Economically Optimal Control of a Cold Room Using an Artificial Neural Network and Dynamic Programming

9th IFAC Conference on Manufacturing Modelling, Management and 9th IFAC Conference on Manufacturing Modelling, Management and Control 9th IFAC Conference on Manufacturing Modelling, Management and Available online at www.sciencedirect.com Control 9th IFAC Conference on Manufacturing Modelling, Management and Berlin, Germany, August 28-30, 2019 Control Berlin, Germany, August 28-30, 2019 Control Berlin, Germany, August 28-30, 2019 Berlin, Germany, August 28-30, 2019

ScienceDirect

IFAC PapersOnLine 52-13 (2019) 2002–2007

Economically Optimal Control of a Cold Economically Optimal Control of a Cold Economically Optimal Control of a Cold Room Using an Artificial Neural Network Economically Optimal Control of a Cold Room Using an Artificial Neural Network Room Using an Artificial Neural Network and Dynamic Programming Room Using an Artificial Neural Network and and Dynamic Dynamic Programming Programming and Dynamic Programming Alnour Ribault ˚,˚˚ Samuel Vercraene ˚ S´ ebastien Henry ˚

Electricity Electricity Electricity Electricity Electricity price price price price price (€/MWh) (€/MWh) (€/MWh) (€/MWh) (€/MWh)

˚ Alnour Ribault ˚,˚˚ Samuel Vercraene ebastien Henry ˚˚ ˚ ˚˚ ˚,˚˚ Ouzrout ˚ S´ Yacine Lucie Peguet Alnour Ribault Samuel Vercraene S´ ebastien Henry ˚ ˚˚ ˚,˚˚ Ouzrout ˚ ˚ Yacine Lucie Peguet ˚ ˚˚ Alnour Ribault Samuel Vercraene S´ ebastien Henry Yacine Ouzrout Lucie Peguet ˚ ˚˚ Yacine Ouzrout Lucie Peguet ˚ ˚ University of Lyon, DISP Laboratory, France Lyon, DISP Laboratory, France of ˚ University ˚˚ University of Lyon, Laboratory, Pool, DISP Chamb´ ery, FranceFrance ˚˚ Energy ˚ Energy Pool, DISP Chamb´ ery, FranceFrance ˚˚ University of Lyon, Laboratory, Energy Pool, Chamb´ ery, France ˚˚ Energy Pool, Chamb´ery, France Abstract: We consider the economically optimal control of a cold store with a single cold Abstract: We consider the economically optimal control of a cold store with a single cold room. The thermal inertia of economically a cold room acts as an energy of storage and can therefore be used Abstract: We consider optimal a coldand store a single cold room. The thermal inertiathe of economically a cold room acts as ancontrol energy of storage canwith therefore be used Abstract: We consider the optimal control a coldand store with a asingle cold for economic optimization in athe presence of aasdynamic electricity price, under bounding room. The thermal inertia of cold room acts an energy storage can therefore be used for economic optimization in the presence of aaasdynamic electricity price, under aa bounding room. The thermal inertia of a cold room acts an energy storage and can therefore be used for economic optimization in the presence of dynamic electricity price, under bounding constraint on the internal temperature of the cold room. However, aa high number of room.electricity However, price, highunder number of frost frost constraint on optimization the internal in temperature of the for economic the premature presence ofwear a cold dynamic a Since bounding production startups may induce of room. the cold store’s compressors. the constraint on the internal temperature of the cold However, a high number of frost store’s acompressors. Since the production on startups may induce premature wear of room. the cold constraint the internal temperature of the cold However, high number of frost thermal losses are a function of the premature internal temperature of the cold room, conventional inventory production startups may induce wear of the cold store’s compressors. Since the thermal losses are a function of the premature internal temperature of the cold room,compressors. conventional Since inventory production startups may induce wear of the cold store’s the management solving techniques are not suited for this problem. In this paper, we use an artificial thermal losses are a function of the internal temperature of the cold room, conventional inventory areinternal not suited for this problem. In this paper, we use aninventory artificial management solving techniques thermal losses are a function of the temperature of the cold room, conventional management solving techniques are not suited for this problem. In this paper, we use an artificial neural network as temperature forecast. A dynamic programming algorithm is used to solve the neural network as temperature forecast. A dynamic algorithm isweused to the management solving techniques are not suited for thisprogramming problem.temperature In this paper, use and an solve artificial model that includes the non-linear artificial neural network forecast a fixed neural as temperature forecast. A dynamic algorithm is used to solve the artificial neural programming model network that includes the non-linear network temperature forecast and a fixed neural network as temperature forecast. A dynamic programming algorithm is used to solve the cost atthat eachincludes compressor startup. This allowsneural us to network solve industrial instances of the problem model the non-linear artificial temperature forecast and a fixed cost atthat eachincludes compressor startup. This allowsneural us to network solve industrial instances of the problem model the non-linear artificial temperature forecast and a fixed optimally and within reasonable time. We showus thetointerest of solving instances the problem optimally as cost at each startup.time. ThisWe allows solve industrial of the problem optimally andcompressor within reasonable show the of the optimally as cost at each compressor startup. ThisWe allows tointerest solve industrial instances the problem optimally within reasonable time. showus the interest of solving solving the problem problem optimally as opposed toand using a conventional hysteresis-based control method, and discuss theofopportunity of opposed toand using a conventional hysteresis-based control method, and discuss the opportunity of optimally within reasonable time. We show the interest ofinstead solving the problem optimally as using an dynamic price based on the electricity market aa traditional contracted opposed to using ahourly conventional hysteresis-based control method, and of discuss the opportunity of using an dynamic hourly price based on the electricity market instead of traditional contracted opposed to using © ahourly conventional hysteresis-based control method, and of discuss the opportunity of price. an Copyright 2019 IFAC using dynamic price based on the electricity market instead a traditional contracted price. Copyright 2019 using dynamic© price based on the electricity market instead of a traditional contracted price. an Copyright ©hourly 2019 IFAC IFAC © 2019,Copyright IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. price. © 2019 IFAC Keywords: Energy systems, Industrial applications of optimal control, Energy Storage Keywords: Energy systems, Industrial applications of optimal control, Energy Storage Operation and Planning, Neural networks, Modeling for control optimization Keywords: Energy systems, Industrial applications of for optimal control, Energy Storage Operation Planning, Neural networks, Modeling control optimization Keywords: Energy systems, Industrial applications of for optimal control, Energy Storage Operation and and Planning, Neural networks, Modeling control optimization Operation and Planning, Neural networks, Modeling for control optimization 1. INTRODUCTION and deterministic for at least the next 12 hours and at 1. INTRODUCTION INTRODUCTION and deterministic for at least the next 12 hours and at 1. mostdeterministic the next 36 hours. and for at least the next 12 hours and at most the 1. INTRODUCTION and deterministic for at least the next 12 hours and at the next next 36 36 hours. hours. In 2002, the total cold storage volume in Europe was most 70 the next 36 hours. In 2002, the total cold storage volume in Europe was most 3 70 In 2002, the total cold storage in Europe estimated to be between 60 and volume 70 millions m3 by was the 70 estimated to between 60 70 m the 60 In 2002, the total coldof storage in Europe was estimated to be be between 60 and and volume 70 millions m3 by 70 International Institute Refrigeration. Cold stores were millions by the 60 3 International Institute of Refrigeration. Cold were 60 3 stores estimated to be between 60 and 70 millions m by the International Institute of Refrigeration. Cold stores were estimated to use between 30 and 50 kWh / m 3 / year. Cold 50 60 estimated to use between 30 and 50 kWh / m / year. Cold 50 3 International Institute of30 Refrigeration. Cold stores were stores in Europe are thus estimated to use between 1800 estimated to use between and 50 kWh / m / year. Cold 50 3 stores in Europe are thus30estimated to use between 1800 40 estimated to use between and 50 kWh / m / year. Cold 50 and 3500 GWh / year. Therefore, the amount of energy 40 stores in Europe are thus estimated to use between 1800 and 3500 GWh /are year. Therefore, the amount of energy 40 stores in Europe thus estimated to use between 1800 consumed by the/cold storage industry isamount considerable and 30 and 3500 GWh year. Therefore, the of energy 40 P/O Winter consumed by the cold storage industry is considerable and 30 and 3500 GWh /cold year. Therefore, theisamount of energy P/O Winter consumed by the storage industry considerable and 30 has an impact on electricity production. SPOT February 11, 2017 P/O Winter has an impact on electricity production. 20 SPOT February 11, 2017 consumed by the storageproduction. industry is considerable and 30 SPOT March 4, 2017 20 SPOT February 11, 2017 has an impact oncold electricity P/O Winter SPOT March 4, 2017 20 SPOT March 4, 2017 has an impact on electricity production. February 11, 2017 12am 06am 12pm 06pm 20 12am 06am 12pm 06pm 4, 2017 SPOT March 12am 06am 12pm 06pm 1.1 Context of electricity production Hour 1.1 Context of electricity production Hour 12am 06am 12pm 06pm 1.1 Context of electricity production Hour 1.1 Context of electricity production Hourprices. Fig. 1. Comparison of electricity The balance between supply and demand is a critical issue Fig. 1. Comparison of electricity prices. The balance between supply and demand is a critical issue Fig. 1. Comparison of electricity prices. on To this balance, production and The balance grid. between supply and a critical issue on electrical electrical To achieve achieve this demand balance,is production and Fig. 1. Comparison of electricity prices. The balance grid. between supply and demand ismodulating a critical issue 1.2 Context consumption are controlled. For instance, the on electrical grid. To achieve this balance, production and Context of of cold cold storage storage consumption are controlled. For instance, modulating the 1.2 on electrical grid. To achieve thisinstance, balance, production and 1.2 Context of cold storage price of electricity is an incentive mechanism that encourconsumption are controlled. For modulating the price of electricity is an incentive mechanism that encourContext of must cold storage stored goods within a given consumption aretocontrolled. For instance, modulating the 1.2cold price of electricity is antheir incentive mechanism that encourages consumers electricity consumption when ages consumers to shift shift electricity consumption when A A cold storage storage must maintain maintain stored goods within a given price of electricity is anistheir incentive mechanism that encourthe price of electricity high. temperature range. This constraint is extremely important ages consumers to shift their electricity consumption when A cold storage must maintain stored within a given the price of electricity istheir high. temperature range. This constraint is goods extremely important ages consumers to shift electricity consumption when A cold storage must maintain stored goods within given for cold storage because a failure to enforce ita might the price of electricity is high. temperature range. This constraint is extremely important for cold storage because a failureis to enforceimportant it might In this paper, we will use two classical incentive prices. The the price of electricity is high. temperature range. This constraint extremely In this paper, we will use two classical incentive prices. The for damage storedbecause goods and the worst case might entail coldthe storage a in failure to enforce it might the stored goods and in the case might entail first one is Peak/Offpeak (P/O), with fixed In this will use twoprice classical incentive The damage for coldhazards storage because aconsumer failure to enforce it might damage the stored goods and in the worst worst casecase might entail first onepaper, is the thewe Peak/Offpeak price (P/O), with a aprices. fixed price price health for the end in the of edible In this paper, we will use two classical incentive prices. The health hazards for the end consumer in the case of edible for daytime and another fixed price for nighttime. The first one is the Peak/Offpeak price (P/O), with a fixed price damage the stored goods and in the worst casecase might entail for daytime and another fixed price for nighttime. The products. health hazards for the end consumer in the of edible first oneone is the Peak/Offpeak price (P/O), a fixed price for daytime and another fixed price forwith nighttime. The products. second is the Day-ahead market price of the European health hazards for the end consumer in the case of edible second one is the Day-ahead market price of the European products. for daytime another fixed priceprice for nighttime. The Cold stores are non-linear and highly complex systems. second one is and the Day-ahead market of the European Power Exchange, which be SPOT price in Power Exchange, which will will market be called called SPOT price in the the products. Cold stores are non-linear and highly complex systems. second one the Day-ahead price of the European rest of Exchange, this ispaper. Figure 1 presents an instance ofinP/O Many stores different exist for coldsystems. rooms. Power which will be called SPOT price the Cold areinstallation non-linear structure and highly complex rest of this paper. Figure 1 presents an instance of P/O Many different structure exist for coldsystems. rooms. Power Exchange, which will be called SPOT price in the Cold areinstallation non-linear and highly complex priceofand 2 paper. SPOT Figure prices. 1The SPOTanprice is published In outstores case we focus on a structure simple setting of acold coldrooms. store rest this presents instance of P/O Many different installation exist for price and 2 SPOT prices. The SPOT price is published In out case we focus on a simple setting of a cold store rest ofand this paper. Figure 1The presents anprice instance of P/O Many different installation structure exist for rooms. price 2PM SPOT prices. SPOT issupply published In out case we focus on a Figure simple2). setting of acold cold store around 12 for the next day based and and with a single coldroom (see First, a heat transfer around 12 PM for the next day based and supply and with a single coldroom (see Figure 2). First, a heat transfer price 2PM SPOT prices. The SPOT price issupply published out case we focus on simple setting of abycold store aroundand 12 for Therefore, the next day based and and In with ais single coldroom (seeastate Figure 2). First, a heat transfer demand forecasts. the SPOT price is known fluid converted from gas to liquid state a cooling demand 12 forecasts. the based SPOT pricesupply is known fluid ais single converted from gas to 2). liquid state by atransfer cooling around PM for Therefore, the next day and with coldroom (seestate Figure First, a heat demand forecasts. Therefore, the SPOT and price is known fluid is converted from gas state to liquid state by a cooling demand Therefore, theFederation SPOT ofprice is known fluid is by converted from state to liquid state by a cooling 2405-8963 forecasts. © 2019, IFAC (International Automatic Control) Hosting Elsevier Ltd. Allgas rights reserved.

Copyright © 2019 IFAC 2037 Copyright 2019 IFAC 2037Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2019 IFAC 2037 10.1016/j.ifacol.2019.11.497 Copyright © 2019 IFAC 2037

2019 IFAC MIM Berlin, Germany, August 28-30, 2019

Alnour Ribault et al. / IFAC PapersOnLine 52-13 (2019) 2002–2007

2003

unit which contains one or several compressors. Once the fluid is converted to liquid, it is pumped towards the cold room. Several evaporators physically present in the coldroom use the liquid fluid to release negative thermal energy (i.e. cold) in the coldroom. The compressors and the condenser works synchronously. They will be denoted as frost production facility in the rest of the paper. The efficiency of a frost production facility is a non-linear, increasing function of its production level.

problem. A hysteresis control policy is presented in Section 5, and the comparison between optimal control and hysteresis control is discussed in Section 6 as well as other numerical results. Conclusion and perspectives are given in Section 7.

Due to thermal phenomena that generate premature wear, compressors are subject to a higher risk of failure when restarting too often. This can entail costly maintenance or the replacement of the machine. In this paper the wear is modeled as a fixed cost when starting a compressor. Moreover, based on professional practices, the production levels of the frost production facility is considered discrete.

2.1 Lot-sizing

The heat exchanges between the cold room and the evaporators, the outside environment, the heat produced by the bodies of the personal, the solar radiation, and other factors, cause the internal temperature of the coldroom to nonlinearly evolve. This last nonlinearity makes the production planning difficult given that moving a production bloc forward or backwards in time may impact the total energy production required. For instance, move forward in time the production of the frost production facility may increase the gradient between internal and external temperatures, so increase the heat exchanges and at the end increases the total energy production required.

2. BIBLIOGRAPHY

By considering the frost production facility as a single production server with fixed setup cost, the cold room as a limited storage, and the frost losses as demands, our problem could be seens as a special single stage, single item capacitated lot-sizing and scheduling problem (CLSP). A review on lot-sizing problems and solving methods can be found in Brahimi et al. [2006] With this parallel the demand (i.e. thermal losses) is a function of the state of the system (e.g. temperature of the coldroom, production level, external temperature, etc.). An example of stock-dependent demand in inventory management can be found in Teng and Chang [2005]. In this paper the demand is proportional to the stored quantity and a continuous time economic production quantity (EPQ) problem is addressed. 2.2 Temperature Control

Electrical network

Compressor

Evaporators

Coldroom

Fig. 2. Representation of a coldroom system. 1.3 Optimization opportunity As presented in Hovgaard et al. [2011], the most common system for controlling the temperature of the coldrooms is a hysteris-based mechanism with the only objective of temperature regulation, but without energy cost optimization. Since the temperature forecast of the coldroom could be efficiently done with an Artificial Neural Network (see Section 6.2) and since the cost of electricity is dynamic and known in advance, the problem could be seen as deterministic. So by using the cold room as a storage of frost energy, the frost production facility planning can be optimized from an economical point of view. The rests of this paper is organized as follows. In Section 2 we discuss the similarities between our problem and the lot-sizing problem, as well as the different approaches found in the litterature for the temperature control problem. In Section 3 we formulate the mathematical model for our problem and present our temperature forecast using artificial neural networks. In Section 4 we explain the dynamic programming algorithm we use to solve the

Some research has been done about the temperature control issue in buildings, either for the industrial [Hovgaard et al., 2011] or the residential [Jia and Tong, 2012] sector, as well as for individual houses [Ha et al., 2006] in the context of smart cities. The temperature forecast is a key element in the temperature control. The detail degrees for the model of temperature forecast are very heteregeneous in the literature. In some works the temperature forecast is a simple linear model [Ha et al., 2006; Sonderegger, 1977]. In other works, the temperature forecast is based on a highly detailed models of the whole frost production system [Hovgaard et al., 2011]. Ruano et al. [2006] use artificial neural networks (ANNs) to predict the internal temperature of a building and argues that ANNs are better suited to produce efficient forecasts for temperature control, while physical models require more time to develop and are therefore rather suited for the conception phase of a building. Some papers aim at optimizing the environmental impact of the process by minimizing the total energy consumption [Lu et al., 2005; Ferreira et al., 2012] while others focus on the economic issue of minimizing the cost of energy consumption [Ha et al., 2006; Hovgaard et al., 2011]. Some papers also include the quality of the service as an optimization criteria, by minimizing the uncomfort felt by the inhabitants of the building [Ferreira et al., 2012; Ha et al., 2006]. A review on the control of energy systems in buildings can be found in [Shaikh et al., 2014].

2038

2019 IFAC MIM 2004 Germany, August 28-30, 2019 Berlin,

Alnour Ribault et al. / IFAC PapersOnLine 52-13 (2019) 2002–2007

2.3 Contributions In the context of temperature control, the concern about the compressor wear, which was expressed to us by a cold store manager, seems to be a novel element in the related research. In this paper, we use an artificial neural network as temperature forecast. A dynamic programming algorithm is used to solve the model that includes the non-linear ANN temperature forecast and a fixed cost at each compressor startup. This allows us to solve industrial instances of the problem optimally and within reasonable time. We show the interest of solving the problem optimally as opposed to using a conventional hysteresisbased control method, and discuss the opportunity of using hourly day-ahead SPOT price instead of a Peak/Offpeak price.

Objective. The objective is to minimize the total cost including maintenance costs, the energy consumption costs for compressors, and the energy consumption cost for evaporators. min f “

« ÿ ÿ ÿ

tPT

lPL l1 PL

3.1 Parameters

Decision variables. ‚ θt P E is the temperature inside the cold room at the beginning of period t P T . ‚ yk,l,t P t0, 1u indicates if the frost production level is set to l P L at the beginning of period t P T . ‚ zl,l1 ,t P t0, 1u indicate if the frost production level changes from level l P L to level l1 P L at the beginning of period t P T .

(1)

@t P T ´T

(2)

yl,t “ 1

@t P T

(3)

yl,t ` yl1 ,t`1 ´ 1 ď zl,l1 ,t`1

@l, l1 P L, t P T ´T

(4)

˜

ÿ

lPL

yl,t Pl , Θext t , θt ÿ

yLk,0 ,0 “ 1 θ 0 “ Θ0

External parameters. Let T “ t0, 1, . . . , T u be the set of time periods. For each period t P T , the energy price is denoted Ct and the external temperature is denoted Θext t .

3.2 Mathematical program

yl,t El

¸

θt`1 “ F

3. MODEL

Note that, the frost power injected Pl and the energy consumption El are not linear in our case. The initial frost production level (during period t “ 0) is L0 . Switching the production from a level l to a level l1 leads to a maintenance cost Ml,l1 . This switch can be done at the beginning of each period, except for period t “ 0. “ ‰ Cold room. Let E “ Θ, Θ be the set of possible temperatures for the coldroom, with Θ and Θ being the maximal and minimal allowed temperature inside the room, respectively. The initial temperature in the cold room is Θ0 . Ś Ś Let F : P R E Ñ R be the temperature forecast function. Given an frost production level l P L, an exterior temperature Θext and a current temperature θt at the t beginning of period t P T , the temperature forecast function F returns the temperature in the cold room at period t ` 1. θt`1 “ F pPl , Θext t , θt q

lPL

ff

Constraints. Let T ´T “ T ztT u be the set of time periods without the last period.

lPL

Frost production. Let L be the set of production levels for the frost production facility. During one period in level l P L, the frost power injected in the coldroom is denoted Pl and the electrical consumption is denoted El . Let P be the set of all possible frost powers : P “ tPl |l P Lu

zl,l1 ,t Ml,l1 ` Ct

ÿ

(5) (6)

Θ ď θt ď Θ

@t P T

(7)

yl,t , zl,l1 ,t P t0, 1u

@l, l1 P L, t P T

(8)

Constraints (2) specify the temperature evolution in all rooms. Constraints (3) impose the unicity of the frost production level. Constraints (4) ensure to pay fixed maintenance cost for the compressor wear when the frost production level changes. Constraints (5) and (6) set the initial state. Constraints (7) limit the temperature in the room. Then, constraints (8) specify the binary decision variables. 3.3 Temperature forecast : Artificial Neural Network As shown in Ruano et al. [2006] and Ferreira et al. [2012], artificial neural networks are an efficient tool for internal temperature forecasting in an energy control problem. In our case, an Artificial Neural Networks has been trained on historical data composed by easy to find data: coldroom’s internal temperature, frost production level, and external temperature. We use a Recurrent Neural Network in order to factor in the dynamic aspect of temperature. Hyperparameter tuning revealed that an ANN with a simple structure yields the better results for this forecasting task. A standard cross-validation procedure has been followed by dividing the data in sets of equal size: a training set, a validation set, and a testing set. An ANN with fixed hyperparameter values is trained on the training set. Then the different ANNs are compared using their prediction performance on the validation set, and the best performing ANN on this set is selected as the final model. Finally, the performance of the final model is assessed using the testing set. The resulting network structure is the following. The input layer has two neurons: one for the evaporator power at timestep t, Pt , and one for the external temperature at timestep t, Θext t . The inputs are normalized before being

2039

2019 IFAC MIM Berlin, Germany, August 28-30, 2019

Alnour Ribault et al. / IFAC PapersOnLine 52-13 (2019) 2002–2007

fed to the input layer. The ANN has one hidden layer, with a number of neurons which can vary between 1 and 10 depending on the coldroom that is being modelled. The hidden layer’s activation function is a logistic function. The output layer has only one neuron and outputs the normalized internal temperature for timestep t ` 1, θt`1 , which is fed back to the hidden layer.

2005

the set of vertices and $ ˇ ˜C G C G¸ ˇ t P T ´T ’ & ˇ t t`1 ˇ ^pθ, θ1 q P E 12 θ , θ1 E“ ˇ ’ ˇ ^pl1 , l2 q P L2 % l1 l2 ˇ ^θ1 “ F 1 pP , Θext , θq l1 t

, / . / -

(10)

the set of edges. With vt “ xt, θt , lt y P V , the weight Hvt ,vt`1 on the edge pvt , vt`1 q P E is defined as (11)

Hvt ,vt`1 “ Ct Elt ` Mlt ,lt`1 .

Finding the cost-optimal production planning for the special case is then equivalent to find the shortest path from v0 “ x0, Θ0 , L0 y to vT that could be any vertex of the last period t “ T . The time complexity of the algorithm is Op|E|q with |E| ď T.|E|.|P|2 . The solution (i.e. the path) can be expressed as an ordered set of vertices. Given the link between the temperature and the production level, a solution can also be expressed entirely with the production level for each period or the temperature of the coldroom for each period.

Fig. 3. Example of an ANN structure used to model internal temperature evolution in a cold room 4. RESOLUTION METHOD : DYNAMIC PROGRAMMING Our problem can be solved with dynamic programming. However, state variables need to be discrete. Therefore, we must discretize the internal temperature variable. We define

ˇ * Θ ´ Θ ˇˇ E “ Θ`n n P t0..N ´ 1u N ´1ˇ with N P N the number of final discretized temperatures. We also define Θ´Θ Θ“Θ` N ´1 and Θ´Θ Θ“Θ´ N ´1 which are forbidden temperatures, and E 1 “ E 1 ` tΘ, Θu the extendedŚ set Ś of discretized temperatures. We then define F 1 : P R E 1 Ñ E 1 the discretized temperature forecast function. Given an frost production level l P L, an exterior temperature Θext and a current temperature θt t at the beginning of period t P T , the temperature forecast function F 1 returns the discretized temperature in the cold room at period t ` 1, such that θt`1 “ F 1 pPl , Θext t , θt q “ $ ext ext &arg minθPE 1 |θ ´ F pPl , Θt , θt q| if F pPl , Θt , θt q P E, Θ if F pPl , Θext t , θt q ą Θ, % Θ if F pPl , Θext t , θt q ă Θ. 1

"

Let G “ pV, Eq be a graph, with

+ #C Gˇ t ˇ ˇ θ ˇ t P T ^ θ P E1 ^ l P L V “ l ˇ

(9)

5. HYSTERESIS POLICY In order to evaluate the interest of our algorithm, we sought to compare it with a largely spread temperature control method, which is hysteresis based. The hysteresis control functions with a binary frost production. Let P max and P min be the maximal and minimal frost production, respectively. Given a high threshold Θ` and a low threshold Θ´ for the internal temperature, the control policy is defined by the following recursive equation:

Pt`1

$ ` ´ & Pt if F 1 pPt´1 , Θext t , θt q P rΘ , Θ s , max 1 ext ` if F pPt´1 , Θ , θt q ą Θ , “ P % P min if F 1 pP , Θtext , θ q ă Θ´ . t´1 t t

In order to get the values of the two parameters of the hysteresis policy (i.e. P max and P min ) for a given price scenario, the dynamic programming algorithm is executed with a constant price E equal to the mean of the price scenario over the 24h optimization horizon, ř Et E “ tPT . T As presented in Figure 4, the solution obtained is approximately a hysteresis profile. To remove edge effects, maximal and minimal threshold values are identified on the middle third of the obtained solution. Θ` “

max

tPt T3 ... 2T 3 u

θt

and

Θ´ “

min

tPt T3 ... 2T 3 u

θt .

6. NUMERICAL RESULTS 6.1 Instances We tested our algorithm over the models of three real cold rooms (I1,I2, and I3), which are part of the same industrial

2040

2019 IFAC MIM 2006 Germany, August 28-30, 2019 Berlin,

Alnour Ribault et al. / IFAC PapersOnLine 52-13 (2019) 2002–2007

where ∆ “ 10 min is the time step, R the equivalent thermal resistance between the inside of the coldroom and the outdoors, and C the thermal capacity of the system.

-18.3

Although this model is linear with respect to the inputs θt , Pl and θtext , it requires quadratic optimization in order to estimate the values of parameters R and C by solving a constrained ordinary least squares problem. As we can see in Figure 5, the linear model fails to capture the temperature evolution.

-18.6 -18.9 -19.2 12am

08am

Hour

04pm

Fig. 4. Estimating Θ` and Θ´ from a pseudo-hysteresis behaviour obtained by applying Dynamic Programming to a constant price. cold store located in the north of France, after training an ANN by cold room to model the evolution of its internal temperature. Table 1. Instances from real cold rooms. Surface (m2 ) Volume (m3 ) Number of evaporators Number of compressors Compressor max power (kW) Maximum temperature (°C) Minimum temperature (°C)

I1 6600 62800 2 1 1050 -18 -19.5

I2 2740 27400 3 1 1050 -18 -21

I3 7440 78000 4 1 1050 -18.5 -21

Function F 1 returns an element of E 1 by computing the output of function F and rounding it to the closest element of E 1 . A way to estimate the impact of the temperature rounding on the final solution is to run the algorithm with two other types of rounding: rounding to the discretized temperature just above or just under the temperature returned by F . Starting with a small value of N , we observe a convergence of the solutions found with these three rounding with rounding to the closest element when N increases. This convergence is complete for N “ 10000 with the three resulting production profiles being equal on all the simulations ran. With N “ 10000 the algorithm is executed in less than 1 second. All optimizations were done with a 10 minutes time step length, for an optimization horizon of 24 hours (144 time steps). In Figure 5, the discretised ANN is compared to the linear model proposed by Sonderegger [1977]: ∆

∆

-20

-22

Real temperature Discretized neural network Linear model 06am

12pm

Hour

06pm

Fig. 5. Discretized ANN prediction compared to real temperature and to the linear model on a given day.

In order to apply the dynamic programming algorithm to our problem, the internal temperature variable must be discretized. Since the size of graph G grows with the size of E 1 , it is important to keep the number of discretized temperatures N as small as possible, or it would impact the execution time of our algorithm. However, N being too small would have an important downside as well, since the rounding error of function F 1 which accumulates over all the time steps would become important and therefore impact the solution given by the algorithm.

∆

-18

12am

6.2 ANN performance

θt`1 “ e RC θt ` Rp1 ´ e RC qPl ` p1 ´ e RC qθtext

Temperature (°C)

Temperature (°C)

-18.0

The ANN, however, gives an overall satisfying prediction for the temperature. Prediction quality varies between different cold rooms, with Root-Mean-Square Error ranging from 0.12°C to 0.26°C and Mean Absolute Error ranging from 0.08°C to 0.18 °C over the test dataset for our three ANNs. 6.3 Contracted Peak/Offpeak price compared with SPOT price Our dynamic program is run to solve the 3 instances with two types of prices, the P/O price and the SPOT price. As presented in Figure 1, on a given day, the SPOT price may be either close to the P/O price, notably lower, or notably higher. In order to compare the interest of choosing either of these prices, we compare the P/O price with 6 years of SPOT price data, from 2012 to 2017. Table 2 contains the average prices for the SPOT prices and the P/O price. We can observe first that the average value of SPOT price varies from one year to another with 34.7e/MWh in 2014 and 47e/MWh in 2012. The second observation is that the average SPOT price over 6 years (40.9e/MWh) is lower than the average P/O price (42e/MWh). Table 3 contains the yearly functioning cost (ke). It appears that generally, SPOT price is more advantageous, the average functioning cost for SPOT price over 6 year is lover than the average functionning cost for P/O price. However the SPOT price seems more risky because in 2012 and 2017 it was more beneficial to select the P/O price and the forecast of the price over one year is nearly impossible (it depends on whether conditions). Then, the last observation is that in 2013 while the mean SPOT price was slightly above the mean P/O price, the

2041

2019 IFAC MIM Berlin, Germany, August 28-30, 2019

Alnour Ribault et al. / IFAC PapersOnLine 52-13 (2019) 2002–2007

optimization still managed to take advantage of the greater variance of the SPOT price. Table 2. Average prices (e/MWh) over a year for SPOT prices and P/O price. Year Price

2012 47.0

2013 43.3

2014 34.7

SPOT 2015 2016 38.6 36.8

2017 45.0

Aver. 40.9

P/O 42.0

Table 3. Yearly expenses (ke) for SPOT price compared to P/O price. Inst. I1 I2 I3

2012 68.0 83.0 39.2

2013 63.3 76.5 36.5

2014 57.7 67.1 33.4

SPOT 2015 61.5 73.4 35.6

2016 59.1 70.4 34.5

2017 66.7 81.7 38.5

Aver. 62.7 75.4 36.3

P/O 66.4 79.4 38.0

6.4 Economic comparison with hysteresis control Since the hysteresis policy is quite naive and doesn’t take into account either the startup costs or the dynamic electricity cost, it performs worse than the dynamic programming. However, the extent of the performance difference is quite variable depending on the cold room that is being optimized. Indeed, we have found the performance gap to vary from 7% to 44% on our three cold rooms. It seems that the gap is smallest on cold rooms which have a small thermal inertia, since those need frost power to be injected very frequently and it is therefore more difficult to take advantage of the price variations by passively storing energy. Figure 6 illustrates the downside of hysteresis control compared to optimal control: on November 8, 2016, the SPOT price peaked over 800 e/MWh between 6 pm and 7 pm, and stayed over 205 e/MWh between 7 pm and 8 pm. The optimal profile manages to avoid all electric consumption between 18:00 and 20:00, whereas the hysteresis control profile does, and therefore sees its cumulated cost rise very sharply. At the end of the optimization horizon, the cost difference between the two profiles is 200 e over 24 hours.

SPOT Cumulated (€/MWh) Cost (€)

Power (kW)

Dynamic Programming 60 40 20 0 12am

Hysteresis

06am

12pm

06pm

400 300 200 100 0 12am 800 600 400 200

06am

12pm

06pm

12am

06am

12pm

06pm

Hour

Fig. 6. Comparison between optimal profile and hysteresis profile on November 8, 2016. 7. CONCLUSION AND PERSPECTIVES

2007

temperature with the novel concern of compressor wear by introducing a fixed cost payment at each compressor startup. We have shown that discretizing the internal temperature variable allows to use a dynamic programming algorithm to obtain an economically optimal production profile in less than 1s. Comparing the optimal production profile to a hysteresis-based, we have illustrated that optimal control could induce savings varying from 7% to 44% in the presence of a dynamic electricity cost. We have also shown that chosing a SPOT price rather than a Peak/Offpeak price may allow for savings, however this is largely dependent on the yearly average SPOT value which can be volatile. Perspectives include the optimization of frost energy production for a multi-cold room cold store and the inclusion of an active storage unit, as opposed to the passive storage allowed by the thermal inertia of the cold room. In industrial cold stores, a coolant tank may act as a dedicated energy storage between frost energy production (by the compressors and condensers) and distribution. REFERENCES Brahimi, N., Dauzere-Peres, S., Najid, N.M., and Nordli, A. (2006). Single item lot sizing problems. European Journal of Operational Research, 168(1), 1–16. Ferreira, P., Ruano, A., Silva, S., and Concei¸ca˜o, E. (2012). Neural networks based predictive control for thermal comfort and energy savings in public buildings. Energy and Buildings, 55, 238–251. Ha, L.D., Ploix, S., Zamai, E., and Jacomino, M. (2006). Tabu search for the optimization of household energy consumption. In Information Reuse and Integration, 2006 IEEE International Conference on, 86–92. IEEE. Hovgaard, T.G., Larsen, L.F.S., and Jorgensen, J.B. (2011). Flexible and cost efficient power consumption using economic MPC a supermarket refrigeration benchmark. 848–854. IEEE. Jia, L. and Tong, L. (2012). Optimal pricing for residential demand response: A stochastic optimization approach. In Communication, Control, and Computing (Allerton), 2012 50th Annual Allerton Conference on, 1879–1884. IEEE. Lu, L., Cai, W., Xie, L., Li, S., and Soh, Y.C. (2005). HVAC system optimization—in-building section. Energy and Buildings, 37(1), 11–22. Ruano, A., Crispim, E., Concei¸c˜ao, E., and L´ ucio, M. (2006). Prediction of building’s temperature using neural networks models. Energy and Buildings, 38(6), 682–694. Shaikh, P.H., Nor, N.B.M., Nallagownden, P., Elamvazuthi, I., and Ibrahim, T. (2014). A review on optimized control systems for building energy and comfort management of smart sustainable buildings. Renewable and Sustainable Energy Reviews, 34, 409–429. Sonderegger, R.C. (1977). Diagnostic tests determining the thermal response of a house. ResearchGate, 84(1). Teng, J.T. and Chang, C.T. (2005). Economic production quantity models for deteriorating items with price- and stock-dependent demand. Computers & Operations Research, 32(2), 297–308.

We have presented a non-linear model of a cold room based on an artificial neural network forecast of the internal 2042