposed to allowbuilding aggregations to systematically schedule the reserve capacity that they canoffer depending on how much bias exists in the SFC signal.This paper includes substantial extensions over our previous work [26], wheresome preliminary results were presented. A major improvement is related to mod-eling of the SFC reserve requests within the reserve scheduling and building controloptimization problems. This paper considers also energy-constrained SFC signalsapart from conventional unconstrained signals, whereas [26] considered only thelatter. In the absence of energy constraints, the worst case reserve request is equalto either the up- or down-reserve capacity along the whole scheduling horizon. Thereserve scheduling problem for this case was cast as a robust optimization problemwith additive uncertainty in [26]. On the contrary, the determination of the worstcase reserve request along the scheduling horizon is more involved with energyconstraints due to time coupling. In this paper, we model an energy-constrainedSFC signal as an uncertain variable that lives in a polyhedral uncertainty set andenters multiplicatively in the constraints. This allows us to reformulate the robustreserve scheduling problem into a deterministic tractable one. Another main ex-tension is related to the number and type of actuators that provide reserves in eachbuilding. In contrast to [26] that assumed either only heating or only cooling ac-tuators per building, the proposed formulations can be used to allocate reservesamong different types of actuators within the same building. Therefore, the pro-posed formulations are more general than the ones in [26]. In addition, the simu-lation studies of this paper contain new material compared with [26]. We analyzerecorded data of the Swiss SFC signal to illustrate that the SFC signal is not neces-sarily approximately zero-energy, but it can be significantly biased over periods ofseveral hours. The performance of the control framework is then evaluated usingan energy-constrained SFC signal, whereas the sensitivity analysis is performedconsidering signals with and without energy constraints.The rest of the paper is organized as follows: Section 2 summarizes someimportant aspects of power system SFC, Sections 3 and 4 introduce the modelingand control framework, the performance of the framework is demonstrated andevaluated in Section 5 in a simulation example, a sensitivity analysis is performedin Section 6, and Section 7 concludes the paper. Typically, a TSO controls frequency in three steps: primary, secondary and tertiarycontrol. Primary control is a distributed, proportional controller that stabilizes thefrequency after a disturbance. Secondary control is a centralized, proportional-4ntegral (PI) controller that restores the frequency to its nominal value and main-tains the desired exchanges between neighboring control areas. Tertiary controlreleases secondary control in case of large disturbances and is typically manuallyactivated.Before continuing, we summarize some important aspects of scheduling andactivation of SFC in power systems [27], which will be particularly relevant to thesensitivity analysis of Section 6. In most of Europe, SFC reserves are procured ina market setting, i.e., the generators bid their reserve capacity and price in weeklyor daily auctions. The requirements of these auctions and the characteristics ofthe SFC reserve products vary from country to country. The minimum bid sizeis typically in the range [1 , MW, e.g., MW in Switzerland [28]. In manycountries, only symmetric reserves, i.e., equal up- and down-reserve capacities,are allowed, whereas in other countries asymmetric reserves are also accepted.The reserve energy is requested from the generators via a signal sent by the TSO,typically every − seconds. There are two main activation rules: (a) the pro-rata activation, where the reserve energy is proportional to the capacity, and (b) themerit-order activation, where the reserves are requested based on the short termmarginal costs of generators. The remuneration of reserve energy is also countrydependent but it is typically separate from the reserve capacity remuneration. Insome countries, e.g., in Switzerland, the reserve energy remuneration is coupledwith the energy price in the spot market. As a final remark, note that we treat theSFC signal as uncertain in this paper because it is unknown at the time when thereserve capacities are determined. We consider buildings with integrated room automation (IRA) systems, whereheating, cooling, ventilation, blinds, and lighting are jointly controlled. IRA istypically used in office buildings because it provides high comfort while being en-ergy efficient [29]. We represent building thermal dynamics using a well-tested th order multiple-input-multiple-output bilinear model [11]. This model is basedon the widespread thermal resistance-capacitance network approach, and was val-idated against the well-known building simulation software TRNSYS [11]. Themodel’s satisfactory accuracy and its relatively low complexity make it suitablefor MPC. A model similar to the one considered here was used in an MPC imple-mentation in a real building, and it was found that it captures the building thermaldynamics well [10]. Furthermore, this model is flexibly customizable and allowsus to perform large-scale simulation studies with different representative buildingtypes.The original model is bilinear between inputs u and disturbances v (e.g., blindposition and solar radiation), as well as between inputs and states x (e.g., blindposition and room temperature). If the disturbances are fixed, e.g., according to5heir predicted values, the bilinearities between u and v vanish and the systembecomes time-varying. However, the bilinearities between x and u remain. Foroptimization purposes, sequential linear programming (SLP) can be applied byiteratively linearizing the bilinear x terms around the most recently calculated x trajectory and solving the resulting linear program (LP) until convergence [29].If SLP is applied, the dynamics of a building b can be described by the lineartime-varying model x bt +1 = A b x bt + B bt u bt + E b v bt + R b ∆ u bt , (1)where x bt ∈ R n x denotes the states at time step t ( n x is the number of states),i.e., the room air temperature as well as the temperatures in different layers in thewalls, floor, and ceiling (all measured in o C). u bt ∈ R n u denotes the IRA controlinputs, namely heating and cooling power, ventilation, blind position, and lighting( n u is the number of actuators). The heating and cooling are represented in thethermal model as heat fluxes affecting the system states and their units are W/m ,i.e., the heat fluxes are normalized by the floor area. The blind position is a num-ber between (fully closed) and (fully open). The lighting is also normalizedby the floor area and measured in W/m . v bt ∈ R n v denotes the disturbances thataffect building’s states, e.g., ambient temperature in o C, solar radiation in W/m ,and internal heat gains by occupants and equipment in W/m ( n v is the numberof disturbances). ∆ u bt ∈ R n r denotes the uncertain change in heat fluxes due tochange in power consumption of the heating/cooling devices during reserve provi-sion, where n r ≤ n u is the number of actuators that are used for reserve provision. R b consists of the columns of B bt that correspond to the heating/cooling actuatorsthat provide reserves. The system’s output equation is y bt = C b x bt + D bt u bt + F b v bt ,where y bt ∈ R n y denotes the room temperature in o C and illuminance in lux.Denote by r b,je,t the (electric) reserve capacity of actuator j of building b at timestep t . Buildings can provide up-reserves by decreasing their consumption, and down-reserves by increasing it . For now, we assume symmetric reserve capacities;asymmetric reserves will be discussed in Section 4. Since the HVAC control inputfor heating and cooling is defined as a heat flux, it is convenient to define also the“thermal” reserve capacity r b,jt that has W/m units. r b,je,t can be obtained from r b,jt by division with the coefficient of performance (COP). For notational convenience,we use the variable r b,jt in the problem formulations and call it simply reservecapacity keeping in mind that it is actually the “thermal” reserve capacity. In thepro-rata activation case, the reserve energy is proportional to the reserve capacitybased on a normalized SFC signal w t ∈ [ − , . Note that the reserve capacity r b,jt is a decision variable for the buildings, whereas the normalized SFC signal w t isuncertain. The primitive uncertainty w t results in an uncertain change in electric The (electric) reserve capacity is the amount of SFC reserves that becomes available to the TSO[30]. In the context of frequency regulation, the term up-reserves denotes increase of a generator’sproduction or decrease of a load’s consumption to increase system frequency. Similarly, the termdown-reserves denotes decrease of a generator’s production or increase of a load’s consumption todecrease system frequency. j , ∆ u b,je,t = r b,je,t w t and the corresponding uncertainchange in heat flux ∆ u b,jt = r b,jt w t .Denote by x bt + k | t ∈ R n x the predicted state of building b for time t + k attime t . The predicted states at time t along a prediction horizon N are assembledin one vector as x bt = [ x bt | t x bt + k | t ... x bt + N | t ] ⊤ ∈ R n x ( N +1) . Adopting the samenotation for inputs and disturbances, the building dynamics along N can be writtenas x bt = A b x b + B bt u bt + E b v bt + R b ∆ u bt and y bt = C b x bt + D bt u bt + F b v bt , wherethe matrices A b , B bt , E b , R b , C b , D bt , and F b are of appropriate dimensions. Theconstraints on outputs (thermal comfort zone) and HVAC control inputs along N are y b min ≤ y bt ≤ y b max , u b min ≤ u bt + H b ∆ u bt ≤ u b max , (2)where H b ∈ R n u × n r has or as entries. By substituting the dynamics in (2),the constraints can be written in terms of the control inputs and uncertainty as G b u bt + S b ∆ u bt ≤ Q b , where the matrices G b , S b , and Q b are defined as G b = G bp I N − G bp − I N , S b = S bp H b − S bp − H b , Q b = y b max − Q bp u b max Q bp − y b min − u b min , (3) G bp = C b B bt + D bt , S bp = C b R b , (4) Q bp = C b ( A b x b + E b v bt ) + F b v bt , (5)and I N is the N-dimensional identity matrix. Using G b , S b , and Q b , the HVACinput and comfort zone constraints along the prediction horizon can be representedcompactly in the optimization problems of Section 4.We are interested in building aggregations for two main reasons. First, thereserve markets typically have requirements on the minimum size of the biddenreserve capacity, which is typically in the range [1 , MW, as mentioned in Sec-tion 2. In most cases, individual commercial buildings cannot meet these minimumbid size requirements, so building aggregations are needed to enable participationin the reserve market. Even in markets with low minimum bid size requirements,e.g., in the range of a few hundred kWs, building aggregations would still be ofinterest in presence of other typical requirements, such as symmetry and minimumduration of the bidden reserve capacity. As shown in [26], aggregating buildingswith different characteristics results in a larger total reserve capacity comparedwith the case where each building participates individually in the market. The sec-ond argument in favor of building aggregations is more practical. An aggregator’sjob would be to determine the reserve capacity, bid it in the reserve market, andinteract with the TSO during reserve activation and for the financial settlement.These tasks are very different to the normal activities of a building manager; there-fore, the aggregator could take over this burden that would otherwise be with thebuilding manager. 7onsider an aggregation of L buildings, i.e., b = { . . . L } . Denote by x t = (cid:2) x t x t ... x Lt (cid:3) ⊤ ∈ R n x L ( N +1) the vector containing all predicted states of allbuildings along N . Using the same notation for inputs and disturbances, the in-put/output constraints of the aggregation can be written as Gu t + S∆ u t ≤ Q ,where G , S , and Q are block diagonal matrices with G b , S b , and Q b on the di-agonal, respectively. Denoting by w t ∈ [ − , N the SFC signal along N , and by r t ∈ R n r LN a collection of the reserve capacities of all actuators and all buildingsalong N , the input/output constraints of the aggregation can be written as ˜ R = R , . . . R , R , . . . ... R N, R N, . . . R N,N , Gu t + S ˜ Rw t ≤ Q , (6)where ∆u t = ˜ Rw t , ˜ R ∈ R ( n r LN ) × N is the reserve capacity matrix , and R i,k ∈ R n r L is a column vector. The diagonal vectors of ˜ R are the actuators’ reservecapacities for every time step t , i.e., R t,t = [ r t . . . r n r Lt ] ⊤ . This lower triangularstructure satisfies causality and allows us to model the effect of past SFC signals.For example, R t,t − accounts for the signal at time t − to determine the reserveat time t . However, to comply with today’s practice we fix R i,k = 0 for i = k inthe following, i.e., the reserve energy depends only on the capacity and the currentSFC signal. Thus, ˜ R is a block diagonal version of r t . In pro-rata activation [28], the reserves are requested by the TSO via a normalizedfrequency signal w t ∈ [ − , . This box constraint represents the power constraintsof the signal. The values w t = − and w t = 1 indicate full activation of up- anddown-reserves, respectively. The uncertainty set along N can be written as W = { w t ∈ R N : || w t || ∞ ≤ } . (7) To facilitate the participation of energy-constrained resources (thermal loads andstorages) in AS, the frequency signals can be distinguished by their energy content.A building aggregation could then choose the preferred product to offer reserves.Energy constraints can be cast as linear constraints on the mean value of the signalas − εT ≤ X t + T − k = t w k ≤ εT , (8)8here ε is the bias coefficient, T is the averaging period, and both are fixed by theTSO for a particular product. Equation (8) implies that the bias of the signal over T is bounded. If we stack (8) along N , we get the polyhedral constraint on theuncertainty A w w t ≤ b w , where A w is a matrix with entries − , or , and b w is a vector with entries εT . The power constraints (7) are still present, since thefull reserve capacity could be requested anytime. Denote by I N the N-dimensionalidentity matrix, and by N the N-dimensional vector with ones. Defining ¯ A w =[ A w ; I N ; − I N ] and ¯ b w = [ b w ; N ; N ] , the uncertainty set for PEC is W = { w t ∈ R N : ¯ A w w t ≤ ¯ b w } . (9) Apart from the SFC signal, in general, there are two other sources of uncertaintyassociated with our problem: (a) weather and occupancy uncertainties, i.e., devia-tions from the predicted values, and (b) electricity and reserve price uncertainties.For simplicity, weather/occupancy uncertainties are not considered in this paper,i.e., the predictions are assumed to be perfect. There exist several approaches in theliterature of building climate control to handle these uncertainties such as stochas-tic MPC with additive Gaussian noise [29], scenario-based MPC [31], or simplyconstraint tightening, which can be relatively easily integrated in our framework.We assume that the building aggregation acquires energy from the retail market,and since the utility company tariffs are typically constant over long periods, thereis no electricity price uncertainty in our case. However, the situation is differentfor the reserve prices. Although the reserve bid price is selected by the aggregator,there is uncertainty involved because the bid might not be accepted in the auction.Incorporating this source of uncertainty requires modeling the market clearing pro-cess and is beyond the scope of this paper, but it may well constitute an interestingtopic for future research.
In this section, we present the hierarchical control scheme for frequency reserveprovision by commercial building aggregations, which is graphically shown inFig. 1. Level (Lv ), the aggregator scheduling , is performed on a daily basiscentrally by an aggregator. There are two conflicting objectives in the problem:(a) minimize energy consumption through energy efficient control, and (b) deviatefrom the energy-optimal solution to leave slack for reserve provision. The optimaltradeoff between these conflicting objectives depends on the monetary incentivesfor reserves, which we call capacity payments, and the cost of electricity. Lv de-termines the reserve capacity, and its allocation among buildings, which achieve9 v1:AggregatorUtilitycompanyReservemarketTSO PlantLv2: MPCLv3: Feedbackcontrol Reserve scheduling & allocation HVAC control &SFC tracking
Electricity tariffStates, predictions, constraintsReserve productcharacteristicsUp- & down-reserves Up- & down-reservesSFC signal HVACPower flow Building b set-pointFiltering Baseline Figure 1: Overview of the proposed hierarchical control scheme.the optimal tradeoff while respecting occupants’ comfort in case the reserve is re-quested. The aggregator solves a robust optimization problem using building statemeasurements, prices of electricity and reserves, as well as predictions of weatherand occupancy. We formulate the robust problem first for PC only and second forPEC. Reliable provision of frequency reserves is critical for power system security,and so TSOs typically require availability of the reserve capacity [28]. Forthis reason, HVAC input constraints are truly hard constraints and robust optimiza-tion is a natural approach to follow.Level (Lv ), the building HVAC control , is a robust MPC [32] that determinesthe energy optimal HVAC control inputs every minutes locally at each building,and leaves enough slack for reserves. Level (Lv ), the frequency signal filteringand tracking , is a feedback controller that tracks the SFC signal by controllingthe power consumption in real-time, e.g., every seconds. In Fig. 1, signals andcontrol actions with thick/solid curves are real-time, the ones with thin/solid curvesare executed every minutes, and the ones with dashed curves are executed oncea day. : Aggregator Scheduling The aggregator’s goal is to determine the optimal amount of reserves ˜ R ∗ to beoffered in the market. Denote by c t ∈ R n u LN and k t ∈ R n r LN the electricitycost and reserve capacity payment vectors, respectively, where N is the predictionhorizon. Note that efficiency factors incorporating actuators’ COP and buildingdistribution system losses are included in c t and k t . Lv can be cast as the robust10P ( u ∗ t , ˜ R ∗ ) := arg min c ⊤ t u t − k ⊤ t ˜ R1 N (10a)s.t. max w t ∈ W ( Gu t + S ˜ Rw t ) ≤ Q (10b) M ˜ R1 N = , (10c)where W is the uncertainty set, i.e., W ∈ { W , W } . Equation (10b) requiresthat input and output constraints are satisfied even in the worst case of uncertaintyrealization. By appropriately selecting M in (10c), we can impose constant reservecapacities over a period of time and/or the block diagonal structure on ˜ R discussedin Section 3.1. The building dynamics in (10b) are decoupled among buildings;however, the coupling comes via (10c) and the objective function.Denote by X ( j ) the j th row of any matrix or vector X . We derive the robustcounterpart of (10) for PC and PEC. Consider the j th row of (10b) max w t ( G ( j ) u t + S ( j ) ˜ Rw t ) ≤ Q ( j ) . The term S ( j ) ˜ Rw t is a scalar and can be written as w ⊤ t F ( j ) ˜ r ,where ˜ r ∈ R n r LN is a column-wise vectorized version of ˜ R and F ( j ) = ( I N ⊗ S ⊤ ( j ) ) ⊤ ∈ R N × ( n r LN ) , where ⊗ is the Kronecker product. Thus, the j th row of(10b) is equivalent to max w t ∈ W ( G ( j ) u t + w ⊤ t F ( j ) ˜ r ) ≤ Q ( j ) . (11) In the presence of PC only, the uncertainty set W is given by (7), i.e., w t isconstrained in an ∞ -norm ball. In this case, we can maximize the left-hand sideof (11) analytically using the dual of ∞ -norm, i.e., the -norm [33]. Followingthis procedure, the deterministic equivalent of (11) is G ( j ) u t + || F ( j ) ˜ r || ≤ Q ( j ) .Repeating this procedure for all rows of (10b), the robust counterpart problem of(10) can be written as ( u ∗ t , ˜ r ∗ ) := arg min c ⊤ t u t − ˜ k ⊤ t ˜ r (12a)s.t. G ( j ) u t + || F ( j ) ˜ r || ≤ Q ( j ) ∀ j (12b) ˜ M ˜ r = , (12c)where ˜ k t ∈ R n r LN and ˜ M are defined by ˜ k ⊤ t ˜ r = k ⊤ t ˜ R1 N and ˜ M ˜ r = M ˜ R1 N .Problem (12) is an LP, but it grows quadratically in N since we need N auxiliaryvariables and N additional constraints for each uncertain constraint (11) to modelthe -norm. A similar reformulation for a general class of linear systems withreserve demands can be found in [34].We now consider the case where either only heating or only cooling actua-tors provide reserves in each building, which is likely in practice to avoid energydumping by simultaneous heating and cooling. Recall from (3), (4) that S is a11lock diagonal collection of S b = [ C b R b ; H b ; − C b R b ; − H b ] . If only heating(cooling) actuators are used for reserve provision, then all entries of R b are non-negative (non-positive). Additionally, all entries of C b and H b are non-negative byconstruction. Therefore, every row of S contains either only non-negative or onlynon-positive entries and, by construction, the same holds for all entries of F ( j ) , i.e., [ F ( j ) ( i, k ) ≥ ∀ i, k ] or [ F ( j ) ( i, k ) ≤ ∀ i, k ] ∀ j . Based on the definitions of F ( j ) , S ( j ) , ˜ r , and r t , and recalling that ˜ r is non-negative, || F ( j ) ˜ r || can be equivalentlywritten as the linear term || F ( j ) ˜ r || = X N i =1 X n el k =1 | F ( j ) ( i, k )˜ r ( k ) | = (13) X N i =1 X n el k =1 | F ( j ) ( i, k ) | ˜ r ( k ) = N | F ( j ) | ⋆ ˜ r = | S ( j ) | ⋆ r t , where n el = n r LN and | · | ⋆ denotes the element-wise absolute value operator. Inthis case, the more general formulation (12) can be simplified to the following LPthat has the same size as (10), and so can be solved efficiently . ( u ∗ t , r ∗ t ) := arg min c ⊤ t u t − k ⊤ t r t (14a)s.t. G ( j ) u t + | S ( j ) | ⋆ r t ≤ Q ( j ) ∀ j (14b) Mr t = . (14c) With PEC, the uncertainty set W is a polyhedron and duality theory can be appliedto derive the robust counterpart problem [33, 35]. We write constraint (11) as anoptimization problem over w t max w t ( G ( j ) u t + w ⊤ t F ( j ) ˜ r ) ≤ Q ( j ) (15a)s.t. ¯ A w w t ≤ ¯ b w . (15b)By deriving the dual problem of (15) for all j , we obtain the following robustcounterpart of (10) ( u ∗ t , ˜ r ∗ , λ ∗ ( j ) ) := arg min c ⊤ t u t − ˜ k ⊤ t ˜ r (16a)s.t. ¯ b ⊤ w λ ( j ) + G ( j ) u t − Q ( j ) ≤ ∀ j (16b) ¯ A ⊤ w λ ( j ) = F ( j ) ˜ r ∀ j (16c) λ ( j ) ≥ ∀ j, (16d) ˜ M ˜ r = , (16e) Note that (14) is a special case of (12), and it is equivalent to formulation (13)-(15) in ourprevious work [26]. λ ( j ) is the vector of dual variables. Problem (16) is an LP, but its sizegrows quadratically in N . Although this dualization technique holds also for PC,we apply analytic maximization in that case since it results in fewer variables andconstraints. Remark:
Asymmetric reserves can be modeled defining ˜ R + , w + t ∈ [0 , N for down- and ˜ R − , w − t ∈ [ − , N for up-reserves. In case of PC only, theuncertainty set remains a polyhedron and a tractable robust counterpart problemcan be derived using analytic maximization. The reader is referred to [26] fordetailed formulations. In case of PEC, the non-linear constraint w − t w + t = 0 isneeded to ensure that up- and down- reserves are not requested simultaneously.Therefore, the uncertainty set is not a polyhedron any more and the dualizationtechnique cannot be applied in this case to derive the robust counterpart problem. : Building HVAC Control Given the optimal reserve allocation from Lv , in Lv the HVAC control inputsare determined every minutes by the robust MPC with prediction horizon N u b, ∗ t := arg min ( c bt ) ⊤ u bt (17a)s.t. max w t ∈ W (cid:16) G b u bt + S b ˜ R b, ∗ w t (cid:17) ≤ Q b , (17b)where c bt and ˜ R b, ∗ are the parts of c t and ˜ R ∗ , respectively, that are relevant forbuilding b . The first input of the optimal control sequence of (17) determines theLv setpoint of the HVAC system for the next minutes , u b, Lv t . Problem (17)formulates an MPC with open-loop predictions, i.e., the optimization is performedexplicitly over the control inputs u bt . MPC with closed-loop predictions, i.e., opti-mization over affine policies of the uncertainty, showed minor or zero performanceimprovement in this case, and so it is not used. In the following, we derive therobust counterparts of (17) for PC and PEC. In this case, the deterministic equivalent of (17) can be obtained by substituting(17b) with G b ( j ) u bt ≤ Q b ( j ) − || F b ( j ) ˜ r b, ∗ || ∀ j , where F b ( j ) is defined similarly to F ( j ) but for a single building, and ˜ r b, ∗ is a column-wise vectorized version of ˜ R b, ∗ .Note that ˜ r b, ∗ is fixed in Lv , and the right-hand side of the inequality is a constant. Although a consumption schedule is calculated in Lv , the MPC of Lv can reduce the costsdue to less uncertainty (recent reserve requests are known and better weather forecasts might beavailable) and, possibly, shorter optimization time steps. In case of plant-model mismatches, MPCadditionally reduces constraint violations due to its closed-loop operation. The MPC schedule isthe building’s baseline consumption, and is communicated to the aggregator. Since the baseline is aby-product of the formulation, baseline prediction methods that have proven to be hard [17] are notrequired. Furthermore, the predictive nature of MPC inherently accounts for rebound effects due toreserve provision when calculating future HVAC setpoints [9]. .3.2 Robust Counterpart for Power and Energy Constraints With PEC, the robust counterpart of (17) is ( u b, ∗ t , λ b, ∗ ( j ) ) := arg min ( c bt ) ⊤ u bt (18a)s.t. ¯ b ⊤ w,t λ b ( j ) + G b ( j ) u bt − Q b ( j ) ≤ ∀ j (18b) ¯ A ⊤ w,t λ b ( j ) = F b ( j ) ˜ r b, ∗ ∀ j, (18c) λ b ( j ) ≥ ∀ j . (18d)Problem (18) is similar to (16), but there are two main differences: first, ˜ r b, ∗ isfixed; and second, ¯ A w,t and ¯ b w,t are time-varying. For a time step t in the averag-ing interval [ t , t ] of length T , (8) can be written as − εT − w p ,t ≤ P t k = t w k ≤ εT − w p ,t , where w p ,t = P t − k = t w k is known because the uncertainty up to t − is realized. Thus, the coupling constraint on { w t , . . . , w t } depends on the energycontent of the SFC signal in the previous time steps of the averaging interval. : Frequency Signal Filtering and Tracking In Lv , the HVAC consumption is controlled around u b, Lv t for reserve provision.We consider water-based HVAC systems that are common in Europe, but the pro-posed reserve scheduling problem applies also to air-based systems [23]. Thepower consumption of water circulation pumps in water-based HVAC systems istypically small, and so we directly control the heating or cooling devices, e.g., heatpumps (HPs), to provide reserves. The desired power consumption of HP j ofbuilding b is u b,j, Lv t = u b,j, Lv t + ∆ u b,jt = u b,j, Lv t + w t r b,j, ∗ t , (19)where r b,j, ∗ t is fixed from Lv . In case of asymmetric reserves, r b,j, ∗ t is equal to thedown-reserve capacity if w t ≥ , and equal to the up-reserve capacity if w t < . Ina fast time scale, and depending on the HVAC system, the HP consumption can becontrolled by modifying either the water temperature setpoint at condenser’s outletor the refrigerant’s flow rate via valves. We rely on the second approach that wasexperimentally shown to be able to track fast reference signals, e.g., SFC signals,in [36]. The desired HP consumption u b,j, Lv t can be tracked using a feedback PIcontroller.With PC only, the normalized reserve request w t sent from the TSO is theoriginal SFC signal, whereas with PEC, w t is a filtered version of that signal.With reasonable modifications, the current operational paradigm at the TSO sidecan integrate multiple SFC signals with different energy contents. In the currentparadigm, there is a single SFC signal that is the output of a PI controller withthe area control error (ACE) as input. In the new paradigm, the TSO provides anumber of reserve products and each reserve provider chooses the product to offer14ts reserve capacity. Assume that the total reserve request (original SFC signal) attime step t is ˜ w t . The TSO will decompose ˜ w t into the desired number of signals(e.g., using a filter bank) and will sent the appropriate components to the providersdepending on the reserve product they offer reserves for. To simulate this processin this paper, we use a causal Chebyshev filter to get the energy-constrained signal w t sent to the building aggregation, but other filters might also be used. Similar fil-tering approaches have been used in previous works on power system applications,e.g., [37]. Note that although w t is a filtered signal, it does not mean that its bias iszero but instead that its bias is bounded. The filter’s transfer function is H ( z ) = P n f i =0 b i z − i P n f i =1 a i z − i , (20)where n f is the filter’s order, and a i , b i are its coefficients that depend on the pass-band edge frequency f c . From a TSO perspective, n f and f c can be chosen such thatthe resulting low-frequency component (LF signal) and high-frequency component(HF signal) have the desired ramping rates and energy contents. In this paper, wefix n f = 3 since it showed good performance in preliminary simulations.It is important to clarify how the SFC signal filtering is taken into account inthe higher levels of the control hierarchy. Recall that the filtered signal is trackedevery few seconds in Lv , whereas the decisions in Lv (reserve scheduling) andLv (determination of optimal building setpoints) are taken every minutes. Dueto this time scale separation, the important information from the filtering of Lv that needs to be conveyed to Lv and Lv is only the integral of the SFC signalover this period, i.e., the bias of the signal. This is needed for example in Lv to schedule the reserve capacities without violating the buildings’ thermal comfortconstraints. By formulating the energy constraint of the SFC signal as the linearinequalities (8), we can directly account for the signal’s bias in the optimizationproblems of Lv and Lv in a tractable way. Note that the bias coefficients ε fordifferent averaging periods T in (8) can be empirically obtained from the filter (20).To do so, one can simply apply the filter on historical data of SFC signals usingdifferent f c = 1 /T .Due to the robust design, any admissible reserve request w t that satisfies (8)will not lead to comfort zone or input constraint violations, provided that thereis no significant plant-model mismatch. The tracking quality of w t depends onHP’s mechanical delays and dead-times, ramping limits, and minimum down-timesand/or run-times. If such dynamics are significant, tests similar to the ones in[36] must be performed to identify upper limits on the frequency content of w t that result in good tracking by the HP. In this case, the building aggregation couldform a coalition with faster resources, e.g., an aggregation of residential electricwater heaters, exclude very high frequency components of the SFC signal using anappropriate band-pass filter, and send them to the faster resources. Since the focusof this paper is on the robust reserve scheduling and allocation side, we assumethat such HP dynamics are negligible, i.e., the reference u b,j, Lv t (and so the SFC15able 1: Bias Coefficients ( ε ) for the SFC Signal and its High-frequency Compo-nent (HF) for Different Averaging Periods ( T ) T h h h h h hSFC .
000 0 .
989 0 .
952 0 .
796 0 .
592 0 . HF .
528 0 .
467 0 .
337 0 .
273 0 .
384 0 . SFC .
927 0 .
781 0 .
674 0 .
624 0 .
553 0 . HF .
382 0 .
300 0 .
317 0 .
290 0 .
237 0 . signal) can be perfectly tracked in our simulations if the comfort zone and inputconstraints are satisfied. We investigate the performance of the proposed control framework in simulationswith an aggregation of typical Swiss office buildings. We consider two HVACsystems: in system A, heating is performed via radiators with coefficient of per-formance (COP) equal to , whereas cooling with cooled ceilings (COP = 3 . );in system B, both heating and cooling are performed using thermally activatedbuilding systems (TABS) with COP = 3 . . We also differentiate between heavy(eh) or light (el) building envelope, high (wh) or low (wl) window area fraction,and high (gh) or low (gl) internal gains. In our simulations, we consider an ag-gregation of large buildings ( m each) with the following characteristics:A = { A,eh,wh,gh } , A = { A,eh,wl,gl } , A = { A,el,wl,gl } , B = { B,eh,wh,gh } ,B = { B,eh,wl,gl } , and B = { B,el,wl,gl } . Typical occupancy profiles were used,whereas weather data were provided by Meteoswiss (the Swiss federal office ofmeteorology and climatology). More information regarding the buildings can befound in [26, 29]. Because of the thermal inertia, heating/cooling actuators can beused to provide frequency reserves. In buildings with the HVAC systems consid-ered here, the fresh air flow rate is usually kept constant because changes wouldbe immediately realized by the occupants. For this reason, we do not use venti-lation for reserve provision. The temperature comfort zone during working hoursis [21 , o C in winter and [22 , o C in summer, which is in accordance with theASHRAE 55-2013 standard [38]. During non-working hours and weekends, thecomfort zone is relaxed to [12 , o C in both seasons. The optimizations are per-formed with a time step of minutes, which is the discretization step of buildingmodels (1), and the prediction horizons of Lv and Lv are fixed to N = 96 ( days) and N = 48 ( day), respectively. We assume symmetric, daily reserves,i.e., constant reserve capacity over a day, and capacity payments higher thanthe electricity price, i.e., k = 1 . c . 16 .2 Parameters of Energy Constraints To apply energy constraints as in (8), we determine appropriate pairs of averagingperiod and bias coefficient ( T, ε ) based on the historical normalized SFC signals w k ∈ [ − , from the Swiss control area for and . We consider sixdifferent averaging periods, T = { , , , , , } hours, and calculate six sets offilter parameters in (20), one for each of the pass-band edge frequencies f c = 1 /T .For each value of T , we filter the historical SFC signals using the correspondingfilter (20) to get the HF signals for and . For each of the four signalsand for each value of T , we calculate ε as the largest absolute mean value of theSFC or HF signal over any period T , i.e., ε = max( | (1 /T ) · P Tk =1 w k | ) . Theresults are summarized in Table 1. Notice that the original SFC signals can besignificantly biased over periods of several hours. Note also that the biases ofthe HF signals are significantly lower than those of the original SFC signals, andthat the signals in are generally less biased than in . This is becausefrom March the ACE of Switzerland is netted with that of other Europeancountries before generating the SFC signal. We use the HF signal of andselect ( T = 2 h , ε = 0 . for the simulations of this section, according to Table 1. We present simulation results for typical weeks in winter and summer. In Fig. 2,we show a -hour extract of the SFC, LF, and HF signals for a sample day, whereasin Fig. 3 we show the -hour moving averages of the signals for the whole day.Although the original SFC signal is mostly negative, its bias is absorbed by the LFsignal, and so the HF signal is approximately zero-mean. The bias ε of the SFCsignal is larger than . , whereas the energy-constrained HF signal has a bias lessthan . .Figure 4 shows the optimal total reserve capacity of the aggregation and itsallocation among buildings for the winter week. The capacity is constant for eachday and ranges from approximately kW on Friday to kW on Saturday withan average weekly value of kW. Note that the capacity is shifted among build-ings in a way that maximizes the total capacity of the aggregation. Interestingly, thebuildings offer higher reserve capacities when they normally consume less power.For example, buildings with system A contribute mainly at night, whereas build-ings with system B offer more reserves during working hours, because they preferto preheat at night. The scheduled total energy consumption of the aggregationis spread throughout the whole week to maximize the reserve potential. Duringthe weekend, the buildings are not occupied and the comfort zone is larger, whichresults in higher reserve capacities compared to working days.The optimal operation of building A based on Lv and Lv is presented inFigs. 5 and 6. Figure 5 shows the HVAC system heating power and Fig. 6 theresulting temperature trajectories. The black-solid curves indicate the scheduledconsumption and temperature by the MPC, whereas the grey-dashed curves corre-17 S i gna l ( − ) Figure 2: A -hour extract of the original SFC signal, its low-frequency (LF), andits high-frequency (HF) components for a sample day. S i gna l m o v i ng a v e r age ( − ) SFC LF signal HF signal
Figure 3: The moving averages of the signals of Fig. 2 for the whole sample day.spond to the final values after tracking the HF signal. The grey envelope showsthe problem’s robust region, which is defined as the region of the thermal comfortzone with the following property. If the temperature is within the robust regionat time step t = 0 , then for any SFC signal with the appropriate energy contentthere exists a feasible control input trajectory that satisfies the thermal comfort andHVAC input constraints (the black-dashed curves in both figures) along the pre-diction horizon t ∈ [0 , N − . In our simulations, the HVAC inputs and roomtemperature stay always within the robust region.In summer, the reserve potential from cooling actuators is also significant. Dur-ing the considered week, the capacities range from approximately kW on Sun-day to kW on Saturday with an average weekly value of kW. We observeda more uniform distribution of reserves among different buildings in summer, com-pared to winter, because cooled ceilings and TABS have similar time constants.Due to space limitations, plots similar to Figs. 4, 5 and 6 are omitted.Based on our results, the considered buildings with floor area m ,average rated heating power W/m , and average rated cooling power W/m ime (h) R e s e r v e c apa c i t y ( k W ) Figure 4: Optimal reserve allocation among buildings (winter week). H VA C c on s u m p t i on ( W / m ) Robust region Lv2 Lv3 Input constraints
Figure 5: HVAC heating power for building A (winter week). R oo m t e m pe r a t u r e ( o C ) Lv2 Lv3Robust regionComfort zone
Figure 6: Room temperature for building A (winter week).provide a reserve capacity equal to kW and kW on average in winter andsummer, respectively. Using simple linear extrapolation, we find that the requiredminimum reserve capacity of MW in Switzerland can be provided by (similar)buildings in winter and buildings in summer, i.e., approximately buildingsin both seasons. Note that this is only a rough estimate as: (a) it is based onthe average reserve capacity values; and (b) the extrapolation might overestimatethe required number of buildings to meet the MW limit, because the larger theaggregation the more flexibility exists in allocating reserves among buildings.
To maximize the reserve potential, the buildings operate in a less energy efficientway. For example, during the considered winter week reserve provision resultedin a consumption higher than that of energy efficient building control, i.e.,19n MPC with objective to minimize electricity cost without offering reserves. Asshown in Fig. 5, in order to provide reserves in both directions, the buildings tryto operate close to the middle of the heating/cooling device’s power range. This isin contrast to an energy efficient operation, where the power consumption wouldbe as close as possible to the minimum value. However, the increase in energyconsumption does not mean that the building control is suboptimal. For the givenelectricity price and capacity payment, this building operation minimizes the totalcost defined as the sum of electricity cost and reserve profit.It is important to note that the consumption increase is in comparisonwith an MPC-based energy efficient building control. However, most buildings to-day operate with supervisory rule-based controllers, i.e., not optimal controllers.Therefore, if the proposed methods are applied to such buildings, the observed in-crease in consumption is expected to be less than the reported value. Althoughenergy efficiency is usually the goal in building control, increasing the energy con-sumption is not necessarily a drawback if this helps accommodate more RES inthe power system, and at the same time the additional electric energy is stored asthermal energy in the buildings.
As mentioned in Section 3.3, the formulations and simulations in this paper as-sumed perfect weather and occupancy predictions. The robust formulation guaran-tees satisfaction of the HVAC and comfort constraints for any admissible reserverequest and perfect weather and occupancy predictions, but it cannot provide math-ematical guarantees on the worst case temperature deviations in case of imperfectpredictions. Empirical guarantees could be provided by Monte Carlo simulations:(i) schedule the reserve capacities using imperfect disturbance predictions; (ii) sim-ulate the building operation under disturbance uncertainty, i.e., the weather and oc-cupancy realizations are different to the predictions; and (iii) analyze the results tokeep track of the number and magnitude of temperature deviations. Alternatively,probabilistic guarantees could be obtained by modeling the weather and occupancyuncertainty via scenarios and then robustifying against the reserve uncertainty sep-arately for each scenario. This is an exciting research direction for future work.However, some intuition can be provided without following any of the abovetwo approaches. Consider a building in energy efficient operation using a determin-istic MPC that relies on an imperfect weather forecast. As explained in Section 5.4,the building would operate close to the minimum power consumption, and thus thetemperature trajectory would stay close to one of the comfort zone boundaries.Now consider the same building operated under reserve provision. As shown inFig. 6, the building would operate closer to the middle of the comfort zone to max-imize the reserve capacity. For this reason, we expect the reserve provision caseto result in smaller and perhaps less frequent comfort zone violations compared tothe energy efficient control case, if the building is exposed to the same weather andoccupancy uncertainty in both cases. 20
Sensitivity Analysis
As explained in Section 4.1, the proposed reserve scheduling methods identifythe optimal tradeoff between minimizing energy consumption and leaving enoughslack for reserve provision. The buildings would not deviate from the energy ef-ficient control and would not offer any reserves if the additional electricity costoccurring due to this deviation were higher than the reward received for the slackprovided as reserve capacity. In principle, the amount of reserves depends on therelationship (ratio) between the capacity payment k , i.e., the remuneration for eachkW of reserve capacity provided, and the electricity price c , i.e., the cost for eachkWh of electricity consumed. In this Section, we investigate this relationship byrunning simulations, similar to the ones in Section 5.3, over -week periods inwinter and summer and for various k/c ratios. The total reserve capacity for eachcase, i.e., the sum of the capacities of each day of the -week period, is presentedin Fig. 7, where the left plot is for ratios k/c > and the right plot is for ratios k/c ≤ . These plots represent the aggregation’s bid curves because they commu-nicate how much capacity the aggregation is willing to bid in the reserve marketdepending on the payment it receives for each kW of the capacity.For k/c > (left plot - Fig. 7a) the simulations are performed for winter andsummer with and without consideration of energy-constrained SFC signals: theblack curves correspond to winter weeks (“win”) and the grey curves to summerweeks (“sum”), whereas the dashed curves are for PC and the solid curves forPEC. Our simulations show that with the same financial incentive and for bothseasons, the buildings are willing to offer up to more reserves compared withPC if energy constraints are considered. Note that the gap between PC and PECis generally larger for lower k/c , particularly in winter. In winter, the capacitysaturates at its maximum value at k/c = 1 . , whereas in summer it increasesmonotonically as the ratio increases up to .The analysis of Fig. 7a focused on k/c > , which is a necessary conditionfor reserve provision with PC. This observation was also made in our previouswork [26]; however, here we provide an explanation by studying the structure ofproblem (12). If k < c , the optimal solution is ( u ∗ t , ˜ r ∗ ) = ( u min t , ) , where u min t is the energy optimal scheduling. If k = c , the optimal cost is and any solutionwithin the feasible range of ˜ r will be optimal. If k > c , the optimal solutionis (˜ r max , ˜ r max ) , where ˜ r max is the upper limit of ˜ r , and the optimal cost will be ( c ⊤ t − k ⊤ t )˜ r max < , i.e., the aggregation earns profit. The limit ˜ r max dependson input/output constraints, and so different solutions are obtained for different k/c > ratios, as shown in Fig. 7a. In case of daily reserves, k/c > neednot to be satisfied point-wise throughout the whole day; instead, k/c < can bechosen during daytime, and k/c > at night when electricity prices might belower. Reserve provision will be triggered if || k || / || c || > , i.e., the capacitypayment is on average higher than the electricity price.21n the other hand, with energy-constrained SFC signals (PEC) reserves canbe provided also with ratios k/c < throughout the whole day. It is easier toexplain this with an example. Assume that the buildings have declared a capacity r ( t ) for day d . Assume also that up regulation (i.e., consumption decrease) ismainly requested during day d . If the signal is energy constrained, only a fractionof the worst case reserve energy R N − t =0 r ( t ) dt will be requested as consumptiondecrease from the buildings. The rest part of R N − t =0 r ( t ) dt will be stored as thermalenergy in the buildings and will reduce the required heating/cooling energy (andthe respective costs) during day d + 1 . For this reason, the buildings are willing toprovide reserves even if k/c < . We present simulation results for PEC in winterand summer in Fig. 7b. The threshold ratio for reserve provision is k/c = 0 . forboth winter and summer, and it depends on ε and T . The capacity increases slowlyin the ratio range [0 . − . , particularly in summer, and then it suddenly jumpsto higher values as k/c approaches to . No results are shown for PC in Fig. 7bbecause the capacity is zero for k/c < , as explained before.Fig. 7a shows that for unconstrained SFC signals (PC) a ratio k/c = 1 . ,which means a capacity payment higher than the electricity price, taps most ofthe reserve potential. In particular, further increasing the ratio up to the maximumconsidered value k/c = 2 increases the reserves only by in winter and no morethan in summer. Assuming an average electricity price of . CHF/MWh,which is the case for consumers who consume more than MWh/year in Zurich,with a ratio k/c = 1 . capacity payments around CHF/MW/h are needed.This is significantly lower than the most expensive accepted bids, but approxi-mately times higher than the average capacity payment in [39]. Fig. 7bshows that energy-constrained SFC signals can reduce the necessary capacity pay-ments down to of the retail price ( k/c = 0 . ), but of course with a largereduction in the reserve capacity. Thus, reserves are actually costly for buildingsalready equipped with MPC for energy efficient (optimal) control, especially if theSFC signal is not energy-constrained. This is in contrast to TCLs with simple hys-teresis control based on a deadband, where reserves can be provided at a lowercost. However, note that our calculations are based on the prevailing case wherethe buildings acquire energy in the retail electricity market. In another market set-ting where the buildings acquired energy directly in the spot market, the buildingscould offer reserves at more competitive prices because the retail electricity pricesare typically significantly higher than the wholesale spot electricity prices. Theanalysis of this section provides intuition on the relationship between the amountof reserves from building aggregations and the capacity payments. In practice, esti-mating the capacity payment is a challenging task that needs to consider additionalcosts, e.g., due to device wear, but also the competition, i.e., the expected bid pricesof generators and/or other load aggregations.22 Capacity payment ratio (k/c) T o t a l r e s e r v e c apa c i t y ( M W ) PC, win PEC, win PC, sum PEC, sum a T o t a l r e s e r v e c apa c i t y ( M W ) PEC, winterPEC, summer b
Figure 7: Bid curves of building aggregation in winter and summer. Left: for PCand PEC, and ratios k/c > . Right: for PEC and ratios k/c ≤ . In this section, we fix k/c = 1 . and investigate the influence of important reserveproduct characteristics on reserve capacities. For PC, we consider reserves withdaily or hourly duration, and symmetric or asymmetric capacities. For PEC, weinvestigate reserves with daily or hourly duration, as well as different ( T , ε ) pairs,based on Table 1. We summarize simulation results with respect to the total reservecapacity for two weeks in winter and summer in Tables 2 and 3, where positivevalues denote up- and negative values down-reserves.For PC and symmetric products , hourly reserves increase capacities by . in winter and . in summer, compared to daily reserves. If asymmetric reservesare allowed, the aggregation provides significantly more down- than up-reservesin summer, whereas in winter no up-reserves are provided at all. Down-reserves(increasing power consumption) are preferable for buildings equipped with MPCfor energy efficient control because the capacity can be offered without increasedbaseline consumption. The energy efficient control tries to stay as close as possibleto the minimum power consumption of the heating/cooling device. In order toprovide up-reserves (consumption decrease), a building needs to be able to reduceits consumption without violating occupant comfort. Therefore, the building mustschedule its operation (baseline) at a power level higher than the energy optimal,which increases energy costs. On the other hand, down-reserves can be providedwhile operating at the energy efficient trajectory because the consumption can onlyincrease will tracking the SFC signal.Compared to PC, PEC increase the reserves for all pairs ( T , ε ) up to . inwinter and . in summer. For given T and ε , adopting hourly instead of dailyreserves increases the capacities up to in winter and . in summer. Notethat the increase is higher for large T . Notice that shorter T are characterizedby smaller εT products, and so constraint (8) on the uncertainty becomes tighter.Therefore, one would intuitively expect that decreasing T increases the reservecapacities monotonically. However, one has to consider that longer T couple moreoptimization periods, which is why a monotonic behavior is not observed in our Note that the results of Table 2 are not directly comparable with the values reported in Tables 1and 2 of [26], because different COP values have been used in the simulations of the two papers. ± . ± . daily asymmetric +0 / − . . / − . hourly symmetric ± . ± . hourly asymmetric +0 / − . . / − . Table 3: Capacity (MW) of Reserve Products (Power & Energy Constr.) T (h) ε (-) εT (h) day/win hour/win day/sum hour/sum .
382 0 . ± . ± . ± . ± .
62 0 .
300 0 . ± . ± . ± . ± .
34 0 .
317 1 . ± . ± . ± . ± .
56 0 .
290 1 . ± . ± . ± . ± .
68 0 .
237 1 . ± . ± . ± . ± .
512 0 .
203 2 . ± . ± . ± . ± . simulation results. To gain deeper insight into the dependence of reserve capacitieson T and ε , we run simulations for two weeks in winter and summer with T ∈{ , , , , , } hours and ε varying from . to . with . increments, i.e., combinations for each season. We assume daily, symmetric reserves and simulateonly Lv , i.e., the reserve scheduling problem, whereas the building HVAC controlis not considered. The reason is that many of the combinations of T and ε arenot achievable by the Chebyshev filter (20), and so Lv will likely be infeasible.However, our analysis provides intuition on the effect of T and ε that could beuseful for filter design in a practical application.In Fig. 8(a), we show the total reserve capacities for each of the simulatedcases in winter. Similar results are obtained for summer, but are omitted here dueto space limitations. The dependence of reserve capacities on T and ε demonstratesa clear pattern: increasing any of the two parameters reduces the reserve capacity.Note that the capacity is more sensitive to ε than to T . To better illustrate this,we present the results of Fig. 8(a) based on the product εT in Fig. 8(b). As ex-pected, increasing εT generally decreases the reserve capacity. The same εT canbe obtained by different ( T , ε ) pairs: for example, both (8 h , . and (4 h , . obtain εT = 2 , but the first achieves a capacity . MW, whereas the secondachieves . MW. In this case, the capacity is more sensitive to ε than to T . Wecompared all cases in winter and summer with the same εT and found out that in of them smaller ε is preferable to smaller T . This means that buildings cancope easier with signals that are significantly energy constrained over long periodsthan signals that are moderately constrained over shorter periods.24 Concluding Remarks
In this paper, we proposed a new framework based on robust optimization and MPCfor scheduling and provision of secondary frequency control (SFC) reserves by theHVAC systems of commercial building aggregations . The framework incorporatestractable methods to account for energy-constrained SFC signals, and relies ondecentralized reserve provision to keep real-time communication requirements lowand preserve privacy. We also presented how the framework can be used to estimatethe SFC reserve potential from commercial buildings.Our analysis was based on four main assumptions: (a) there is no plant-modelmismatch; (b) the predictions of weather and occupancy are perfect; (c) all buildingstates can be measured; and (d) the reaction of heating/cooling devices is fast anddoes not cause any wear. Therefore, the reported results provide an upper boundon the amount of reserves from buildings. For a real implementation, additionalcare must be taken for (a)-(d): accounting for modeling and weather/occupancyprediction errors, use of state estimators, and modeling of the fast dynamics ofheating/cooling devices. If multi-zone building models with hundreds of states areavailable, model reduction techniques [10] can be used to reduce the optimizationproblem’s size. The reserve scheduling problem considered capacity payments, butneglected revenues from reserve energy utilization. Incorporating the latter in theframework is possible if SFC signal scenarios can be generated based on histori-cal data, and is expected to reduce the necessary capacity payments. Additionally,comfort constraints can be relaxed as chance constraints allowing comfort zoneviolations with a small probability, which is typical for building climate control,while keeping HVAC input constraints robust. Such combination of robust andchance-constrained optimization is likely to increase reserve capacities. In the fu-ture, we plan to include these aspects in the framework, implement and test it on areal building.Overall, our results show that significant amounts of SFC reserves can be reli-ably offered by an aggregation of ∼ commercial buildings without loss of oc-cupant comfort. We found that with traditional unconstrained SFC signals asym-metric reserves are preferable for buildings, that energy-constrained SFC signalsreduce the necessary capacity payments and increase reserves by up to ∼ com-pared to traditional SFC signals, and that reducing the duration of reserve productsfrom day to hour increases reserves by up to ∼ . References [1] Y. Makarov, C. Loutan, J. Ma, P. De Mello, Operational impacts of wind gen-eration on California power systems, IEEE Transactions on Power Systems24 (2) (2009) 1039–1050.[2] D. Callaway, I. Hiskens, Achieving controllability of electric loads, Proceed-ings of the IEEE 99 (1) (2011) 184–199.25 .1 0.2 0.3 0.4 0.5 1 2 4 6 8 12 ε (−) T ( h )
106 108 110 112a ε T (h) T o t a l r e s e r v e c apa c i t y ( M W ) WinterSummer (T, ε ,R) = (4 h, 0.5, 93.61 MW)(T, ε ,R) = (8 h, 0.25, 96.72 MW)b Figure 8: Left: sensitivity of reserve capacity (MW) on averaging period T andbias ε in winter. Right: capacity’s dependence on εT ∼∼