Increasing Energy Resiliency to Hurricanes with Battery and Rooftop Solar Through Intelligent Control
IIncreasing Energy Resiliency to Hurricanes with Battery and RooftopSolar Through Intelligent Control
Ninad Gaikwad, Naren Srivaths Raman, and Prabir Barooah
Abstract — Rooftop solar photovoltaic (PV) panels togetherwith batteries can provide resiliency to blackouts during naturaldisasters such as hurricanes. Without intelligent and automateddecision making that can trade off conflicting requirements,a large PV system and a large battery is needed to providemeaningful resiliency. By utilizing the flexibility of varioushousehold demands, an intelligent system can ensure thatcritical loads are serviced longer than a non-intelligent system.As a result a smaller (and thus lower cost) system can providethe same energy resilience that a much larger system will beneeded otherwise.In this paper we propose such an intelligent control systemthat uses a model predictive control (MPC) architecture. Theoptimization problem is formulated as a MILP (mixed integerlinear program) due to the on/off decisions for the loads.Performance is compared with two rule based controllers, asimple all-or-none controller that mimics what is availablenow commercially, and a Rule-Based controller that uses thesame information that the MPC controller uses. The controllersare tested through simulation on a PV-battery system chosencarefully for a small single family house in Florida. Simulationsare conducted for a one week period during hurricane Irma in2017. Simulations show that the size of the PV+battery systemto provide a certain resiliency performance can be halved bythe proposed control system.
I. I
NTRODUCTION
Atlantic hurricanes are occurring with increasing fre-quency [1]. Loss of electricity supply for long periods isa common outcome of hurricanes. Rooftop solar PV panelscan provide a resilient energy supply to hurricane-inducedblackouts since the sky is often clear immediately after thehurricane. Serving the entire household load from an on-sitePV+battery system may require a large system depending onthe size of the house, driving up cost. Even serving only themost critical loads, such as refrigeration and a few lights andfans, during a long blackout in the face of solar generationuncertainty may still require a large PV+battery system.We argue that the size—and thus, cost—of the PV+batterysystem to provide resiliency can be reduced with the help ofan intelligent control system. The key is to exploit flexibilityin demand as well as in the supply in conjunction withforecasts. Flexibility in demand comes from the fact that,after a disaster not some loads are more critical than others.Among the critical loads that need to be served during ablackout, refrigeration for food and medicine is the mostimportant [2]. Next comes lights, and then fans. Fans can
The authors are with the Department of Mechanical and AerospaceEngineering, University of Florida, Gainesville, Florida 32611, USA. [email protected], [email protected],[email protected].
The research reported here is partiallysupported by NSF awards 1646229 and 1934322. serve as temporary replacements for air conditioners toprovide thermal comfort, and are much less energy intensivethan air conditioners. An intelligent controller can prioritizethe refrigerator demand over light and fan demand, and turnoff all other loads. Flexibility in supply comes from the factthat the charging rate of batteries is variable; a battery can befast charged to prepare for a forecasted low solar irradianceevent, though at some cost to the battery’s health.Thus, by using forecast of solar generation and house-hold demand, an intelligent decision maker can operate theequipment (battery, primary loads and secondary loads) toensure that a higher energy resiliency is obtained. Withoutsuch intelligence, a much larger system will be needed todeliver the same energy resilience.This paper presents such an intelligent control system fora home in which the critical loads are a refrigerator, a fewlights and fans, and a small PV+battery system. Among thecritical loads that need to be served with on-site energyduring an outage, the refrigerator is deemed primary loadwhile the combined load of the lights and fans is called the secondary load. It is more important to serve the primarythan the secondary load. The goal of the control system is tokeep the refrigerator temperature within a band while servingthe secondary demand as much as possible. Although thereare many more electrical loads in a typical household, othersare not critical for health and well being after a disaster.For instance, though many homes use electric cook-tops forcooking in Florida, people often use outdoor gas grills tocook food after hurricanes [3], [4].A model predictive control (MPC) architecture is usedthat uses available measurements and forecasts to makeoptimal decisions in real time. The optimization problem isformulated as a MILP (mixed integer linear program); theinteger valued variables are for the on/off status of powersupply to the primary and secondary loads (refrigerator andthe aggregate demand of lights and fans). Performance iscompared with two rule based controllers, a simple baseline controller that mimics what is available now commercially,and another controller – that we call Rule-Based controller– that uses the same information that the MPC controlleruses. Simulations show that the proposed MPC controller isable to service the primary load (refrigerator) throughout thesimulation period (7 days after hurricane Irma in 2017) whilethe baseline controller is unable to do so for several hourseach day. In addition, it is able to service the secondaryload a little more than the baseline. We measure primaryresiliency performance of a control system as the averagedaily duration that the system is able to meet demand from a r X i v : . [ ee ss . S Y ] F e b he primary load. A simulation based study indicates that tomeet a specific primary resiliency performance, the cost ofthe PV+battery system needed by the baseline controller is twice that of that needed by the MPC controller. The costof energy resiliency can therefore be halved by the MPCcontrol system.All three controllers utilizes the same hardware, and anymodification to the household electrical wiring to implementthem is the same. The difference is in the sophisticationof their decision making and available sensing. The MPCand Rule-Based controller use two more sensors (refrigeratortemperature and house temperature) and uses solar irradianceforecasts compared to the baseline controller.In this work we only focus on the control algorithm forpost-disaster scenario in which grid supply has been lost. It isassumed that when grid supply is restored, the software willswitch to a “normal operating” mode. The normal operatingmode may also be a sophisticated controller that seeksto, for instance, minimize the utility bill of the consumerby controlling the PV+battery system. There is a plethoraof work in that direction; see [5], [6], [7], [8] and [9].Therefore we do not consider that problem here. Works oncontrolling the PV+battery system to maximize resiliencyperformance in a post-disaster scenario, the focus of thispaper, is extremely limited. To the best of our knowledge,only [10] and [11] consider the problem of operation forresiliency. However, both [10] and [11] ignore the mixed-integer nature of the optimization problem, and ignores thecapability of a battery to vary charging rate which can beexploited during contingency situations like power outage.A preliminary version of this work was presented in [12].We have made several improvements in this paper. (i) Weintroduce a Rule-Based controller in this paper that uses thesame sensors and forecasts, and even the dynamic models,that MPC uses to make decisions. The purpose is to checkwhether the performance of the MPC controller is achiev-able with a simpler controller that does not need real-timeoptimization. Results indicate it is unlikely. (ii) The MPCcontroller in [12] needed a dynamic model of the house(indoor) temperature - apart from a sensor - for prediction.Obtaining parameters of such models is challenging [12].The MPC controller proposed here uses a simpler parametricmodel of the indoor temperature that only needs a temper-ature sensor and historical data on outdoor temperature forpredicting indoor temperature. (iii) The controller in [12]assumed the ability to turn the refrigerator on and off. Whileturning off is easy, forcing the compressor on is difficultwith an existing refrigerator without expensive retrofit. Thecontroller in this paper assumes only the ability to interruptpower supply to the refrigerator, not the ability to force iton. Such actuation is far cheaper in practice, with a wirelesscontrolled smart switch, than the ability to command therefrigerator compressor to turn on. (iv) We have reportedhere several sensitivity studies, while the simulations in [12]were only for one specific set of parameters.The rest of the paper is organized as follows: the systemdescription and mathematical models of the system compo- nents are described in Section II. The formulation of MPCcontroller, and the description of Rule-Based and baselinecontrollers are provided in Section III. The simulation setup,computation and simulation parameters discussed in Sec-tion IV. The results of the simulation study are presentedand discussed in Section V. Finally, the main conclusionsare provided in Section VI.II. S YSTEM D ESCRIPTION
Figure 1 shows the schematic of a house with solar PVpanels, a battery energy storage system, a primary load(refrigerator), and secondary loads (lights+fans). An intelli-gent controller needs to trade-off primary-secondary demandand battery life to provide ’best’ service possible duringan extended outage within certain constraints. The primarygoal is to maintain the refrigerator temperature within theprescribed limits, and a secondary goal is to service thesecondary load during times that are pre-decided by theoccupants.In order to achieve these goals, the intelligent controllerneeds to control the following: (i) power supply to the refrig-erator, (ii) on/off state of the secondary load (aggregate oflights and fans), (iii) charging/discharging state of the battery,and (iv) when charging the battery, the charging mode ofthe battery. The battery has two charging modes: normaland fast. The refrigerator power supply is actuated using asmart switch. The intelligent controller can control only thepower supply to the smart switch to which the refrigerator isconnected, but not the actual on-off of the refrigerator com-pressor which is controlled by the refrigerator thermostat.The secondary load on-off is actuated through a secondarycircuit coupled with a smart switch. The battery actuationcan be done through the smart charge controller. Moreover,the intelligent controller requires the following additionalinformation: (i) forecasted irradiance, (ii) measured housetemperature, (iii) measured internal refrigerator temperature,and (iv) measured battery state of charge. Forecasts for theirradiance are obtained from a provider over the internet. Itis assumed that the controller stores forecasts received overat least a week, which can be used to fit a simple time seriesmodel to provide local forecasts of the irradiance in casethe internet is down during the extended outage. The houseand internal refrigerator temperatures are sensed using smarttemperature sensors which communicate with the controllerover the air, while the battery state of charge can be obtainedover the air from a smart charge controller.Mathematical models of each of these components aredescribed in the subsections below. Time is discrete, with k = 0 , , , . . . denoting the time index and ∆ t s denoting theinterval (hours or minutes) between k and k +1 . In the sequel, E ( k ) ( W h ) will denote the energy consumed/generatedduring the time interval between time indices k and k + 1 ,with the subscript specifying the source or consumer of theenergy. The dependence on k will be often omitted, e.g., wewill say x instead of x ( k ) .The plant used for closed loop simulations consists of thedynamic models: Battery energy storage system model, re- o l a r P V P a n e l B a tt e r y C h a r g e C on t r o ll e r I nv e r t e r R e fr i g e r a t o r F a n L i gh t s C on t r o ll e r T r an sf o r m e r Fig. 1: Hardware involved in the intelligent control system.frigerator thermal model and the energy consumption modelsof the household electric loads, presented in Section III, andthe interactions between the PV panels, the battery, and theloads (both primary and secondary). These interactions arerepresented mathematically using the following equations: E hl ( k ) = u fr ( k ) E fr ( k ) + u s ( k ) E s ( k ) η inv , (1) E cbat ( k ) = c ( k ) min (cid:8) E pv ( k ) − E hl ( k ) , ¯ E bat − E bat ( k ) , x bat ( k ) ¯ E cbat (cid:9) , (2) E dcbat ( k ) = d ( k ) min (cid:8) E hl ( k ) − E pv ( k ) ,E bat ( k ) − ¯ E bat , ¯ E dcbat (cid:9) . (3)Eq. (1) shows that the total house load ( E hl ) is composed ofthe energy used by the refrigerator ( E fr ) and the secondaryloads ( E s ), where η inv is the inverter efficiency, u fr is therefrigerator power supply on-off control command, and u s is the on-off command signal for the secondary load. Eq. (2)shows that the battery charging energy ( E cbat ) is such that itnever charges beyond the maximum limit ( ¯ E bat ), where E bat is the battery energy level, x bat is either 1 or 2 (1 - normalcharging and 2 - fast charging), ¯ E cbat is the maximum batterycharging energy, c is the battery charging control command( c = 1 for charging, c = 0 for not charging), and E pv is maximum available energy that can be produced by thePV panels. Eq. (3) shows that the battery never dischargesbelow the minimum limit ( ¯ E bat ), where E dcbat is the batterydischarging energy, ¯ E dcbat is the maximum battery dischargingenergy, and d is the battery discharging control command( d = 1 for discharging, d = 0 for not discharging).III. C ONTROL A LGORITHMS
A. Model Predictive Control (MPC)
The control decisions are computed at discrete time steps k = 1 , . . . N with ∆ t s as the sampling period, and N is the total number of time steps in the planning/predictionhorizon. The decision variables for the optimization problemelemental to the MPC controller are as follows: the statesof the process x ( k ) = [ E bat ( k ) , T fr ( k )] T , where T fr ( k ) isthe refrigerator internal temperature; the control commands u ( k ) = [Γ( k ) , u fr ( k ) , u s ( k )] T , where Γ( k ) is the fraction of the normal battery charging energy; the internal variables v ( k ) = [ g ( k ) , ζ fr ( k )] T , where g ( k ) is the energy producedby the PV panels, and ζ fr ( k ) is a slack variable for re-frigerator temperature to ensure feasibility. The exogenousinputs whose predictions are assumed to be known for the N time steps w ( k ) = [ E pv ( k ) , T house ( k ) , E s ( k )] T , where E pv ( k ) is the available energy from the PV panels computedusing (9), E s ( k ) is the total secondary load and is assumedto be known ahead of time and can be decided by theoccupants, and T house ( k ) is the internal house temperaturewhich is estimated using a simple parametric model de-scribed in (5). Hence, the complete decision vector for theoptimization problem is given as [ X, U, V ] T , where X :=[ x ( k + 1) , . . . , x ( k + N )] T , U := [ u ( k ) , . . . , u ( k + N − T and V := [ v ( k ) , . . . , v ( k + N − T .A constrained optimization problem is solved, to generatecontrol commands for N time steps, which tries to maximizethe primary and secondary loads, while minimizing thebattery degradation and maximizing system life. This objec-tive is achieved subject to constraints on dynamics, energybalance, states and control commands. The optimizationproblem at any time index j is: min U j + N − (cid:88) k = j (cid:20) λ ( N − k ) ζ fr ( k ) − λ E bat ( k )+ (4a) λ Γ( k ) − λ ( N − k ) u s ( k ) (cid:21) , (4b)subject to: T fr ( k + 1) = AT fr ( k ) + Bu fr ( k ) Q fr + DT house ( k ) , (4c) E bat ( k + 1) = E bat ( k ) + Γ( k ) η c,dc,conbat ¯ E cbat , (4d) u fr ( k ) E fr + Γ( k ) ¯ E cbat + u s ( k ) E s ( k ) = g ( k ) , (4e) ¯ T fr ≤ T fr ( k ) ≤ ¯ T fr + ζ fr ( k ) , (4f) ζ fr ( k ) ≥ , (4g) ¯ E bat ≤ E bat ( k ) ≤ ¯ E bat , (4h) ¯ u s ≤ u s ( k ) ≤ ¯ u s ( k ) , (4i) ¯Γ ≤ Γ( k ) ≤ ¯Γ , (4j) ≤ g ( k ) ≤ E pv ( k ) , (4k)and only the j th control is implemented. The process isrepeated at j + 1 , · · · .The cost function in (4b) consists of four terms. The firstterm, λ ( N − k ) ζ fr ( k ) , penalizes the refrigerator temperatureslack variable which in turn tries to keep the refrigeratorinternal temperature within prescribed bounds. The timevarying weighing factor N − k puts a higher penalty onthe temperature violations at earlier times and less weighton violations during later times of the planning horizon. Thesecond term, − λ E bat ( k ) , penalizes a low state of chargewhich helps in extending the life-time of the system. Thethird term, λ Γ( k ) , penalizes fast charging, as it is unde-sirable since it degrades battery life. Γ models the fractionof the charging/discharging energy of the battery which arecontinuous and variable; moreover, it depends on the amountf energy available from the PV panels during charging,and the amount of the load demand to be supplied duringdischarging. The fourth term, − λ ( N − k ) u s ( k ) , along withthe inequality constraint (4i) maximizes the operation ofthe secondary loads when desired. The reason for the timevarying weight in this term is similar to that in the first term.The parameters λ , λ , λ , and λ are designer specifiedwights.The equality constraint (4c) is due to the thermal dynamicsof the refrigerator. This is the discretized form of the con-tinuous time refrigerator thermal dynamic model presentedin [13]. Where, Q fr ( W ) is the thermal power rejected bythe refrigerator to the ambient when the compressor is on,and T house ( ◦ C ) is the average internal house temperature. T house is computed using a data driven parametric modelgiven in (5) rather than relying on a house thermal dynamicsmodel, which increases the complexity of implementation.In (5), T meas is the the internal house temperature measuredusing a sensor, and T hist is the averaged historical ambienttemperature of the location in which the house is situatedobtained from a credible weather data repository. The av-eraged historical ambient temperature is created as a singledaily file for each day of the year. T house ( k ) = T meas , if k = 1( T meas − T hist ( k ))+ T hist ( k ) , if k ∈ , , .., N (5)In addition, Q fr = COP P ratedfr , where
COP is thecoefficient of performance, and P ratedfr is the rated powerconsumption of the refrigerator. A , B , and D are the discretetime equivalents of the continuous time model given in [13],which are functions of the thermal resistance ( R fr ) andthermal capacitance ( C fr ) of the refrigerator. The plant uti-lizes the exact same model for refrigerator thermal dynamics,hence there is no plat-model mismatch.The equality constraint (4d) is due to the battery energydynamics, where ¯ E cbat is the maximum battery chargingand discharging energy (which are assumed to be equal formodeling simplicity) in the normal mode, and η c,dc,conbat isthe charging-discharging efficiency of battery used in thecontroller. This battery dynamics differ from plant batterydynamics given in eq. (6) as it models the battery chargingand discharging energies with a single continuous variable( Γ ), hence there is no plat-model mismatch. The batterystorage system for the plant is simply modeled as a bucketof energy and its dynamics are as follows: E bat ( k + 1) = E bat ( k ) + η cbat E cbat ( k ) − E dcbat ( k ) η dcbat , (6)where, η cbat and η dcbat are the battery charging efficiencyand the battery discharging efficiency, respectively. The bat-tery energy level is bounded between the minimum ( ¯ E bat )and maximum ( ¯ E bat ) battery energy limits, i.e. E bat ∈ [¯ E bat , ¯ E bat ] . The charging and discharging energies for thebattery are constrained by the maximum charging and dis-charging energies as E cbat ∈ [0 , ¯ E cbat ] and E dcbat ∈ [0 , ¯ E dcbat ] . The equality constraint (4e) is the energy balance equation,where the electrical energy consumed by the primary (refrig-erator) and secondary (lights and fans) loads is simply theintegral of their rated powers times the number of individualload units.The inequality constraint (4f) is to maintain the refrig-erator temperature within the lower ( ¯ T fr ) and upper ( ¯ T fr )temperature limits. The inequality constraint (4g) is presentto not allow the refrigerator temperature slack to becomenegative. The inequality constraint (4h) bounds the batteryenergy between the minimum ( ¯ E bat ) and maximum ( ¯ E bat )battery energy limits. The inequality constraint (4i) is presentto force the secondary load control command to be zerowhen secondary loads are not desired to be turned on bythe occupants, where ¯ u s and ¯ u s are the lower and upperbound on u s respectively, and are defined as follows: ¯ u s ( k ) = (cid:40) , if E s ( k ) > , if E s ( k ) = 0 (7) ¯ u s ( k ) =0 , ∀ k = 1 , , . . . , N. (8)The inequality constraint (4j) bounds the fraction of batterycharging/discharging energy ( Γ ) between a minimum ( ¯Γ = − ) and maximum ( ¯Γ = 2 ) value. For negative valuesthe battery discharges; whereas for positive values till 1it charges in normal mode with a battery charging energyof ¯ E cbat as the maximum, for values above 1 it charges infast mode with twice the normal battery charging energy of × ¯ E cbat as the maximum.The inequality constraint (4k) bounds the energy producedby the PV panels such that it cannot be negative and is alwaysless than or equal to the maximum available PV energy( E pv ). E pv is estimated using the following: E pv ( k ) = N pv P ratedpv (cid:18) G ( k ) G std (cid:19) × ∆ t s , (9)where N pv is the number of PV panels, P ratedpv ( W ) isthe rated power output of PV module, G std ( W/m ) is thesolar irradiance standard test condition respectively, and G ( W/m ) is the current solar irradiance. Eq. (9) is a modifiedversion of that used in [14].The control variables u fr and u s are modeled as binaryinteger variables, taking values in { , } to turn the loadson and off respectively. This binary nature of the controlcommands makes this problem a Mixed Integer LinearProgram (MILP).The implementation of the control commands computedby the MPC Controller is as follows. The control commands u fr and u s are directly applied to the plant, turning therefrigerator and the secondary loads on and off dependingon whether u fr and u s are 1 and 0 respectively. However, Γ is converted into appropriate discrete decisions, c , d and x bat , which are then applied to the charge controller in theollowing manner: c ( k ) = (cid:40) , if Γ( k ) > , if Γ( k ) ≤ (10) d ( k ) = (cid:40) , if Γ( k ) < , if Γ( k ) ≥ (11) x bat ( k ) = , if < Γ( k ) ≤ , if < Γ( k ) ≤ , otherwise . (12) B. Baseline Controller
The baseline controller is what is commercially availablenow when one installs a PV+battery backup system. It simplysupplies power to the primary and secondary loads as longas there is enough supply from the PV and/or the battery.Otherwise it turns off supply to the loads. The thermostatcontrols the on-off ( u fr ) of the refrigerator compressor. Itturns the compressor ’on’ when the refrigerator internaltemperature ( T fr ) goes above the maximum limit ( ¯ T fr ),turns it ’off’ when T fr goes below the minimum limit ( ¯ T fr ),and uses the previous control command otherwise. Note thatturning the fridge on is only possible if there is power supply,otherwise a “on” decision has no effect. The charging ( c ) anddischarging ( d ) of the battery is controlled by the chargecontroller as: charging is turned ’on’ ( c = 1 ) when there isexcess energy available from PV after servicing both primaryand secondary load demands and is turned ’off’ ( c = 0 )otherwise; discharging is turned ’on’ ( d = 1 ) when theenergy available from PV is not sufficient to satisfy theprimary and secondary load demands and is turned ’off’( d = 0 ) otherwise. However, the amount of charging energydepends on the surplus energy production from the PV panelsafter the house loads have been serviced and the batteryenergy level as given in (2); and the amount of dischargeenergy depends on the house load energy not served by thePV panels and the battery energy level as given in (3). Thebaseline controller uses only the normal charging mode; itdoes not employ fast charging. C. Rule-Based Controller
The Rule-Based controller is an attempt to try and dupli-cate the MPC controller using rule based logic avoiding thesophisticated optimization approach. It consists of the follow-ing three sub-units which compute the control commands atevery time step;
1) N Time Steps Simulation Model:
It consists of theplant model, which utilizes the rule based baseline controllerwith fast charging capability, see Section III-C.3 . Thefast charging is limited to a certain number hours/day ,which is determined by the designer to minimize batterydegradation. It performs a N steps closed loop simulation ofthe plant; while utilizing the same information as the MPCcontroller. The refrigerator power supply control command( u fr ( k ) ) for the current time step is computed through thethermostat logic. Moreover, it computes the total house loaddemand serviced ( E total demand serviced ), with which the energy mismatch ( E Mis = E total demand serviced − E total demand desired ) iscomputed. E Mis is utilized to make the control decision forsecondary load as described in Section III-C.2.
2) Secondary Load Logic Controller:
It turns the sec-ondary load on ( u s ( k ) = 1 ) when there is no mismatch( E Mis ( k ) = 0 ) as computed from the N time stepssimulation model. Moreover, when mismatch is present( E Mis ( k ) < ) (i.e. generation is lower than the desiredtotal house load) it turns the secondary load on in such away that the mismatch is shedded through the secondaryload alone. Finally, when mismatch is larger than that canbe shed through the secondary load, it turns the secondaryload off ( u s ( k ) = 0 ).
3) Battery Logic Controller:
Once u fr ( k ) and u s ( k ) are decided as per Section III-C.1 and Section III-C.2respectively, total house load ( E hl ) is computed as E hl ( k ) = u fr ( k ) E fr + u s ( k ) E s ( k ) and the battery control commands( c ) and ( d ) are computed using the baseline battery controllerlogic given in Section III-B. While fast charging command( x bat ) is computed by augmenting this logic, where fastcharging is commanded when excess energy from the PVallows for it i.e. E pv ( k ) − E hl ( k ) > ¯ E cbat .IV. S IMULATION S TUDY S ETUP
The period selected for simulation is the time hurricaneIrma passed over Gainesville, FL, USA, starting from itslandfall on Sept. 11, 2017, to Sept. 17, 2017. Weather data isobtained from National Solar Radiation Database ( nsrdb.nrel.gov ). The simulations are run for 7 days starting at00:00 hours (midnight) at day 1 (September 11, 2017) witha planning horizon of 3 hours and a time step of 10 minutes( ∆ t s = 10 mins, N = 18 ) with battery initial state at ¯ E bat (i.e., E bat (0) = ¯ E bat ) and the refrigerator initial temperatureat 2 ◦ C (i.e., T fr (0) = 2 ◦ C ). A. Simulation Parameters
The house described in [15] consists of four bedrooms, aliving room, and a kitchen. Hence, during and post hurricaneperiod when power from grid is not available, the minimumload which will provide habitable conditions was decided tobe an LED light for each room, a fan for each bedroom, andone refrigerator in the kitchen. Fig. 2 illustrates the secondaryload trajectory for a given day which is composed of: LEDlights being on from 18:00 hours to 00:00 hours and fansrunning from 21:00 hours to 09:00 hours.
Fig. 2: Secondary load demand (daily trajectory).We selected the Canadian Solar CS6K-285 polycrystallinepanel ( $100 /panel ), and Trojan SPRE 12 225 (lead acidtype) solar battery unit ( $400 /unit ). Lead acid battery isselected over Lithium-Ion (Li-ion) battery despite the latteraving performance advantages over the former in orderto reduce cost, since Li-ion batteries are four times moreexpensive than lead acid batteries per kW h [16]. The systemwas sized to service the total house load of the refrigerator,fans and lights for one day, which yielded a system consistingof 3 PV panels ( W ) connected in parallel, 2 units ofbattery ( W h ) connected in series.The internal house temperature, T house ( k ) , for the plantis computed using the linear th order ODE model givenby [15], which models a typical, detached, two-story housein the USA.The parameters for the plant components; PV panels: N pv = 3 and P ratedpv = 285 W , G std = 1000 W/m ;Battery: ¯ E bat = 1080 W h , ¯ E bat = 5400 W h , ¯ E cbat = 810 W h , ¯ E dcbat = 844 . W h , η cbat = 0 . and η dcbat = 0 . andLoads: Refrigerator - P ratedfr = 250 W , ¯ T fr = 0 ◦ C , ¯ T fr = 4 ◦ C , Lights - N l = 6 , P ratedl = 8 W Fans - N f = 4 , P ratedf = 65 W . The inverter efficiency is η inv = 0 . .The parameters for the refrigerator thermal model are C fr = 8 . × J/ ◦ C , R fr = 1 . ◦ C/W and
COP = 0 . .The parameters for the MPC are λ = 1 , λ = 1 , λ = 1 , λ = 10 , η c,dc,conbat = 1 , ¯Γ = − and ¯Γ = 2 . B. Computation
The plant is simulated in MATLAB. The optimizationproblem is solved using GUROBI [17], a mixed integer linearprogramming solver, on a Desktop Linux computer with 8GBRAM and a 3.60 GHz × ESULTS AND D ISCUSSION
A. Performance Metrics for Controller Comparison:
The performance of a controller is quantified by how wellthe primary and secondary loads were serviced during anextended outage. Hence, two metrics have been designed tocompare the performance of the controllers. The PrimaryResiliency Metric (PRM) is defined as
PRM = 1 − (cid:82) T sim T fr ( t ) > ¯ T fr +2 dtT sim , (13)where T sim is the total simulation time, expressed inhours/day. It is a complement of the average hours per daythe refrigerator temperature was above the tolerable upperlimit. The Secondary Resiliency Metric (SRM) is defined as SRM = (cid:82) T sim u s ( t ) .E s ( t )= E s ( t ) dtT sim , (14)it is the percentage time secondary load was serviced com-pared to the desired secondary load trajectory. For bothmetrics, higher value means better performance. B. Performance Comparison of Controllers:
Figure 3 shows the simulation results when using theMPC, baseline and the Rule-Based controllers. The MPCand Rule-Based controllers keep the refrigerator temperaturewithin the prescribed limits for the entire 7 days with minorexcursions; see Figures 3a and 3c. In contrast, the baseline controller fails to do so for long periods; see Figure 3b.ThePRM of baseline controller is 15.64 hours/day, while itis well over 20 hours/day for the MPC and Rule-Basedcontrollers; see Table I. The Centers for Disease Controland Prevention state that perishable foods (including meat,poultry, fish, eggs and leftovers) in the refrigerator shouldbe thrown away if the power has been off for 4 hours ormore [2]. Meaning, a PRM of at least 20 hours/day is needed.Thus, while the MPC and Rule-Based controllers will be ableto keep perishable foods fresh for the entire seven days ofthe outage, with the baseline controller, the stored food willget spoiled after the very first day without grid power; of thethree controllers MPC has the best PRM.TABLE I: Performance comparison of MPC, baseline andRule-Based controllers.
Controller Type PRM ( h/day ) SRM (%)
MPC 24 57.69Baseline 15.64 47.94Rule-Based 22.04 48.21
Figures 3d, 3e and 3f show the trajectories of the sec-ondary loads serviced by the baseline and MPC controllersrespectively. It can be seen that none of the controllers areable to meet the secondary loads for the desired duration.However, the SRM of MPC controller is better (10%) thanthe other two controllers, while the baseline and Rule-Basedcontrollers perform similarly; see Table I.It is important to note that the SRM is poorest for theRule-Based controller as the refrigerator on-off decision ismade using the thermostat control and all other decisions aretaken based on it i.e. the primary load is always favored overthe secondary load.Hence, the MPC controller demonstrates superior perfor-mance in servicing both primary and secondary loads ascompared to the baseline and Rule-Based controllers. Thesuperior performance of the MPC controller is attributed to(i) its taking into account forecasts of disturbances (solarenergy available, and desired trajectory for the secondaryloads) in making decisions and (ii) making the trade offbetween various conflicting requirements by solving an op-timization problem. While the baseline controller operateswith the information consisting of just the present states( T fr ( k ) and E bat ( k ) ) of the system and its decision makingis simple (rule based). And, even with equivalent information(sensing, forecasts and plant model) as the MPC controller,the Rule-Based controller performs just slightly poorly thanMPC controller in terms of PRM, while performing poorlyin terms of SRM. This poor performance can be attributedto simple rule based decision making logic; which eventhough has access to same information as the MPC controller,is inadequate to make intelligent decisions to achieve thecontrol goals. As the ability to make excellent trade-offsinvolving conflicting requirements is almost impossible withrule based logic in this problem, which makes the MPCcontrol framework a rational choice for this problem.
24 48 72 96 120 144 168-505101520253035 020406080100 (a) MPC: T fr and SoC. (b) Baseline: T fr and SoC. (c) Rule-Based: T fr and SoC. (d) MPC: secondary performance. (e) Baseline: secondary performance. (f) Rule-Based: secondary perfor-mance. Fig. 3: Comparison of MPC, Baseline and Rule-Based controllers’ performances for the week after hurricane Irma (Sept.2017) in Gainesville, FL.
C. Effect of System Size on Controllers Performance:
Table II lists the various PV+battery sizes for whichsimulations were conducted.TABLE II: System size, and description
System size DescriptionA 3 PV panels + 2 Battery unitsB 3 PV panels + 4 Battery unitsC 4 PV panels + 2 Battery unitsD 4 PV panels + 4 Battery unitsE 5 PV panels + 4 Battery unitsF 6 PV panels + 4 Battery units
Fig 4 shows that for the baseline controller to achievea similar PRM as MPC controller, the system cost/sizehas to be doubled. It should be noted that the Rule-Based controller’s PRM is slightly lower as compared toMPC controller’s PRM at smaller system sizes; however, itachieves a similar PRM performance as the MPC controllerwith a / th increase in system cost/size demonstrating itsedge over the baseline controller. It can be noticed thatSRM is directly proportional to the system size. SRM ofMPC controller is always higher than that of the baseline(9.41% on average), while it is higher than that of the Rule-Based controller (6.53% on average) on all but the biggestsystem size where they are equal. Hence, the equipment costrequired to achieve a similar level of resiliency performancecan be halved by the MPC controller as compared to the A B C D E F152025 A B C D E F50100
Fig. 4: Resiliency performance metric of MPC, baseline andRule-Based controllers with system size.baseline controller.
D. Effect of House Temperature Parametric Model on MPCController:
It is observed that the resiliency performance of the MPCcontroller is unaffected by the choice of house temperaturemodel; see Table III. This demonstrates that the simpleparametric house temperature model developed in (5) doesot deteriorate the performance of the MPC controller ascompared to the complex linear th order ODE thermalmodel of the house given in [15]. Moreover, it helps usavoid this complex thermal model of the house to be part ofthe MPC controller. This is desirable as this thermal modelis designed for a typical detached two-story house in theUSA and would have required the additional effort of modelestimation for any other house.TABLE III: Performance comparison of MPC controller withhouse temperature models. House temperature model PRM ( h/day ) SRM ( % )Parametric 24 57.69Linear th order ODE 24 57.42 E. Effect of Fast Charging Hours on Rule-Based Controller:
For the MPC controller, fast charging is limited (whichlimits battery degradation) through cost function design. Onthe other hand in the Rule-Based controller this is done bysetting the maximum allowed fast charging hours for a givenday. It can be seen that the PRM of Rule-Based controller isnot affected much by increasing the maximum allowed fastcharging hours, while the SRM generally increases slightlyand plateaus at 5 hours of fast charging per day; see Table IV.TABLE IV: Performance of Rule-Based controller with fastcharging hours.
Fast charging hours ( hours ) PRM ( h/day ) SRM ( % )1 22.31 41.212 23 43.273 22.98 45.194 22.73 46.025 23 45.886 23 45.88 F. Effect of Planning Horizon on MPC and Rule-BasedControllers:
Generally, longer the planning horizon larger the futureinformation data is provided to the controller, and betterthe performance becomes. However, the computational costincreases with planning horizon for solving the underlyingoptimization problem in case of MPC based controllers.TABLE V: Performance comparison of MPC and Rule-Basedcontroller with planning horizons.
Controller MPC Rule-BasedPH ( hours ) PRM SRM PRM SRM1 13.08 17.86 19.75 50.413 24 57.69 22.04 48.216 23.96 56.32 22.98 47.1212 23.79 52.75 22.98 47.1224 23.75 53.02 23.81 37.77
It is observed from the Table V that the performance of theMPC controller is poorest with the smallest planning horizon(1 hour) indicating that future information is necessaryfor better performance, and performance peaks at 3 hours.However, further increase in planning horizon deterioratesthe performance owing to the GUROBI MILP solver notbeing successful in converging to a solution 100% times butrather stalling, see Table VI leading to utilization of a sub-optimal solution.TABLE VI: Computational performance of MPC controllerwith planning horizons.
Controller MPCPH ( hours ) Average solver time ( sec ) Solver success ( % )1 0.02 1003 0.06 1006 0.21 10012 5.97 97.8324 60 91.49 On the other hand, it can be observed that the PRM of theRule-Based controller improves with planning horizon, whileits SRM decreases; see Table VI. This demonstrates that theRule-Based controller is able to use the additional futureinformation well for the primary load, but is not intelligentenough to use it effectively for the secondary load.VI. C
ONCLUSION
The study provides further verification of the premise thatintelligence can reduce cost of energy resiliency to hurricaneinduced blackouts. Our preliminary work [12] showed thatintelligent controller using MPC performs better than asimple baseline controller that simply supplies demand untilenergy runs out. That was not surprising, but it was not clearwhether similar performance could be obtained by a littlemore intelligent decision making instead of using MPC ifit too can avail of the sensing and forecast information thatMPC uses. The results here shows that such an intelligentRule-Based controller requires a increase in system size,while the baseline controller requires a increase in sys-tem size, to provide the same level of resiliency performanceas the proposed MPC controller. For a fixed PV-battery size,the MPC controller moderately outperforms the Rule-Basedcontroller in servicing the primary load (refrigerator) andsignificantly outperforms it in servicing the secondary load.This observation indicates that intelligent and automateddecision making that is rooted in real-time optimizationprovides better performance compared to complex rule basedlogic even when the two are provided the same sensingand forecast information. The sensitivity studies conductedprovide further confidence on the performance of the MPCcontroller.This study opens up many directions for future research.These include analysis of sensitivity to forecast errors, reduc-ing the information requirements (both sensing and forecasts)of the control algorithm, optimal sizing of a PV-batteryystem taking into account resiliency during disasters andenergy savings during normal times.R
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