Distributed Multi-Building Coordination for Demand Response
Junyan Su, Yuning Jiang, Altug Bitlislioglu, Colin N. Jones, Boris Houska
aa r X i v : . [ m a t h . O C ] J a n Distributed Multi-Building Coordinationfor Demand Response ⋆ Junyan Su ∗ Yuning Jiang ∗ Altu˘g Bitlislio˘glu ∗∗ Colin N. Jones ∗∗ Boris Houska ∗∗ School of Information Science and Technology, ShanghaiTechUniversity, Shanghai, China(email: sujy, jiangyn, [email protected]). ∗∗ Automatic Control Laboratory, Ecole Polytechnique F´ed´erale deLausanne (EPFL), Lausanne, Switzerland(e-mail: [email protected]fl.ch, colin.jones@epfl.ch).
Abstract:
This paper presents a distributed optimization algorithm tailored for solving optimalcontrol problems arising in multi-building coordination. The buildings coordinated by a gridoperator, join a demand response program to balance the voltage surge by using an energycost defined criterion. In order to model the hierarchical structure of the building network,we formulate a distributed convex optimization problem with separable objectives and coupledaffine equality constraints. A variant of the Augmented Lagrangian based Alternating DirectionInexact Newton (ALADIN) method for solving the considered class of problems is then presentedalong with a convergence guarantee. To illustrate the effectiveness of the proposed method, wecompare it to the Alternating Direction Method of Multipliers (ADMM) by running both anALADIN and an ADMM based model predictive controller on a benchmark case study.
Keywords:
Distributed control, Smart power applications, Predictive control, Structuraloptimization1. INTRODUCTIONEnergy generation is undergoing a rapid transition fromfossil fuels to renewable sources (Liserre et al. (2010)),which poses a challenge to balance the unpredictable gen-eration demand due to the highly stochastic nature ofrenewable energy sources, and requiring advanced ancil-lary service providers. Recently, Demand Response (DR)programs utilizing the flexibility of power demand to pro-vide services have been considered in the power systemscommunity (Siano (2014)). These programs cover collec-tive load shifting, real time power regulation for loadbalancing and capacity firming, which has been appliedto mitigate the uncertainty in renewable power genera-tion effectively (Bitlislioglu (2018)). Because commercialbuildings, which are equipped with available heating, ven-tilation and air conditioning (HVAC) systems, have a po-tential to collectively offer ancillary services to the powergrid (Oldewurtel et al. (2012)). Smart grids connectingmultiple commercial buildings were developed recently inthe DR program to match the increasing power scale. Inthis setting, individual buildings are coupled via the gridoperator. This yields a coordination problem, which can beput in the generic framework of multi-agent optimizationand control (Bitlislioglu (2018)).Typically, in order to meet the real-time requirement,these multi-agent coordination problems are embedded ina Model Predictive Control (MPC) framework (Rawlingset al. (2017)), where the resulting problems have to be ⋆ The first two authors contributed equally. solved once during each sampling time, which requiresan efficient online solver. For this purpose, distributedalgorithms have already been developed (Bitlislio˘glu et al.(2017); Boyd et al. (2011); Braun et al. (2018)). A classof these approaches is based on decomposition methods,including primal and dual decomposition. In (Rantzer(2009); Richter et al. (2011)), gradient-based dual decom-position methods are used to solve the concave dual prob-lem. Alternatively, semi-smooth Newton methods (Fraschet al. (2015)) can be applied. In (Bitlislio˘glu et al. (2017)),an interior point method based on primal decompositionwas proposed, which writes all the inequality constraintsinto the objective by using a primal barrier function (Boydand Vandenberghe (2004)). As a follow-up, (Bitlislio˘gluand Jones (2017)) proposed a primal-dual interior pointmethod, which further decomposes the resulting Newton-step. However, such Newton-type methods are in gen-eral only convergent if they are equipped with additionalsmoothing heuristics and line-search routines. Comparedto the decomposition method, the Alternating DirectionMethod of Multipliers (ADMM) has more reliable con-vergence properties (Boyd et al. (2011); Hong and Luo(2017)). Many variants of ADMM exploit the hierarchicalstructure (Boyd et al. (2011); Goldstein et al. (2014)),but, in practice, a heuristic pre-conditioner is required toenhance convergence (O’Donoghue et al. (2016)).This paper considers the case that the grid operator coor-dinates a group of commercial buildings, which joins a DRprogram. Section 2 introduces the problem formulationbased on (Bitlislio˘glu et al. (2017)). Then, we reformulatehe original problem by exploiting the decomposed struc-ture of the building network, in which the local variablesare hidden. A distributed optimization problem is thusyielded, where the decoupled objectives are non-smoothPiece-Wise Quadratic (PWQ) functions with linear cou-pling constraints. For solving this problem in the con-text of MPC, Section 3 proposes a tailored AugmentedLagrangian based Alternating Direction Inexact Newton(ALADIN) method (Houska et al. (2016)), which comesalong with a convergence guarantee. ALADIN recentlyhas been proposed to solve multi-agent optimization prob-lems (Jiang et al. (2017); Engelmann et al. (2019)). Similarto ADMM, it requires the local agents to solve small-scaledecoupled problems and the central entity to deal with alinear equation in every iteration. For this variant, a warm-start strategy is further proposed to improve its onlineperformance. As a result, compared to ADMM, ALADINtakes the same communication effort per iteration whilerequiring much fewer iterations to converge to a desiredaccuracy. This is illustrated by running both an ALADINand an ADMM based MPC controller in a benchmark casestudy.
Notation:
The set of symmetric, positive (semi-)definitematrices in R n × n is denoted by ( S n + ) S n ++ . We use notation n = [1 . . . ⊤ ∈ R n for all n ∈ N . For a given matrixΣ ∈ S n + the notation k x k Σ = √ x ⊤ Σ x is used. Moreover, we call a function f : R n → R ∪ {∞} strongly convex with Σ ∈ S n + , if the inequality f ( tx + (1 − t ) y ) ≤ tf ( x ) + (1 − t ) f ( y ) − t (1 − t ) k x − y k is satisfied for all x, y ∈ R n and all t ∈ [0 , A ∈ R k × l and B ∈ R m × n is given by A ⊗ B = ( a ij B ) i,j ∈ R km × ln .
2. PROBLEM FORMULATIONThis section introduces a hierarchical optimal controlproblem for coordinating a group of commercial buildings,which join a Demand Response (DR) program.
This paper considers a building network in which eachbuilding has a central heating control system. The dy-namics of the i -th building can be described by a linearinput-output system, x i,k +1 = A i x i,k + B i u i,k + w i,k ,y i,k = C i x i,k + D i u i,k , (1)where state x i,k ∈ R n xi denotes the temperatures ofthe thermal zones of the i -th building, u i,k ∈ R n ui thethermal cooling energy input to the building at time step k and w i,k ∈ R n wi the system disturbance. The output y i,k ∈ R n yi denotes the mean zone temperatures andcoefficient matrices A i , B i , C i , D i depend on the specificbuildings.For a given reference room temperature y ref i , the followingtracking optimal control problem can be constructed, min u i N − X k =0 (cid:16)(cid:13)(cid:13) y i,k − y ref i (cid:13)(cid:13) Q i + k u i,k k R i (cid:17) s.t. ∀ k ∈ { , ..., N − } x i,k +1 = A i x i,k + B i u i,k + w i,k ,y i,k = C i x i,k + D i u i,k x i, = ˆ x i ,y i ≤ y i,k ≤ y i ,u i ≤ u i,k ≤ u i (2)with u i = [ u ⊤ i, . . . u ⊤ i,N − ] ⊤ . Here ˆ x i is the initial state,[ y i , y i ] and [ u i , u i ] denote box constraints on the systemoutputs and control inputs, respectively. The matrices Q i ∈ S n xi + and R i ∈ S n ui ++ are symmetric positive semi-definite and positive definite such that Problem (2) isa strongly convex quadratic programming (QP) prob-lem (Borrelli et al. (2003)). In the following, we representthe states and outputs as an affine function of the initialstate ˆ x i and control inputs u i by using the recursivederivation x i,k +1 = A k +1 i ˆ x i + k X j =0 A k − ji ( B i u i,j + w i,k )for k = 0 , ..., N − x i = A i ˆ x i + B ui u i + B wi w i y i = C i x i + D i u i with x i , y i and w i defined analogously to u i . Thus, we canrewrite the objective of (2) as a quadratic cost f i ( u i ) = u ⊤ i H i u i + 2 h ⊤ i u i with matrix H i ≻ h i given by H i = ( C i B ui + D i ) ⊤ Q i ( C i B ui + D i ) + R i ,h i = H i ( C i A i ˆ x i + C i B wi w i − N ⊗ y ref i ) , and the constraints as a polyhedral set u i ∈ U i := { u ∈ R Nn x | E i u i ≤ e i } with matrix E i and vector e i given by E i = C i B ui + D i −C i B ui − D i I − I , e i = N ⊗ y i − C i A i ˆ x i − C i B wi w i C i A i ˆ x i + C i B wi w i − N ⊗ y i N ⊗ u i − N ⊗ u i . Here, we use notation Q i = diag( Q i , ..., Q i ) and R i =diag( R i , ..., R i ). The constraint set U i is convex and com-pact Braun et al. (2018) such that Problem (2) has aunique solution with respect to a given initial state ˆ x i .Next, we will present the coordination problem basedon (2). In this paper, we consider a smart grid with M commercialbuildings, which join a DR program and are coordinatedby a grid operator. The goal of coordination is to balancethe voltage surge caused by a large renewable energygeneration such as solar plant. We denote by θ i,k the activepower injection or consumption from the i -th building. Itis a linear function of u i,k , θ i,k = F i u i,k , (3)ith energy transfer matrix F i for all k = 0 , , ..., N − v k ∈ R can berepresented as an affine map of θ i,k , i = 1 , ..., M given by v k = M X i =1 G i θ i,k + ˜ v k (4)for all k = 0 , ..., N −
1. Here, ˜ v k is given by the predictedvoltage magnitudes at time step k , in p.u., G i models theeffect of the injections from the i -th building on the overallvoltage magnitudes. In addition, v k needs to satisfy thebox constraints v ≤ v k ≤ v (5)for all k = 0 , ..., N −
1. The optimal coordination problemcan be formulated asmin p,θ,u M X i =1 f i ( u i ) + N − X k =0 π k θ i,k ! s.t. ∀ k ∈ { , , ..., N − } θ i,k = F i u i,k , i = 1 , ..., Mv k = M X i =1 G i θ i,k + ˜ v k ,v ≤ v k ≤ v ,u i ∈ U i , i = 1 , ..., M . (6)The used demand response criterion is defined by the sec-ond term in the objective, which is the economic cost of theelectricity. Here, π k denotes the price of the electricity. Ifwe substitute (3) into the objective and the constraints, (6)becomes min u,v M X i =1 f i ( u i ) + N − X k =0 π k F i u i,k ! s.t. ∀ k ∈ { , ..., N − } v k = M X i =1 G i F i u i,k + ˜ v k ,v ≤ v k ≤ vu i ∈ U i , i = 1 , ..., M . (7)The coupling between the buildings is modeled by theglobal variable v = [ v ⊤ · · · v ⊤ N − ] ⊤ in (4). In order todesign an efficient distributed optimization algorithm, inthe following, we will eliminate this variable and reformu-late (7) into the standard distributed form, which is onlywith local variables coupled by an affine equality. Let us introduce auxiliary variables s i,k ∈ R for i =1 , ..., M , k = 0 , ..., N −
1. The constraints (4) and (5) canthen be reformulated as inequality constraints g i ( u i , s i ) ≤ g i ( u i , s i ) = ˜ v k − vM + G i F i u i,k v − ˜ v k M − G i F i u i,k − s i,k k ∈{ ,...,N − } for all i = 1 , ..., M and the equality affine constraints M X i =1 s i = 0 (9)with s i = [ s ⊤ i, . . . s ⊤ i,N − ] ⊤ . As a result, Problem (7) canbe rewritten asmin z M X i =1 ˜ f ( u i ) s.t. M X i =1 s i | λz i ∈ Z i , i = 1 , ..., M (10)with stacked variables z i = [ u ⊤ i s ⊤ i ] ⊤ . Here, λ denotes theLagrangian multipliers of the affine equality constraints.The decoupled objectives are given by˜ f i ( u i ) = f i ( u i ) + N − X k =0 π k F i u i,k and the constraint sets are denoted by Z i = (cid:26) (cid:20) u i s i (cid:21) ∈ R Nn ui +2 N (cid:12)(cid:12)(cid:12)(cid:12) u i ∈ U i , g i ( u i , s i ) ≤ (cid:27) . Since (8) are affine inequalities, sets Z i are convex poly-topes. In the following, the equivalence between (7)and (10) is established. Proposition 1.
If Problem (7) is feasible with solution u ∗ ,Problem (10) has a solution ˆ z = (ˆ u, ˆ s ) with ˆ u = u ∗ .Reversely, if Problem (10) is feasible with solution ˆ z =(ˆ u, ˆ s ), ˆ u is the minimizer of Problem (7). Proof.
Let u ∗ be a minimizers of Problem (7). Then, wecan construct a feasible point ˆ z = ( u ∗ , ˆ s ) of (10) asˆ s i,k = G i F i ˆ u i,k − M M X i =1 G i F i ˆ u i,k − G i F i ˆ u i,k + 1 M M X i =1 G i F i ˆ u i,k . (11)For any feasible point z = ( u, s ) of (10), u is also feasiblefor (7). Thus, we have M X i =1 ˜ f ( u ∗ i ) ≤ M X i =1 ˜ f ( u i ) . This shows that ˆ z is a minimizer of (10). Similarly, forthe other direction, let ˆ z = (ˆ u, ˆ s ) be a minimizer of (10).For any feasible point u of (7), we can construct a feasiblepoint z = ( u, s ) of (10) based on (11) such that M X i =1 ˜ f (ˆ u i ) ≤ M X i =1 ˜ f ( u i ) . Therefore, ˆ u is a minimizer of Problem (7). (cid:4) Concerning Problem (10), due to the strong convexityof f i ( · ) and compact polyhedron Z , the optimal solution( u ∗ , s ∗ ) of (10) is unique with respect to u ∗ but not s ∗ .Therefore, we further introduce a least-squares regulariza-tion of s i in the decoupled objective, F i ( z i ) = ˜ f i ( u i ) + µ k s i k (12)with a sufficiently small constant µ >
0. This regulariza-tion enforces strong convexity of the problem and thus,uniqueness of s i . Note that in practice, this small regu-larization does not lead to large changes of the optimalsolution, which will be numerically illustrated later.ext, we introduce the function Ψ i : R N → R ≥ given bythe following multi-parametric QP (mpQP) problemΨ i ( s i ) = min u i F i ( z i ) s.t. z i ∈ Z i (13)for all i = 1 , ..., M . According to Theorem 2 in (Alessio andBemporad (2009)), Ψ i is a strongly convex PWQ functionand its solution map u ⋆i : R N → R Nn ui is Piece-WiseAffine (PWA). In the following, we use the notationΨ i ( s i ) = 12 s ⊤ i S i ( s i ) s i + r i ( s i ) ⊤ s i , where matrix S i ( · ) and vector r i ( · ) are piece-wise constantwith respect to s i inside different critical regions (Borrelliet al. (2003)). For a given s i , we denote the active con-straints at u ⋆i ( s i ) by[ P i, ( s i ) P i, ( s i )] (cid:20) u ⋆i ( s i ) s i (cid:21) + p i ( s i ) = 0with active Jacobian [ P i, ( s i ) P i, ( s i )], the matrices S i ( s i )are given by S i ( s i ) = µI + P i, ( s i ) ⊤ [ P i, ( s i ) H − i P i, ( s i ) ⊤ ] − P i, ( s i ) (14)with S i ( s i ) ∈ S N ++ . Accordingly, the multi-building coor-dination problem can be written asmin s M X i =1 Ψ i ( s i ) s.t. M X i =1 s i = 0 , | λ (15)which is a strongly convex but non-smooth optimizationproblem. In the following, we will design an algorithm forsolving (15) in a distributed manner.3. ALGORITHMThis section proposes a distributed algorithm based onALADIN (Houska et al. (2016)) for multi-building coordi-nation. Algorithm 1
ALADIN for solving (10)
Initialization:
Initial guess ( s, λ ), choose symmetric scal-ing matrices Σ i ≻ ǫ > Repeat:
1) Each building solves the decoupled QPmin ξ i F i ( ξ i ) + λ ⊤ ξ si + 12 k ξ si − s i k i s.t. ξ i = ( ξ ui , ξ si ) ∈ Z i (16)for i = 1 , ..., M in parallel and send solution ξ si to thegrid operator.2) Terminate if k s − ξ s k ≤ ǫ .3) The grid operator collects ξ si and solve the equalityconstrained QPmin s + M X i =1 (cid:13)(cid:13) s + i − ξ si + s i (cid:13)(cid:13) i s.t. M X i =1 s + i = 0 | ∆ λ + (17)Then, update λ + = λ + ∆ λ + and spread ( s + i , λ + ) to i -th building for all i = 1 , ..., M . Algorithm 1 outlines to solve (15) by using a tailoredALADIN algorithm. Similar to the standard ALADINmethod, Algorithm 1 alternates between solving (16) inparallel and dealing with the equality constrained QPproblem (17) for consensus. Here, Problems (16) are equiv-alent to min ξ si Ψ i ( ξ si ) + λ ⊤ ξ si + 12 k ξ si − s i k i , (18)which are also mpQPs with input parameters ( λ, s i ). Thus,the solution maps of (18), denoted by ξ ⋆i ( λ, s i ) , i = 1 , ..., M , are piece-wise affine (Alessio and Bemporad (2009)). Dueto QP (17) without inequality constraints, the solution canbe worked out analytically,∆ λ + = 2Λ − M X i =1 ξ si ! with Λ = M X i =1 Σ − i (19a) s + i = 2 ξ si − s i − Σ − i ∆ λ + , i = 1 , ..., M. (19b)Here, it is clear that the grid operator only needs to collectthe local solution ξ si and spread ∆ λ + to each building.Compared to ADMM, Algorithm 1 takes exactly the samecommunication effort as ADMM per iteration (Houskaet al. (2016)). As discussed in (Jiang et al. (2019)), the iterates of Algo-rithm 1 converges globally with a linear rate. Furthermore,if the scaling matrices are chosen as Σ i = S i ( s i ) with S i ( s i ) = S i ( s ∗ i ) the exact Hessian of Ψ i at the optimal s ∗ i ,Algorithm 1 can further achieve a local one-step conver-gence under a regularity condition (Frasch et al. (2015)).Note that this choice of Σ i requires prior knowledge of theoptimality of (15), which is in general impractical. How-ever, in the context of Model Predictive Control (MPC),the result of the last MPC iteration can be used to chooseΣ i online, which has a potential to satisfy Σ i = S i ( s ∗ i ),and thus, local convergence can be improved. In order to arrive at an efficient implementation, thestructure of (16) and (17) can be exploited as follows.
Online solver:
When we apply Algorithm 1 as an onlinesolver for MPC, we can move the primal update (19b)into the local phases such that a simplified version ofAlgorithm 1 is given byParallel Step λ + = λ + ∆ λ ,s + i = 2 ξ si − s i − Σ − i ∆ λ ,ξ + i = ξ ⋆i ( λ + , s + i ) , (20a)Consensus Step ∆ λ + = 2Λ − M X i =1 ξ si + ! . (20b) Warm-start:
In an MPC scheme, the initial guess ofAlgorithm 1 can be initialized by shifting the horizon, s i = ( s ∗ i, , . . . , s ∗ i,N − , , i = 1 , ..., M ,λ = ( λ ∗ , . . . , λ ∗ N − , . his strategy has been used in the context of an ADMMbased model predictive controller for smart grids (Braunet al. (2018)). Here, ( s ∗ , λ ∗ ) denotes the optimal solution ofthe current MPC problem. Furthermore, for Algorithm 1,we can set Σ i = S i ( s ∗ i ), which potentially improves thelocal convergence as discussed in Section 3.2. Note that inpractice, matrix S i ( s ∗ i ) might be ill-conditioned such thata iterative linear equation solver is required to deal withthe equality constrained QP (17).4. NUMERICAL RESULTSThis section illustrates the effectiveness of Algorithm 1 inthe MPC scheme by comparing it to the state-of-the-artmethod ADMM.In our implementation, both algorithms are executed asthe online solver for MPC. And the warm-start strategydiscussed in Section 3.3 is applied. Furthermore, in or-der to obtain a fair comparison, we implemented a pre-conditioner for ADMM by performing a modified Ruizequilibration (Ruiz (2001)) on the decoupled constraintmatrices E i .The data used to generate the benchmark is obtained byusing the EnergyPlus toolkit (Crawley et al. (2000)) andthe thermal model of buildings are generated with the
OpenBuild toolbox (Gorecki et al. (2015)). The length ofprediction horizon is chosen by N = 14 with sampling time0 . z i
280 98 70dimension of e i v ( k ) is given by v k ∈ [0 . , .
05] in p.u., andthe price π k = 1 is fixed for all k = 0 , ..., N − u ∗ ( µ ) and u ∗ (0) in [kW]. We consider a benchmark case study from (Bitlislio˘gluet al. (2017)) with a mix of 12 commercial buildingsincluding 2 Large, 7 Middle and 3 Small such that thereare 1456 variables and 4816 affine inequality constraintsin total. Regarding the choice of µ , Fig. 1 illustratesthe difference between the optimal solutions of (15) withdifferent µ = 0 and the result with µ = 0. The gap k u ∗ ( µ ) − u ∗ (0) k ∞ increases linearly with µ increasing. Weset the accuracy of the online solver as 10 − such that wechoose µ = 10 − in our implementation.Fig. 2 and Fig. 3 show the convergence comparison be-tween ALADIN and ADMM for two different online cases.In the first one, the optimal active set is almost the sameas the previous MPC iteration. On the contrary, there aresome changes of the optimal active set arising in the secondcase.Fig. 2. Convergence comparison of Case I: ALADIN vsADMM.Fig. 3. Convergence comparison of Case II: ALADIN vsADMM.ig. 2 shows the comparison of an example of the firstcase. For solving this particular problem, when we setthe tolerance to 10 − , ALADIN is five times faster thanADMM. In this case study, 252 constraints are active atthe optimal solution. The warm-start strategy for initializ-ing Σ i improves the local convergence of ALADIN. Fig. 3shows a comparison for the second case. In this example,42 constraints are active at the optimal solution. ALADINjust achieves a global linear convergence and is only threetimes faster than ADMM.5. CONCLUSIONThis paper analyzed an optimization problem for coor-dinating multiple commercial buildings. The problem bal-ances the voltage surge of the building network by using anenergy cost defined demand response criterion. By intro-ducing an auxiliary variable, the problem was reformulatedinto a standard distributed form with decoupled PWQ ob-jectives and coupled affine equality constraints. For solvingthis non-smooth convex problem in an MPC scheme, weproposed a tailored ALADIN method, which can warmlystart online and thus its convergence can be sped up. Ournumerical results illustrated the effectiveness of the warm-start strategy and show that the ALADIN based MPCcontroller outperforms the ADMM based controller.ACKNOWLEDGEMENTS JS, YJ and BH were supported by ShanghaiTech University underGrant-Nr. F-0203-14-012. CJ was supported from the Swiss NationalScience Foundation under the RISK project (Risk Aware DataDriven Demand Response, Grant-Nr. 200021 175627).
REFERENCESAlessio, A. and Bemporad, A. (2009). A survey on explicitmodel predictive control. In
Nonlinear model predictivecontrol , 345–369. Springer.Bitlislioglu, A. (2018). Coordinated optimization andcontrol for smart grids. Technical report, Ecole Poly-technique F´ed´erale de Lausanne (EPFL).Bitlislio˘glu, A., Pejcic, I., and Jones, C.N. (2017). Inte-rior point decomposition for multi-agent optimization.
IFAC-PapersOnLine , 50(1), 233–238.Bitlislio˘glu, A. and Jones, C.N. (2017). On coordinatedprimal-dual interior-point methods for multi-agent op-timization. In , 3531–3536.Borrelli, F., Bemporad, A., and Morari, M. (2003). Ge-ometric algorithm for multiparametric linear program-ming.
Journal of optimization theory and applications ,118(3), 515–540.Boyd, S., Parikh, N., Chu, E., Peleato, B., and Eckstein, J.(2011). Distributed optimization and statistical learningvia the alternating direction method of multipliers.
Foundations and Trends in Machine learning , 3(1), 1–122.Boyd, S. and Vandenberghe, L. (2004).
Convex optimiza-tion . Cambridge University Press.Braun, P., Faulwasser, T., Gr¨une, L., Kellett, C.M.,Weller, S.R., and Worthmann, K. (2018). Hierarchicaldistributed ADMM for predictive control with applica-tions in power networks.
IFAC Journal of Systems andControl , 3, 10–22. Crawley, D.B., Pedersen, C.O., Lawrie, L.K., and Winkel-mann, F.C. (2000). Energyplus: Energy simulation pro-gram.
ASHRAE Journal , 42, 49–56.Engelmann, A., Jiang, Y., Houska, B., and Faulwasser, T.(2019). Towards distributed OPF using ALADIN.
IEEETransactions on Power Systems , 34(1), 584–594.Frasch, J.V., Sager, S., and Diehl, M. (2015). A parallelquadratic programming method for dynamic optimiza-tion problems.
Mathematical Programming Computa-tion , 7(3), 289–329.Goldstein, T., O’Donoghue, B., Setzer, S., and Baraniuk,R. (2014). Fast alternating direction optimization meth-ods.
SIAM Journal on Imaging Sciences , 7(3), 1588–1623.Gorecki, T., Qureshi, F., and Jones, C.N. (2015). Open-build : An integrated simulation environment for build-ing control. In , 1522–1527.Hong, M. and Luo, Z. (2017). On the linear convergence ofthe alternating direction method of multipliers.
Mathe-matical Programming , 162(1-2), 165–199.Houska, B., Frasch, J., and Diehl, M. (2016). An aug-mented Lagrangian based algorithm for distributed non-convex optimization.
SIAM Journal on Optimization ,26(2), 1101–1127.Jiang, Y., Oravec, J., Houska, B., and Kvasnica, M. (2019).Parallel explicit model predictive control. arXiv preprintarXiv:1903.06790 .Jiang, Y., Zanon, M., Hult, R., and Houska, B. (2017).Distributed algorithm for optimal vehicle coordinationat traffic intersections. In
In Proceedings of the 20thIFAC World Congress, Toulouse, France , 12082–12087.Liserre, M., Sauter, T., and Hung, J.Y. (2010). Futureenergy systems: Integrating renewable energy sourcesinto the smart power grid through industrial electronics.
IEEE industrial electronics magazine , 4(1), 18–37.O’Donoghue, B., Chu, E., Parikh, N., and Boyd, S. (2016).Conic optimization via operator splitting and homo-geneous self-dual embedding.
Journal of OptimizationTheory and Applications , 169(3), 1042–1068.Oldewurtel, F., Parisio, A., Jones, C.N., Gyalistras, D.,Gwerder, M., Stauch, V., Lehmann, B., and Morari, M.(2012). Use of model predictive control and weatherforecasts for energy efficient building climate control.
Energy and Buildings , 45, 15–27.Rantzer, A. (2009). Dynamic dual decomposition for dis-tributed control. In ,884–888. IEEE.Rawlings, J., Mayne, D.Q., and Diehl, M. (2017).
ModelPredictive Control: Theory, Computation, and Design .Nob Hill Publishing.Richter, S., Morari, M., and Jones, C.N. (2011). Towardscomputational complexity certification for constrainedMPC based on Lagrange relaxation and the fast gradientmethod. In , 5223–5229.IEEE.Ruiz, D. (2001). A scaling algorithm to equilibrate bothrows and columns norms in matrices. Technical report,Rutherford Appleton Laboratorie.Siano, P. (2014). Demand response and smart grids—asurvey.