GRADIENT-BASED OPTIMAL CONTROL OF BATTERIES … · GRADIENT-BASED OPTIMAL CONTROL OF BATTERIES AND HVAC IN DISTRICT ENERGY ... A comprehensive description of the generalized opti-

GRADIENT-BASED OPTIMAL CONTROL OF BATTERIESAND HVAC IN DISTRICT ENERGY SYSTEMS

Marco Bonvini1, Michael Wetter11Lawrence Berkeley National Laboratory, Berkeley, CA

ABSTRACTResidential and commercial buildings use nearly 75%of the overall electrical energy in the U.S., and theamount of renewable energy in the grid keeps increas-ing. Buildings can be an important contributor to en-sure a stable grid operation because they can shift theirloads to reduce peak demand and flatten the ramps ofload increase and decrease. Simulation models thataccount for building loads and building dynamics, aswell as their impact on the electricity distribution grid,are essential to assess different design and control op-tions for buildings and electrical systems. This paperpresents the use of such simulation models of dynamicbuilding loads coupled to electrical models within atool chain that allows efficient numerical solution ofnonlinear optimization problems that aim at control-ling the voltage stability and thermal comfort whileminimizing energy use or energy cost. The optimiza-tion problems are formulated using the open-sourceJModelica software that convert them to a form inwhich they are solved using the nonlinear program-ming solver IPOPT. In this formulation, JModelicaconverts an infinite-dimensional optimization prob-lem, defined on the solution of the differential equa-tions of the simulation model, to a finite-dimensionalnonlinear programming problem using computer alge-bra and collocation methods.

INTRODUCTIONResidential and commercial buildings use nearly 75%of the overall electricity energy in the U.S. The in-crease of PVs and intermittent loads such as electricvehicles towards a new generation of net zero energybuildings is posing a challenge to the stability of thegrid, impacting the reliability of traditional electricitydelivery (Ipakchi and Albuyeh, 2009). To avoid prob-lems, efficient transactions between buildings and thegrid are needed (Palensky and Dietrich, 2011). Simu-lation models that account for buildings and their im-pact on the electricity distribution grid are essential tosupport the design and operation of the distributiongrid (Van Roy et al., 2013). Such multi-disciplinaryand multi-domain models have to describe both the in-teractions and the dynamics affecting these systems.Moreover, once these interactions and dynamics aredefined, they should be leveraged to identify optimalcontrol strategies that help stabilizing the grid.This paper shows how to solve such optimization prob-lems that consider both dynamics and interactions be-tween buildings and the electrical grid. As examples,this paper investigates multiple optimization problemsthat involve the optimal cooling as well as the charging

and discharging of batteries in a commercial buildingwith and without on-site renewable energy sources.The optimization problems are further extended to in-vestigate the impact that buildings have on the voltagequality in a small neighborhood and how batteries canbe controlled to improve it. The models used withinthe optimization problem have been adapted from theElectrical package (Bonvini et al., 2014) of the Mod-elica Buildings library (Wetter et al., 2014).The optimization approach we use differs from othermethods in the literature (Guan et al., 2010; Stadleret al., 2009) that use mixed integer linear program-ming (MILP) techniques. With MILP, it is possibleto approximate nonlinearities in the cost function withpiecewise linear approximations. The approach weuse is based on nonlinear programming (NLP) meth-ods and is able to handle nonlinear dynamics as well asnonlinear constraints on variables of the system. Forexample, it can be used to minimize the total energyuse of a district energy system, or its total electricitycosts or CO2 emissions, while keeping the root meansquare (RMS) voltages within the required limits. Another application could be optimal cooling load shiftwhile maintaining thermal comfort within the requiredlimits.The paper also describes the toolchain based on theopen source tool JModelica1 that allows to reuse sim-ulation models to solve optimization problems. Sucha toolchain enables the users to design and operatebuildings and the distribution grid more efficiently.While the examples in this paper use first-principlephysical models, the methodology works also withgrey box models and with models that are identifiedfrom data.

MOTIVATIONBuildings, renewables energy generation plants anddistribution grids are engineered systems which aresteadily growing in complexity. Modeling and sim-ulation tools are fundamental to support their designand operation. Traditional simulation tools have lim-ited capabilities when it comes to modeling integratedsystems. Different legacy tools are available to simu-late the various domains (e.g., buildings and electricalgrid). However, most building simulation programssuch as EnergyPlus (Crawley et al., 2000) do notmodel the electrical grid, while tools that focus on theelectrical distribution grid, such as OpenDSS (Dugan,2012) and GridLab-D (Chassin et al., 2008) includeonly simplified building and heating models, or takeload curves as input. Hence, if such tools were to

1See www.jmodelica.org.

Proceedings of BS2015: 14th Conference of International Building Performance Simulation Association, Hyderabad, India, Dec. 7-9, 2015.

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www.jmodelica.org

be used together, one would require the use of co-simulation (Makhmalbaf et al., 2014; Chatzivasileiadiset al., 2015). An example of building simulation pro-gram that allows to model electrical models is ESP-r(Eng, 1998).Solving design optimization or optimal control prob-lems with many decision variables is difficult if eval-uating the cost or constraint functions requires co-simulation of legacy code. The reason is that solv-ing large optimization problems is most efficient iffirst or second order derivatives of the cost and con-straint functions with respect to the design parame-ters exist and are accessible. However, legacy codeis often written in such a way that state trajectoriesare not differentiable with respect to the independentparameters. Moreover, in the cases where gradientsexist, legacy tools often do not make them accessible,thereby requiring the optimization algorithm to com-pute numerical approximations of the gradient. Thisis computationally costly and leads to more complexalgorithms (Polak, 1997). Furthermore, optimizationalgorithms have been shown to fail to converge if thecost function is discontinuous, as identified in Energy-Plus (Wetter and Wright, 2004).The increased integration of building and district en-ergy systems leads to complex dynamic systems thatrequire new approaches for design and operational op-timization. The Modelica Buildings library (Wetteret al., 2014) and the IDEAS library (Van Roy et al.,2013) are examples of modeling libraries that havebeen used to overcome the aforementioned limitations.These libraries are based on Modelica, an equation-based modeling language for dynamic, multi-physicsengineered systems (Mattsson et al., 1998). As Mod-elica is a declarative language, models expressed inthis language can be symbolically analyzed and ma-nipulated to bring them in a form that is more amend-able for simulation and optimization. Therefore, Mod-elica is well suited to express models for large scaleoptimization (Akesson et al., 2010). Figure 1 showsthe information flow for equation-based models whenused to solve optimization problems. The engineersand designers simulate models to investigate the be-havior of the system for different design alternatives.To solve an optimal control problem, the models areaugmented by adding objective and constraint func-tions. Next, the models are symbolically manipulatedusing computer algebra to bring the optimization prob-lem into a form that allows an efficient computation ofthe optimal control function. The optimal control sig-nal is then sent to the building system. Using measureddata, models can be adapted and current values of ob-servable states can be estimated to align the states ofthe models with the states of the building system.Such an approach can be done using the open sourcetool JModelica (Akesson et al., 2009) which we usedin our experiments for this paper. JModelica usesthe equation-based structure of the model to con-

Figure 1: Model-based optimization toolchain basedon JModelica.

vert an infinite-dimensional optimization problem intoa finite-dimensional nonlinear programming problemthat is then solves using a nonlinear solver (e.g.,IPOPT). The following sections provide an overviewof the systems that can be used with JModelica and themethods that automatically generate an optimizationproblem starting from a simulation model, constraintsand an objective function.

METHODOLOGYThe systems considered in this paper can be describedby a system of differential algebraic equations (DAE)of index less or equal to one. The general form is

F (t, x(t), x(t), u(t), y(t),Θ) = 0, (1)

where t ∈ [t0, tf ] is time for some initial and fi-nal time t0 and tf , x(t) ∈ Rnx is the state vector,u(t) ∈ Rnu is the input vector, y(t) ∈ Rny is the vec-tor of algebraic variables, and Θ ∈ Rp is the vector ofparameters. The initial conditions for (1) can be im-plicitly specified as

F0(x(t0), x(t0), u(t0), y(t0),Θ) = 0. (2)

A comprehensive description of the generalized opti-mization problems that can be solved with JModelicais

minimizez(·)∈Z,Θ∈Rp

f(t, z(t),Θ) (3a)

subject to F (t, z(t),Θ) = 0, (3b)F0(z(t0),Θ) = 0, (3c)zL ≤ z(t) ≤ zU , (3d)pL ≤ Θ ≤ pU , (3e)h(t, z(t),Θ) = 0, (3f)g(t, z(t),Θ) ≤ 0, (3g)H(t, Zh,Θ) = 0, (3h)G(t, Zg,Θ) ≤ 0, (3i)∀t ∈ [t0, tf ],


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where z(·) = [x(·), y(·), u(·)], f(·, ·, ·) : R ×Rnx+ny+nu ×Rp → R is the cost function, the equal-ities (3b) and (3c) describe the dynamics of the systemand its initial conditions, (3d) and (3e) define the up-per and lower bounds for the time dependent variablesand the parameters, (3f) and (3g) are equality and in-equality path constraints, (3h) and (3i) are the equal-ity and inequality point constraints, where Zh and Ze

are the points where these constraints are imposed.To establish second order optimality conditions and tofind a first order optimal solution, for (3) the functionsf(·, ·, ·), F (·, ·, ·) and F0(·, ·) have to be twice contin-uously differentiable.

The general optimization problem (3) covers a largeclass of problems, that include optimal set point track-ing, and parameter calibration. The problem is infi-nite dimensional because the optimal solution z(·) is afunctional in the set of admissible trajectories Z . Thetechniques and methods described by Biegler (2010)allows to convert an infinite dimensional optimizationproblem of the form (3) to a finite dimensional nonlin-ear programming problem of the form

minimizew∈Rnw

f(w),

subject to wL ≤ w ≤ wU ,

g(w) = 0,

h(w) ≤ 0,

(4)

where w ∈ Rnw is a finite dimensional variable.JModelica uses the direct collocation method to con-vert the infinite dimensional problem (3) into its equiv-alent finite dimensional form (4). The method usespolynomials defined over a finite number of supportpoints, the so-called collocation points, to approxi-mate the trajectories of the dynamic system. Be-fore assigning the collocation points, the time hori-zon [t0, tf ] is divided into ne elements. Next, withineach element, the time-dependent variable z(t) is ap-proximated using a vector valued polynomial zi(t) =(xi(t), xi(t), ui(t), yi(t)). The collocation polynomi-als are formed by choosing a number of collocationpoints nc, within each of the ne elements. The numberof collocation points nc is assumed to be the same foreach element. The collocation polynomials are createdusing Lagrange interpolating polynomials that use thecollocation points as interpolation points. The collo-cation points are selected using the Radau collocationmethod, which place a collocation point at the end ofeach element and then select the others to maximizeaccuracy. In addition to the collocation points, the startpoint of every element is added to impose constraintson the continuity of the state trajectories between dif-ferent elements. The collocation polynomials that ap-

proximate z(t) in the element i are

xi(τ) =

nc∑k=0

xi,k lk(τ), (5a)

ui(τ) =

nc∑k=1

ui,k lk(τ), (5b)

yi(τ) =

nc∑k=1

yi,k lk(τ), (5c)

where τ ∈ [0, 1] is the normalized time in each el-ement i ∈ {1, . . . , ne}, lk(τ) is the Lagrange basispolynomial and lk(τ) is the Lagrange basis polyno-mial that includes the first point to ensure continuityof the state trajectories. Since the time is normal-ized in all the elements, the basis polynomials are thesame for every element. The polynomial approxima-tion of the derivative xi(τ) is the derivative of (5a).The infinite dimensional optimization problem (3) isconverted into a finite dimensional problem that canbe solved using a NLP solver. The conversion is doneby replacing the continuous variables z(t) and t withtheir discretized versions, zi(t) and ti. For a more de-tailed overview see Magnusson and Akesson (2012).

EXAMPLEWe present a series of examples that investigate theoptimal cooling, charge and discharge of batteries in aneighborhood with commercial buildings and on-siterenewable energy sources. The series starts with theoptimization of a single building. Next, we increasecomplexity by adding more buildings, PVs and batter-ies. The series concludes with an optimization prob-lem where batteries in different buildings are coordi-nated to improve the voltage quality of the neighbor-hood while satisfying thermal comfort constraints.

BUILDING MODEL

Figure 2: Comparison of cooling load computed bya reduced order model and the original EnergyPlusmodel. The green, yellow and red colored area repre-sent a relative error of 5%, 10% and 15%.


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The building model used in the example is a 54,000ft2 two story steel-frame office and laboratory build-ing, located in Berkeley, CA. An EnergyPlus modelof the building was available, which we convertedto an equivalent resistor-capacitor (RC) model us-ing the Building Resistance-Capacitance Modeling(BRCM) toolbox (Sturzenegger et al., 2014). Thelinear RC model constitutes a simplification of thewhole-building EnergyPlus model. The number ofoutputs of the building model has been limited to theaverage return air temperatures TR(t) of the thermalzones. The model is

x(t) = Ax(t) +Bvv(t), (6a)y(t) = Cx(t), (6b)

C =[

V1

Vtot, · · · , Vnz

Vtot, 0, · · · , 0

],

where x(t) ∈ Rnx is the state vector containing allthe temperatures of the zones, internal masses andwall layers, v(t) ∈ Rnv are the predicted disturbances(such as external air temperature, solar radiation andinternal heat gains), y(t) ∈ Rny , with ny = 1, is thereturn air temperature, C ∈ R×Rnx is the output ma-trix, nz is the number of thermal zones (the first nz ele-ments of the state vector x(t)), Vi for i ∈ {1, . . . , nz}is the volume of the i-th thermal zone, and Vtot =∑nz

i=1 Vi is the sum of all the volumes. The vector ofknown disturbances v(t) and outputs y(t) are definedas

v(t) =[Qihg(t), Tamb(t), Tgnd(t), Se(t),

Sw(t), Sn(t), Ss(t)]T ,

y(t) = [TR(t)] ,

where Qihg(t) is the internal heat gain, Tamb(t) is theoutside air temperature, Tgnd(t) is the ground temper-ature and Si(t), with i ∈ {n, s, w, e}, are the global di-rect plus the diffuse solar radiation on the north, south,west and east directions. For this study, we further re-duce the size of the initial RC model using a methodthat eliminates the state variables characterized by thesmallest Henkel singular values (Glover, 1984). Withsuch a method we reduced the number of states of theinitial RC model from 106 to 8 state variables. Fig-ure 2 shows a comparison between the cooling loadpredicted by the reduced order model with respect tothe original EnergyPlus model over the summer pe-riod. The higher relative errors that occur in the lowerleft corner are due to the presence of nonlinear effectssuch as radiation to the sky during night time.The HVAC model provides a simple description of theoverall performance of the cooling system and is de-scribed by the following equations

COP (t) = f(Tamb(t)), (8a)Qihg(t) = Po(t) + Pl(t) + Pp(t) + Pcool(t), (8b)

Pel(t) = Pl(t) + Pp(t) +Pcool(t)

COP (t), (8c)

where we used for the coefficient of performanceCOP (t) a linear function of the outside air temper-ature, Po(t), Pl(t) and Pp(t) are the internal heatgains due to occupants, lights and plug loads, Pel(t) isthe electrical power consumption of the building andHVAC system and Pcool(t) is the cooling power thatis needed to maintain the thermal comfort inside thebuilding. Pcool(t) is also one of the decision variablesof the optimization problem.

ELECTRIC MODELS

In the electrical domain, the building is represented byan inductive load with PVs and a battery connected toit. The electric models use the quasi-stationary phaso-rial representation, i.e. voltages and currents sinusoidshave no transients and thus can be represented by vec-tors in the complex plane. The apparent power of theinductive load representing the building is

Sbui(t) = Pel(t)(1 + j tanφ), (9)

where φ is the phase shift between the voltage and cur-rent phasors. If φ > 0, the load is inductive. Themodel of the battery is

Ebattd

dtSOC(t) = Pbatt(t), (10)

where Ebatt is the storage capacity of the battery,SOC(t) is the state of charge and Pbatt(t) is the powerstored into or drawn from the battery. Pbatt(t) is thesecond decision variable of the optimization problem.The battery is connected to a DC/AC converter and weassume that this conversion does not introduce reactivepower. The PV model is

Ppv(t) =(Se(t)Ae + Sw(t)Aw+

Sn(t)An + Ss(t)As) ηPV , (11a)

where Ppv(t) is the power produced by the PVs, Ae,Aw, An, and As are the areas of the PVs panels ori-ented towards east, west, south and north, and ηPV isa parameter that accounts for the efficiency of the PVmodules and the efficiency of the inverter. The reactivepower generated by the DC/AC converter connected tothe PV is assumed to be equal to zero.The quasi-stationary assumption allows to model theelectric loads, PVs, and lines with algebraic equationsthat are coupled to the differential equations of thebuildings and electric storages. Such representationhelps the collocation method because it does not haveto approximate on the same time grid both fast electri-cal transients and slow thermal dynamics.

CONSTRAINTS AND COST FUNCTIONS

All the optimization problems either minimize the en-ergy used by the building E(·, ·) : R × R → R or theenergy cost M(·, ·) : R× R → R. The cost functions


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are defined over a period of five days as

E(t0, tf ) =∫ tf

t0

max(0, Pel(t) + Pbatt(t) + Ppv(t))dt, (12a)

M(t0, tf ) =∫ tf

t0

p(t)(Pel(t) + Pbatt(t) + Ppv(t))dt, (12b)

where p(t) is the time-varying price signal, t0 and tfare the initial and final times of the optimization pe-riod. The price is assumed to be the same for ei-ther buying or selling energy and it varies between0.11$/kWh and 0.24$/kWh. Note that the max(·, ·)function, implemented using a smooth approximation,incentivizes each building to use its electric or thermalenergy storage. The reason is that feeding power tothe grid does not decrease the cost function E(t0, tf ),therefore it disincentives the building from feedingpower back to the grid reducing possible overvolt-ages. Later in this paper, we will impose inequalityconstraints on the maximum voltage. The constraintsincluded in the optimization problem define both thethermal comfort requirements as well the limits of op-eration of the electric storage and the HVAC system.The constraints are

Pmaxbatt ≤ Pbatt(t) ≤ Pmax

batt , (13a)Pmaxcool ≤ Pcool(t) ≤ 0, (13b)

TminR ≤ TR(t) ≤ Tmax

R , (13c)

SOCmin ≤ SOC(t) ≤ SOCmax, (13d)

where Pmaxbatt and Pmax

batt are the maximum dischargingand charging power of the battery, Pmax

cool is the maxi-mum cooling capacity of the HVAC system, Tmin

R andTmaxR are the comfort constraints on the return air tem-

peratures and SOCmin and SOCmax are the allowedminimum and maximum charge of the battery.We initialized all the optimization problems using asolution based on a relay controller that keeps the re-turn air temperature TR(t) within the limits specifiedby (13). Such sub-optimal and feasible initial solutionhelps the nonlinear solver to find the optimal trajec-tory.

SCENARIO 1

In this scenario, the building has neither PVs nor a bat-tery. The cooling power profiles are computed in or-der to minimize either the energy use or the energycost. Figure 3 shows the results of the optimization.When the controller minimizes energy use E(t0, tf ),the room temperature is always at the higher limit ofthe comfort range (the red line). When minimizing en-ergy cost M(t0, tf ), because of the higher price dur-ing the day, the controller shifts the cooling load to thenight and causes the building to pre-cool (black line).

Figure 3: Scenario 1 – The figure shows the temper-ature in the building when controlled to minimize en-ergy (red line) and to minimize cost (black line). Theoutside air temperature and the comfort temperaturerange are shown in green and dashed blue lines. Thedarker the background color, the higher the energyprice.

SCENARIO 2In this scenario the building has PVs but no battery. Asin scenario 1, the only degree of freedom for the con-troller is to decide the optimal allocation of the cool-ing power. Figure 4 shows the results for minimizingenergy or cost. When minimizing energy use, the con-troller decides to use the power provided by the PVsto cool the building during the day (red line) as feed-ing electricity to the grid does not reduce the objec-tive function E(t0, tf ). However, when minimizingthe energy cost M(t0, tf ), the controller, as in sce-nario 1, shifts the cooling to the night and pre-coolsthe building (black line).


SCENARIO 3In the third scenario we add a battery to the system,thus the optimization problem has an extra degree offreedom with respect to scenario 2. In this case, thecontroller can decide the optimal cooling power deliv-


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Figure 5: Scenario 3 – The figure shows the temper-ature in the building when controlled while minimiz-ing the energy (red line) and while minimizing the cost(black line). The outside air temperature and the com-fort temperature range (green and dashed blue lines).The darker the background color, the higher the energyprice.


ery as well the optimal charge and discharge scheduleof the battery. Figures 5 and 6 show the results of thisscenario. Compared to scenario 2, when minimizingenergy, the temperature of the building decreases lesssince part of the power produced by the PVs is storedin the batteries and then used during nighttime whenthe COP is higher and the PVs do not generate power.When optimizing the cost of energy, the variabilityof the energy price and the thermal constraints playa dominant role. The controller pre-cools the build-ing as much as possible when the energy is cheap inorder to minimize the electricity cost during the high-price period. Figure 6 shows how the extra degree offreedom provided by the battery is utilized. The con-troller that minimizes the energy useE(t0, tf ) chargesthe battery during the day when the PVs are producingpower (red line). The controller that minimizing thecost of energy M(t0, tf ) charges the battery at nightusing cheap energy and discharges the battery during

the day (black line). It turns out that the controller thatminimizes energy does not fully charge and dischargethe battery. In contrast, the controller that minimizesthe cost of energy stores as much energy as possiblewhen it is cheap and then uses it during the peak time.

SCENARIO 4

Figure 7: Schematic representation of the neighbor-hood.

In this scenario, we consider three buildings. Themodels used in the former scenari have beenparametrized with different values to represent threecommercial buildings as shown in Figure 7. The pa-rameters that have been modified are the distributionof the PVs along the different directions, and the nom-inal power of the PVs of each building. This changecauses diversity on the magnitude and shape of thegenerated power among the buildings. In this sce-nario, we also consider the grid voltage, and one op-timization scenario includes inequality constraints onthe RMS voltage at the building to grid connections.For every building, the constraints defined in (13) arestill applied, and additional constraints are

V rmsA (t) ≤ V max, (14a)V rmsB (t) ≤ V max, (14b)V rmsC (t) ≤ V max, (14c)

where V rmsA (t), V rms

B (t) and V rmsC (t) are the RMS

voltages of the buildings, V max = 1.02Vn is the max-imum allowed voltage, and Vn is the nominal voltageof the network. It is important to note that the voltageconstraints (14) are nonlinear since the RMS voltageis defined as

V rms(t) =√VRe(t)2 + V 2

Im(t), (15)

where VRe(t) and VIm(t) are the real and imaginarycomponents of the voltage phasors.Figure 8 shows the voltage variations in the differ-ent nodes of the neighborhood under the optimal con-trol function of scenario 3. The blue lines show thevoltages when the cost function of the optimizationis the energy used E(t0, tf ). In this case, the volt-ages do not reach the red shaded area that correspondsto a RMS voltage that is 2% higher than the nominal


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value of Vn = 4.8 kW. Overvoltages happen when thebuildings are controlled to minimize the cost of energyM(t0, tf ) (red lines). This happens because the build-ings charge their batteries during the night and sell thepower generated by the PVs during the day. The greenlines in Figure 8 show the results of the new optimiza-tion problem that also takes into account the voltageconstraints (14).

Figure 8: RMS voltages of the buildings A, B and C.The blue lines are the voltages when the buildings areindividually controlled to minimize the energy use, thered lines when they are individually controlled to min-imize their energy cost, and the green lines when allthe buildings are controlled to minimize energy costwhile satisfying thermal and voltage constraints. Thered shaded area indicates the voltage violation region.

RESULTS AND DISCUSSIONTable 1 compares the results of scenario 1, 2 and 3.As expected, when PVs and batteries are added, thevalue of the cost function is reduced. An interestingfact happens when transitioning from PVs only to PVsand battery, i.e., from scenario 2 to 3. For examplewhen minimizing with respect to the cost of energyM(t0, tf ), the energy use E(t0, tf ) increases despitethe lower cost. The same is true when optimizing withrespect to the energy use E(t0, tf ): the energy use isless but its cost is higher.

Table 1: Comparison of total energy use and cost ofenergy in the different cases. Opt. M are results whenminimizing the cost of energy M(t0, tf ) and Opt. Eare the results when minimizing energy cost from thegrid E(t0, tf ).

Energy [kWh] Cost [$]Scenario Opt. E Opt. M Opt. E. Opt. M.

1 25.7 26.0 3978 39482 11.8 13.0 810 6843 9.7 15.3 913 420

When taking into account the whole neighborhood,the optimization strategy that minimizes the energyused from the grid keeps voltage levels low since eachbuilding tries to use as much power produced by thePVs as possible. Hence, this strategy attenuates thechances of over voltages in the network. Optimiz-ing the cost of energy introduces high grid voltages

as the buildings try to reduce their energy use duringpeak time and selling as much solar power as pos-sible. A coordinated action, obtained by solving anoptimization problem with both, thermal and voltageconstraints, solve this issue. However, the cost of thisvoltage-aware control action impacts the total cost ofoperation. The economic impact affects in particularthe buildings that are more distant from the source, in-creasing the cost of energy up to 17% for building Cwhile only 12% for building B.This optimization scenario includes six independentfunctions to be varied, which are the cooling power ofthe buildings and the charge or discharge power of thebatteries. The total number of variables of the finite-dimensional optimization problem solved by IPOPTdepends on the number of elements ne and collocationpoints nc. Different number of elements have beenused to evaluate the performances of the optimizationmethod. The number of collocation points in each el-ement has been kept constant nc = 3. Table 2 showsthe results of the analysis. The optimization problemshave been solved on a Linux virtual machine with 2GBof memory and 4 cores, using Virtual Box as virtualmachine manager. The host machine is a MacBookPro with a 3 GHz Intel Core i7 processor and 16 GBof memory. Despite the number of variables of thefinite-dimensional optimization problem, the numberof iterations and the time needed to converge to an op-timal solution indicate that the use of gradient basedmethods are an efficient way to handle such optimiza-tion problems.

Table 2: Performances of the optimization method inscenario 4 using different number of elements ne.

ne Variables Iterations Time [s]24 14094 136 23.8848 28038 253 56.7296 55926 443 1164.74

CONCLUSIONSThis paper demonstrated how simulation models canbe reused to generate optimization problems using atoolchain that leverages the Modelica modeling lan-guage and nonlinear programming algorithms. Mod-els adapted from the Modelica Buildings library havebeen used in conjunction with JModelica that solvedthe optimization problems. The example demonstratedhow different optimization problems can be created byincrementally increasing the complexity of the model.The example also demonstrated how nonlinear opti-mization problems that involve both thermal and elec-trical domains in presence of nonlinear cost and con-straint functions can be solved. The performances ofthe optimization method have been tested by dividingthe optimization interval with different number of el-ements. The results show that gradient-based meth-ods are efficient in solving optimal control problemsfor building and grid integration with large number of


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variables.

ACKNOWLEDGEMENTThis research was supported by the Assistant Secretaryfor Energy Efficiency and Renewable Energy, Officeof Building Technologies of the U.S. Department ofEnergy, under Contract No. DE-AC02-05CH11231.This work emerged from the Annex 60 project, an in-ternational project conducted under the umbrella of theInternational Energy Agency (IEA) within the Energyin Buildings and Communities (EBC) Programme.Annex 60 will develop and demonstrate new gener-ation computational tools for building and commu-nity energy systems based on Modelica, FunctionalMockup Interface and BIM standards.

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