Online Combinatorial Optimization forInterconnected Refrigeration Systems:

Linear Approximation and Submodularity ?

Insoon Yang ∗

∗ Electrical Engineering Department, University of SouthernCalifornia, Los Angeles, CA 90089 USA (e-mail: insoonya@usc.edu)

Abstract: This paper proposes a new control method that can save the energy consumptionof multi-case supermarket refrigerators by explicitly taking into account their interconnectedand switched system dynamics. Its novelty is a bilevel combinatorial optimization formulation togenerate ON/OFF control actions for expansion valves and compressors. The inner optimizationmodule keeps display case temperatures in a desirable range and the outer optimization moduleminimizes energy consumption. In addition to its energy-saving capability, the proposed con-troller significantly reduces the frequency of compressor switchings by employing a conservativecompressor control strategy. However, solving this bilevel optimization problem associated withinterconnected and switched systems is a computationally challenging task. To solve the problemin near real time, we propose two approximation algorithms that can solve both the inner andouter optimization problems at once. The first algorithm uses a linear approximation, and thesecond is based on the submodular structure of the optimization problem. Both are (polynomial-time) scalable algorithms and generate near-optimal solutions with performance guarantees. Ourwork complements existing optimization-based control methods (e.g., MPC) for commercialrefrigerators, as our algorithms can be adopted as a tool for solving combinatorial optimizationproblems arising in these methods.

1. INTRODUCTION

Commercial refrigeration systems account for 7% of thetotal commercial energy consumption in the United States(U.S. Department of Energy [2012]). Therefore, there is astrong need for energy-efficient refrigeration systems, butresearch and development have focused on improving hard-ware rather than software, including control systems. Tra-ditionally, hysteresis and set point-based controllers havebeen used to maintain the display case temperature in adesirable range without considering system dynamics andenergy consumption. Over the past decade, however, moreadvanced control systems have been developed to saveenergy consumption using real-time sensor measurementsand optimization algorithms (see Section 1.1). Advancesin new technologies, such as the Internet of Things andcyber-physical systems, enhance the practicality of suchan advanced control system with their sensing, communi-cation, and computing capabilities (Graziano and Pritoni[2014]).

Supermarkets are one of the most important commercialsectors in which energy-efficient refrigeration systems areneeded. The primary reasons are twofold. First, supermar-ket refrigerators consume 56% of energy consumed by com-mercial refrigeration systems (Navigant Consulting, Inc.[2009]). Second, supermarkets operate with very thin profitmargins (on the order of 1%), and energy savings thussignificantly help their business: the U.S. EnvironmentalProtection Agency estimates that reducing energy costs

? This work was supported in part by the NSF under CRII:CPS(CNS1657100).

1 2 3 4 5 6 7 8 9 10

refrigerator

ambient air

display case1

Fig. 1. A supermarket refrigerator, which has 10 displaycases. Evaporator i controls the temperature of dis-play case i. Lines with arrows represent heat trans-fers between neighboring display cases or between adisplay case and ambient air.

by $1 is equivalent to increasing sales by $59 (ES2 [2008]).However, improving the energy efficiency of supermarketrefrigerators is a challenging task because food productsmust be stored at proper temperatures. Failure to do sowill increase food safety risks. The most popular refrig-erators in supermarkets are multi-display case units. Anexample is illustrated in Fig. 1. Each display case hasan evaporator controlled by an expansion valve, and aunit’s suction pressure is controlled by a compressor rack,as shown in Fig. 2. In typical supermarket refrigerators,controllers turn ON and OFF expansion valves and com-pressors to keep display case temperatures in a specificrange. Importantly, there are heat transfers between dis-play cases due to the interconnection among them. Notethat traditional hysteresis or set point-based controllersdo not take into account such heat transfers and thereforeperform in a suboptimal way.

This paper proposes a new control method that canimprove the energy efficiency of multi-case supermarket

display case 1

display case n

...

suct

ion

man

ifold

evaporator 1expansion valve 1

compressor rack

condenser

refri

gera

nt

(liqu

id)

Fig. 2. Schematic diagram of a supermarket refrigerator.

refrigerators by explicitly taking into account the inter-connected and switched dynamics of display case tem-peratures. The proposed controller receives sensor mea-surements and optimizes ON/OFF control actions for ex-pansion valves and compressors in near real time. Thenovelty of this work is a bilevel combinatorial optimizationformulation to generate such ON/OFF control signals inwhich (i) the inner combinatorial optimization module isresponsible for maintaining display case temperatures in adesirable range, and (ii) the outer combinatorial optimiza-tion module minimizes energy consumption. The primaryadvantage of the proposed approach is its energy savings.Because the controller explicitly takes into account the sys-tem dynamics and heat transfers, it effectively uses statemeasurements and optimizes control actions to save energywhile guaranteeing desired temperature profiles. In ourcase studies, the proposed control method saves 7.5–8% ofenergy compared to a traditional approach. The secondarybenefit of the proposed method is to reduce the frequencyof compressor switchings. It is known that frequent switch-ings of compressors accelerate their mechanical wear. Wepropose a conservative compressor control approach thatreduces fluctuations in suction pressure and thus decreasesthe compressor switching frequency. In our case studies us-ing a benchmark refrigeration system model, the proposedmethod reduces the switching frequency by 54–71.6%.

The proposed control method, however, presents a theoret-ical and algorithmic challenge because a bilevel combinato-rial optimization associated with a dynamical system mustbe solved in near real time. To overcome this challenge, wesuggest two approximation algorithms that can solve bothof the inner and outer optimization problems at once. Thefirst algorithm uses the linear approximation method de-veloped in our previous work (Yang et al. [2016]). The ap-proximate problem is a linear binary program, which canbe solved by an efficient and scalable single-pass algorithm.In addition, it simulates the dynamical system model onlyonce to generate control actions at each time point. Wealso show that the approximate solution obtained by thismethod has a provable suboptimality bound. The secondalgorithm is based on the submodular structure in the opti-mization problem. The inner optimization’s objective func-tion is submodular because opening an expansion valvewhen a smaller set of valves are opened gives a greatermarginal benefit than opening it when a larger set of valvesare already opened. We prove this intuitive submodularityproperty. Therefore, a greedy algorithm can be adopted toobtain a (1− 1

e )-optimal solution (Nemhauser et al. [1978]).In our case studies, the actual performance of the proposed

controller using these two algorithms is 98.9–99.5% of theoptimal controller.

1.1 Related Work

Several optimization-based control methods for commer-cial refrigerators have been developed over the past decade.One of the most popular methods is model predictivecontrol (MPC) although it is computationally challengingto apply standard MPC due to the switched dynamics ofrefrigeration systems. It is shown that the mixed logicaldynamical framework is useful to solve small-size prob-lems with a piecewise affine approximation of a systemmodel (Bemporad and Morari [1999], Larsen et al. [2005]).However, the practicality of this method is questionabledue to the high dimensionality of practical problems forsupermarket refrigerators, except for limited cases. Toovercome this limitation, Sarabia et al. [2009] carefully se-lects and parametrizes optimization variables to formulatethe problem as nonlinear MPC instead of hybrid MPC.Nonetheless this approach is computationally expensivebecause a nonlinear program with many variables mustbe solved in each MPC iteration. An alternative approachusing hierarchical MPC is proposed in Sonntag et al.[2008]. This method separates time scales into two: in everynonlinear MPC iteration, low-level temperature controllerswere employed, and the high-level optimization task isto determine optimal parameters for these controllers.However, this approach still presents the combinatorialgrowth of the search space. More recently, a sequentialconvex programming-based method is shown to be compu-tationally efficient in several case studies (Hovgaard et al.[2013]). It iteratively solves an optimization problem usingconvex programming, replacing the nonconvex cost func-tion with a convex approximation. In several numericalexperiments, this heuristic method generates high-qualitycontrol signals although it gives no theoretical performanceguarantee. We believe that our work is complementary tothe aforementioned methods. One of our main contribu-tions is to develop two efficient and scalable algorithms forresolving the computational challenge in discrete optimiza-tion problems associated with supermarket refrigerationsystems. These algorithms can be adopted as a tool forsolving combinatorial optimization problems in the afore-mentioned methods. We propose one of the most efficientcontrol architectures that use the algorithms.

2. SWITCHED DYNAMICS OF SUPERMARKETREFRIGERATION SYSTEMS

We consider a supermarket refrigerator in which multipledisplay cases are interconnected with one another. Forexample, Fig. 1 shows a refrigerator that has 10 displaycases. The temperature of each display case is controlledby an evaporator unit, where the refrigerant evaporatesabsorbing heat from the display case. Let evaporator i bein charge of display case i for i = 1, · · · , n, where n isthe number of display cases in all the refrigerators. Severaldynamic models of supermarket refrigeration systems havebeen proposed (Rasmussen and Alleyne [2006], Larsenet al. [2007], Li and Alleyne [2010], Rasmussen [2012],Shafiei et al. [2013] (see also the references therein)).Among those, we use the benchmark model of a typical

supermarket refrigeration system proposed in Larsen et al.[2007] and widely used in Sarabia et al. [2009], Sonntaget al. [2008], Yang et al. [2011], Vinther et al. [2015]. Thismodel is useful for simulating display case temperaturesand evaluating the performances of several controllers.

2.1 Display Cases and Evaporators

Display cases store food products and keep them refriger-ated. This refrigeration is due to the heat transfer betweenthe food product and the cold air in the display cases.Let Tfood,i and Tair,i denote the temperatures of the foodproduct and the air in display case i. The heat transferQfood→air,i between the food product and the air in displaycase i can then be modeled as

mfood,icfood,iTfood,i = −Qfood→air,i

= −kfood−air(Tfood,i − Tair,i),(1)

where mfood,i is the mass of the food product, cfood,i is theheat capacity of the food product and kfood−air is the heattransfer coefficient between the food product and the air.

The display case air temperature is affected by the heattransfers from the food product (Qfood→air,i), the ambientair (Qamb→air,i), the evaporator (−Qair→evap,i) and theneighboring display case air (

∑nj=1Qj→i). The refrigerant

flow into an evaporator is controlled by its expansion valve.Let ui be the valve control variable for evaporator i suchthat

ui(t) :=

0 if expansion valve i is closed at t1 otherwise.

Expansion valve i controls the refrigerant injection intoevaporator i and decreases the pressure of the refrigerantif it is open, as shown in Fig. 2. Then, the dynamics ofthe display case air temperature can be modeled as thefollowing switched interconnected system:

mair,icair,iTair,i

= Qfood→air,i +Qamb→air,i −Qair→wall,i +

n∑j=1

Qj→i

= kfood−air(Tfood,i − Tair,i) + kamb−air(Tamb − Tair,i)

− kair−evap(Tair,i − Tevapui) +

n∑j=1

ki,j(Tair,j − Tair,i),

(2)

where Tamb is the ambient air temperature, Tevap is therefrigerant’s evaporation temperature, kamb−air is the heattransfer coefficient between the ambient air and the displaycase air and ki,j is the heat transfer coefficient betweendisplay case i’s air and display case j’s air. Note thatki,j = 0 if display cases i and j are not neighbors. Fora more detailed model, one can separately consider thedynamics of the evaporator wall temperature Sarabia et al.[2009]. 1 However, the proposed model is a good approx-imation because the heat transfer coefficient between theevaporator wall and the refrigerant is five to ten timeshigher than other heat transfer coefficients Larsen et al.[2007].

The mass flow out of the evaporator can be computed as

fi :=1

∆tmref,i,

1 Alternatively, one can introduce a delay parameter, τ , and replaceTevapui(t) with Tevapui(t − τ) to explicitly take into account theeffect of the evaporator wall temperature.

where the refrigerant mass in the evaporator is controlledby the electronic expansion valve switching

mref,i =

mmax

ref if ui = 10 if ui = 0.

Depending on the specification of refrigerators, it takesa nontrivial amount of time to fill up the evaporator byrefrigerant. In this case, the dynamics of the refrigerantmass in the evaporator can be explicitly taken into accountSarabia et al. [2009]. Alternatively, one can introduce adelay-time constant, τ , and let mref,i(t) = mmax

ref ui(t − τ)to model the effect of the time to fill up the evaporator.

2.2 Suction Manifold and Compressor Rack

As shown in Fig. 2, the evaporated refrigerant with lowpressure from the outlet of the evaporator is compressedby the electric motors in the compressor bank. Eachrefrigerator could have multiple compressors and eachcompressor is switched ON or OFF. For example, all thecompressors are turned ON when maximal compression isneeded. The compressor bank is conventionally controlledby a PI controller to maintain the suction pressure withina bandwidth.

The suction manifold pressure Psuc evolves with the fol-lowing dynamics:

Psuc =1

Vsucrsuc

(n∑i=1

fi − ρsucnc∑i=1

Fc,i

), (3)

where Vsuc is the volume of the suction manifold, ρsuc isthe density of the refrigerant in the suction manifold, andrsuc := dρsuc/dPsuc. The variable Fc,i denotes the volumeflow out of the suction manifold controlled by compressori. Let uc,i be the control variable for compressor i, whereuc,i = 0 represents that compressor i is OFF and uc,i = 1represents that compressor i is ON. The volume flow Fc,iis then given by

Fc,i = kcuc,i :=ηVcomp

nuc,i,

where η is the volumetric efficiency of each compressor,and Vcomp denotes the compressor volume.

The total power consumption by the compressor rack isgiven by

p = ρsuc(hoc − hic)nc∑i=1

Fc,i,

where hic and hoc are the enthalpies of the refrigerantflowing into and out of the compressor, respectively. Thecompressed refrigerant flows to the condenser and is lique-fied by generating heat, as shown in Fig. 2. The liquefiedrefrigerant flows to the expansion valve, and as a result,the refrigeration circuit is closed.

2.3 Traditional Set-Point/PI-Based Control

A widely used control method consists of (i) a set-pointbased control of expansion valves, and (ii) a PI controlof compressors Larsen et al. [2007], Sarabia et al. [2009].The specific simulation setting used in this paper is con-tained in the extended version of this paper (Yang [2016]).We perturbed the mass of food products in each displaycase by ±20% from the nominal value mfood,i. Despite

Fig. 3. (a) The food temperatures in display cases 1–5, operated by the PI controller over 8 hours. Thetemperatures in all display cases are almost identical.(b) The number of ON compressors operated by thePI controller.

this heterogeneity, the set point-based controller almostidentically turns ON and OFF all the expansion valvesand therefore all the display case temperatures have al-most the same trajectory as shown in Fig. 3 (a). Thissynchronization is due to the decentralized nature of theset point-based controller: the control decision for expan-sion valve i depends only on its local temperature. Intu-itively, this decentralized controller is suboptimal becauseit does not actively take into account the heat transferbetween neighboring display cases. This inefficiency ofthe traditional control approach motivates us to developa new optimization-based control method that explicitlyconsiders the interdependency of display case temperaturedynamics.

Another disadvantage resulting from the synchronizationof expansion valves is the significant fluctuation of suctionpressure. Since the PI controller integrates the deviationof suction pressure from its reference, the output uPI(t)presents large and frequent variations. As a result, thenumber of ON compressors frequently varies as shown inFig. 3 (b). A frequent switching of compressors is a seriousproblem because it accelerates the mechanical degradationof the compressors. Our strategy to discourage frequentcompressor switchings is twofold: (i) our conservative com-pressor control method tries to maintain Psuc(t) = Psuc,not fully utilizing the pressure bandwidth ±DB, and (ii)our online optimization-based controller indirectly desyn-chronizes the ON/OFF operation of expansion valves. Thedetails about the two control methods are presented in thefollowing sections.

3. ONLINE COMBINATORIAL OPTIMIZATION

3.1 Conservative Compressor Control

We control the compressor rack to (approximately) main-tain the suction pressure as the reference Psuc, i.e.,

Psuc(t) ≈ Psuc ∀t.In other words, to make Psuc ≡ 0 in (3), we set uc :=(uc,1, · · · , uc,nc

) such that the refrigerant outflow from thesuction manifold is equal to the inflow:

ρsuc

nc∑i=1

kcuc,i ≈n∑i=1

fi. (4)

In practice, we may not be able to exactly satisfy thisequality because each uc,i is either 0 or 1. However, weassume that the compressor control action uc can be cho-sen to make the difference between the outflow and theinflow negligible. This compressor control rule is subopti-mal: it induces a conservative operation of the compressorrack that does not fully utilize the pressure bandwidth.However, this conservative control approach has a practi-cal advantage: it does not create significant compressorswitchings. Therefore, it can potentially decelerate themechanical wear of compressors. Under this compressorcontrol rule, the total power consumption can be computedas

p = (hoc − hic)n∑i=1

fi =(hoc − hic)mmax

ref

∆t

n∑i=1

ui. (5)

3.2 Bilevel Optimization Formulation

We consider a receding-horizon online optimization ap-proach to generate control signals for expansion valvesand compressors. Let t0, t1, · · · , tk, tk+1, · · · be the timesteps at which the control action is optimized. For thesake of simplicity, we describe a one-step look-ahead op-timization method; however, this approach can be easilyextended to multiple-step look-ahead optimization (seeRemark 2).

Inner problem for temperature management At time tk,we control the expansion valves to minimize the followingquadratic deviation from the upper-bound Tmax

i , i =1, · · · , n:

J(α) =

n∑i=1

∫ tk+1

tk

(Tair,i − Tmaxi )2

+dt,

where (a)2+ = a2 · 1a≥0, assuming Tair is evolving with

(1) and (2). Specifically, the expansion valve action at tkis generated as a solution to the following combinatorialoptimization problem:

minα∈0,1n

J(α) (6a)

s.t. x = Ax+Bu+ C, x(tk) = xmeas (6b)

u(t) = α, t ∈ (tk, tk+1] (6c)

‖α‖0 =

n∑i=1

αi ≤ K. (6d)

Here, x := (Tfood, Tair) and (6b) gives a linear systemrepresentation of the dynamics (1) and (2). Note that xmeas

represents (Tfood, Tair) measured at t = tk. 2 As specifiedin (6c), the control action over (tk, tk+1] is fixed as thesolution α. The last constraint (6d) essentially limits thepower consumed by the refrigeration system as K(hoc −hic)m

maxref /∆t due to (5). Therefore, the choice of K is

important to save energy: as K decreases, the powerconsumption lessens.

Outer problem for energy efficiency To generate anenergy-saving control action, we minimize the numberK of open expansion valves while guaranteeing that thequadratic deviation J(α) from the upper-bound Tmax

i , i =

2 If Tfood is not directly measured, an observer needs to be employedto estimate the state. Then, the control system uses the estimateT estfood instead of its actual measurement.

1, · · · , n is bounded by the threshold ∆. More precisely,we consider the following outer optimization problem:

minK ∈ 0, · · · , n | J(αopt(K)) ≤ ∆, (7)

where αopt(K) is a solution to the expansion valve opti-mization problem (6). Let Kopt be a solution to this prob-lem. Then, αopt(Kopt) is the expansion valve control actionthat saves energy the most while limiting the violation ofthe food temperature upper-bound Tmax

i , i = 1, · · · , n.This outer optimization problem can be easily solved bysearching K from 0 in an increasing order. Once we findK such that J(αopt(K)) ≤ ∆, we terminate the search

and obtain the solution as Kopt := K. In the followingsection, we will show that this procedure can be integratedinto approximation algorithms for the inner optimizationproblem.

Then, as specified in (6c), the controller chooses uopt(t) :=αopt(Kopt) for t ∈ (tk, tk+1]. Furthermore, it determines

the compressor control signal uoptc such that∑nc

i=1 uoptc,i ≈

mmaxref /(ρsuckc∆t) using (4). If Psuc(t) < Psuc, the con-

troller rounds mmaxref /(ρsuckc∆t) to the next smaller inte-

ger and then determines the number of ON compressorsas the integer. If Psuc(t) ≥ Psuc, the controller roundsmmax

ref /(ρsuckc∆t) to the nearest integer greater than orequal to it.

Remark 1. Our objective function J(α) only takes intoaccount the violation of temperature upper-bounds. Thischoice is motivated by the fact that the food temperaturein each display case increases as we close more expansionvalves, which is summarized in Proposition 1. In otherwords, as we reduce the number K of open valves in theouter optimization problem, the possibility of violatingtemperature upper-bounds increases, while it is less likelyto violate temperature lower-bounds. This monotonicityproperty of food temperatures justifies our focus on tem-perature upper-bounds.

Proposition 1. Let Tαfood,j and Tαair,j denote the food andair temperatures in display case j when the control actionα is applied. Then, for any α, β ∈ Rn such that

αi ≤ βi, i = 1, · · · , n,we have

Tαfood,j ≥ Tβfood,j and Tαair,j ≥ T

βair,j , j = 1, · · · , n.

Its proof can be found in Appendix A.

4. APPROXIMATION ALGORITHMS

We present two approximation methods for the inneroptimization problem. One is based on linear approxi-mation, and another utilizes submodularity. These willgive approximate solutions with guaranteed suboptimalitybounds. We further show that, by simply modifying theseapproximation algorithms, we can obtain a near-optimalsolution to the outer optimization problem.

4.1 Linear Approximation

We first consider a linear approximation-based approachto the inner combinatorial optimization problem (6). It isconvenient to work with the following value function:

V (α) = J(0)− J(α). (8)

Initialization:α← 0;

Construction of d:Compute DV (0);Sort the entries of DV (0) in descending order;Construct d : 1, · · · , n → 1, · · · , n satisfying (10);

Solution of (9):while [DV (0)]d(i) > 0 and i ≤ Kαd(i) ← 1;i← i+ 1;end

Algorithm 1. Algorithm for the approximate problem (9)

The value V (α) represents the reduction in the quadraticdeviation from from the upper-bound Tmax

i , i = 1, · · · , n,when expansion valve j is chosen to be open only for j suchthat αj = 1. Note that this value function is normalizedsuch that V (0) = 0. The Taylor expansion of V at 0gives V (α) = DV (0)>α + O(α2) assuming the derivativeDV is well-defined. This motivates us to consider thefollowing first-order approximation of the expansion valveoptimization problem (6):

maxα∈0,1n

DV (0)>α : ‖α‖0 ≤ K. (9)

The ith entry [DV (0)]i of the derivative represents themarginal benefit of opening expansion valve i. There-fore, the approximate problem (9) can be interpreted asmaximizing the marginal benefit of valve operation whileguaranteeing that the number of open valves is less than orequal to K. A detailed method to define and compute thederivative can be found in Yang et al. [2016]. Computingthe derivative should also take into account the depen-dency of the state x on the binary decision variable α. Forexample, an adjoint-based approach can be used to handlethis dependency (Kokotovic and Heller [1967]).

The first advantage of the proposed approximation ap-proach is that it gives an approximate solution with aprovable suboptimality bound. The bound is a posteriori,which does not require the globally optimal solution αopt

but the solution α? of (9).

Theorem 1. (Yang et al. [2016]). Let α? be a solution tothe approximate problem (9). If DV (0)>α? 6= 0, then thefollowing suboptimality bound holds:

ρV (αopt) ≤ V (α?),

where

ρ =V (α?)

DV (0)>α?≤ 1.

If DV (0)>α? = 0, then V (αopt) = V (0) = 0, i.e., 0 is anoptimal solution.

Its proof is contained in Appendix B. This theoremsuggests that the approximate solution’s performance isgreater than (ρ× 100)% of the globally optimal solution’sperformance.

The second advantage of the proposed method is that ityields an efficient algorithm to solve the approximate prob-lem (9). Specifically, we design a very simple algorithmbased on the ordering of the entries of DV (0). Let d(·)denote the map from 1, · · · , n to 1, · · · , n such that

[DV (0)]d(i) ≥ [DV (0)]d(j) (10)

for any i, j ∈ 1, · · · , n such that i ≤ j. Such a map canbe constructed using a sorting algorithm with O(n log n)complexity. Such a map may not be unique. We letαd(i) = 1 for i = 1, · · · ,K if [DV (0)]d(i) > 0. A moredetailed algorithm to solve this problem is presented inAlgorithm 1. Note that it is a single-pass algorithm, i.e.,does not require multiple iterations. For large problems,one may be able to use an O(n) algorithm without sorting(Balas and Zemel [1980]).

Remark 2. The proposed linear approximation methodis applicable to multi-period optimization problems, in

which the objective is given by J(α) :=∑Nperiod

k=1 Jk(αk)and the control variable is time-varying, i.e., α =(α1, · · · , αNperiod) ∈ Rn×Nperiod . In such a case, we com-pute the derivative DVk of Vk(αk) := Jk(0) − Jk(αk) foreach k. The objective function can be approximated as∑Nperiod

k=1 DVk(0)>αk, which is still linear in α.

4.2 Submodularity

The second approach gives another approximate solutionof the expansion valve optimization problem (6) with asuboptimality bound. This solution is generally differentfrom the solution obtained by the first approach. LetΩ := 1, · · · , n be the set of expansion valves to becontrolled. We define a set function, V : 2Ω → R, as

V(X) = V (I(X)),

where the value function V is defined as (11) and I(X) :=(I1(X), · · · , In(X)) ∈ 0, 1n is the indicator vector of thesetX such that Ii(X) := 0 if i /∈ X and Ii(X) := 1 if i ∈ X.In other words, V is a set function representation of V . Theexpansion valve optimization problem (6) is equivalent toselecting the set X ⊆ Ω such that |X| ≤ K to maximizethe value function V(X), i.e.,

maxX∈2Ω

V(X) : |X| ≤ K. (11)

We observe that the value function V has a useful struc-ture, which is called the submodularity. It represents a di-minishing return property such that opening an expansionvalve when a smaller set of valves is opened gives a greatermarginal benefit than opening it when a larger set of valvesis already opened.

Theorem 2. The set function V : 2Ω → R is submodular,i.e., for any X ⊆ Y ⊆ Ω and any a ∈ Ω \ Y ,

V(X ∪ a)− V(X) ≥ V(Y ∪ a)− V(Y ).

Furthermore, it is monotone, i.e., for any X ⊆ Y ⊆ Ω

V(X) ≤ V(Y ).

See Appendix C for a proof.

Initialization:X ← ∅;Greedy algorithm:while i ≤ Ka∗ ∈ arg maxa∈Ω\X V(X ∪ a);X ← X ∪ a∗;i← i+ 1;end

Algorithm 2. Greedy algorithm for (11)

The submodularity of V guarantees that Algorithm 2,which is a greedy algorithm, provides an (1 − 1

e )-optimal

solution. In other words, the approximate solution’s per-formance is greater than (1 − 1

e ) ≈ 63% of the oracle’sperformance. In our case study, the actual submodularityis 98.9%, which is significantly greater than this theoreticalbound.

Theorem 3. (Nemhauser et al. [1978]). Algorithm 2 is a(1− 1

e

)-approximation algorithm. In other words, if we

let X? be the solution obtained by this greedy algorithm,then the following suboptimality bound holds:(

1− 1

e

)V(Xopt) ≤ V(X?),

where Xopt is an optimal solution to (11).

Algorithm 2 makes a locally optimal choice at each iter-ation. Therefore, it significantly reduces the search space,i.e., it does not search over all possible combinations ofopen expansion valves.

while [DV (0)]d(i) > 0 and J(α) > ∆αd(i) ← 1;i← i+ 1;end

Algorithm 3. Modified version of Algorithm 1 for the outer opti-mization problem (7)

4.3 Modified Algorithms for the Outer Problem

We now modify the two approximation algorithms for theinner problem (6) to solve the full bilevel optimizationproblem. In both Algorithms 1 and 2, the expansionvalve chosen to be open at iteration i is independent ofthe selections at later iterations. This independency playsan essential role in incorporating the outer optimizationproblem into the algorithms. To be more precise, wecompare the cases of K = l and K = l+1. Let αl and αl+1

be the solutions in the two cases obtained by Algorithm 1.Since the expansion valve selected to be open at iterationl+1 does not affect the choices at earlier iterations, we haveαld(i) = αl+1

d(i) for i = 1, · · · , l. Therefore, we do not have to

re-solve the entire inner optimization problem for K = l+1if we already have the solution for K = l; it suffices to runone more iteration for i = l + 1 to obtain αl+1

d(l+1). This

observation allows us to simply modify the last part ofAlgorithm 1 as Algorithm 3. We select expansion valvesto be open until the temperature upper-bound violationJ(α) is less than or equal to the threshold ∆. Similarly,we modify the last part of Algorithm 2 as Algorithm 4 tosolve the outer problem.

while J(I(X)) > ∆a∗ ∈ arg maxa∈Ω\X V(X ∪ a);X ← X ∪ a∗;i← i+ 1; end

Algorithm 4. Modified version of Algorithm 2 for the outer opti-mization problem (7)

5. CASE STUDIES

In this section, we examine the performance of the pro-posed online optimization-based controllers. For fair com-parisons with the traditional controller, we use the param-eter data reported in the extended version (Yang [2016]).

Fig. 4. The food temperatures (in 5 display cases out of10) controlled by (a) the linear approximation-basedalgorithm (Algorithm 3), and (b) the submodularoptimization algorithm (Algorithm 4).

5.1 Energy Efficiency

As opposed to the synchronized food temperature profilescontrolled by the traditional method (see Fig. 3 (a)), theproposed controllers induce alternating patterns of thetemperatures as shown in Fig. 4. Such patterns result fromthe explicit consideration of heat transfers between neigh-boring display cases in the optimization module throughthe constraint (6b), which represents the interconnectedtemperature dynamics. Using the spatial heat transfers,the proposed controllers do not turn ON or OFF all theexpansion valves at the same time. Instead, they predictthe temperature evolution for a short period and selectsthe valves to turn ON that are effective to minimize thedeviation from the desirable temperature range during theperiod. As a result, the ON/OFF operation of expansionvalves is desynchronized, unlike in the case of the tra-ditional controller. This desynchronization maintains thetemperatures near the upper-bound Tmax reducing tem-perature fluctuations. Therefore, it intuitively improvesenergy efficiency. As summarized in Table 1, the proposedcontrollers save 7.5–8% of energy. Note that the outeroptimization module minimizes the total energy consump-tion while the inner optimization module is responsible formaintaining the temperature profiles in a desirable range.When the bilevel combinatorial problem is exactly solvedfor all time, the average power consumption is 10.29kW.Therefore, the two proposed controllers’ performances are99.5% and 98.9% of the optimal controller although theirtheoretical suboptimality bounds are 39% and 63%.

Table 1: Energy savings by the proposed controllersPI linear submodular

average kW 11.24 10.34 10.40energy saving – 8.0% 7.5%suboptimality 90.1% 99.5% 98.9%

5.2 Reduced Compressor Switching

Another advantage of the proposed controllers is the con-siderable reduction on the number of compressor switchinginstances. By desynchronizing the switching instances ofexpansion valves in the inner optimization module, theproposed controllers significantly reduce the variation ofsuction pressure. Our conservative compressor control ap-proach presented in Section 4 also helps to minimize the

Fig. 5. The number of ON compressors controlled by (a)the linear approximation-based algorithm (Algorithm3), and (b) the submodular optimization algorithm(Algorithm 4).

deviation of the suction pressure from its reference. As aresult, the controllers significantly reduce the fluctuationson the number of ON compressors as shown in Fig. 5. First,the maximum number of ON compressors is decreasedfrom six to two. This reduction suggests that a mechan-ically more compact compressor or a smaller number ofcompressors in the rack may be enough if the proposedcontrollers are adopted. Second, the proposed controllersreduce the number of compressor switching instances by54.0–71.6% as summarized in Table 2. These infrequentcompressor operation strategies are beneficial to deceleratethe mechanical degradation of compressors.

Table 2: Compressor switching reductions by the proposedcontrollers

PI linear submodular# of switchings 324 92 149reduction – 71.6% 54.0%

6. CONCLUSIONS

The proposed controller explicitly takes into account theswitched and interconnected dynamics, and is thereforeis suitable for multi-case supermarket refrigeration sys-tems. However, it has to solve a bilevel combinatorialoptimization problem in near real time, which is a chal-lenging task. To overcome this difficulty, we proposed twopolynomial-time approximation algorithms that are basedon the structural properties of this optimization problem.We demonstrated the performance of the proposed con-trollers through case studies using a benchmark refrigera-tion system model and found that (i) they improve energyefficiency by 7.5–8% and (ii) they reduce the number ofcompressor switchings by 54–71.6%.

Appendix A. PROOF OF PROPOSITION 1

We use the linear system representation (6b) of the foodand air temperature dynamics (Equations (1) and (2)).We first notice that Ai,j ≥ 0 ∀i 6= j, where Ai,j rep-resents the (i, j)th entry of the matrix A. Furthermore,kair−evapTevap ≤ 0 due to the non-positive evaporator tem-perature. Hence, we have Bi,j ≤ 0 ∀i, j. Using PropositionIII.2 in Angeli and Sontag [2003], we conclude that the

system (6b) is input-monotone such that for any α, β ∈ Rnwith αi ≤ βi, i = 1, · · · , n,

xαi ≥ xβi , i = 1, · · · , n,

where xα denotes the solution of the system (6b) when itsinput is chosen as α.

Appendix B. PROOF OF THEOREM 1

In (6b), we notice that

x(t) = eA(t−tk)xmeas +

∫ t

tk

eA(t−s)Bαds,

which implies that x(t) is linear in α. Therefore, V isconcave with respect to α in a continuously relaxed space,Rn. Then, the result follows from Theorem 2 in Yang et al.[2016].

Appendix C. PROOF OF THEOREM 2

Let TXfood,i denote the temperature of the food product indisplay case i given that the expansion valves in X areopen. Due to the linearity of the system dynamics (1) and(2), TXfood,i is modular, i.e.,

TXfood,i =∑a∈X

Tafood,i.

Therefore, for any X ⊆ Y ⊆ Ω and any a ∈ Ω \ Y

TX∪afood,i − T

Xfood,i = T

Y ∪afood,i − T

Yfood,i.

Furthermore, Proposition 1 yields the following mono-tonicity result: for any X ⊆ Y ⊆ Ω

TXfood,i ≥ TYfood,i,i.e., as we open more expansion valves, the food temper-ature decreases. Lastly, the concavity of V(X) = V(∅) −∑ni=1

∫ tk+1

tk(TXfood,i − Tmax

i )2+dt in TXfood,i implies that X ⊆

Y ⊆ Ω and any a ∈ Ω \ YV(X ∪ a)− V(X) ≥ V(Y ∪ a)− V(Y ).

Therefore, V is submodular. Its monotonicity follows fromProposition 1.

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