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A bottom-up customizable Markov model for householdappliance electric consumption

Chuanyong Shao, Eric Chauveau

To cite this version:Chuanyong Shao, Eric Chauveau. A bottom-up customizable Markov model for household applianceelectric consumption. sifwict, Jun 2019, Nantes, France. �hal-02495241�

A bottom-up customizable Markov model for household applianceelectric consumption

Chuanyong SHAO1 and Eric CHAUVEAU2

Abstract— Residence load is an important part of the totalpower consumption. This paper develops a bottom-up non-intrusive method to build a predictive model for the residenceload using a Markov model. Taking the fridge as the experi-mental subject, the total load is decomposed into three sub-systems. The applicability of this Markov model is proved bythe simulation in the MatlabTM platform. To do this, a pairof Markov chain is then created to predict the load powerconsumption for historical days. Then the average power ofsimulation and measurement are compared, as well as thestate transitions along time. The results clearly shows thatMarkov chains can simulate the behavior of the residence loadin a good way, but considering the influencing factors of theenvironment, this Markov model should be improved to obtaina better accuracy.

Key words: residence load, household appliance consump-tion, Markov model, Markov chain, influencing factors

I. INTRODUCTION

One of the main problem in the power system is to keepthe balance in the power providing and consuming. Loadprediction plays an important part of this problem. Modellinghome energy consumption is necessary for studying demandside response, transformer sizing, and distribution networksimulation [1].

Household appliances, generally the energy intensive ofthe load in the power system, lack of the Human-MachineInteraction process. This increases challenges to the systemscheduling. Nowadays, several revolutions in electric fieldleads to fast power system development, thus facing newproblems with the renewable sources, smart grid, and etc.Because of advanced technique in information field, smartelectric meter has been put into use widely, to help managersand consumers to keep power balance in hand. On the otherhand, the grid can achieve to get the real-time feedback fromthe demand side.

Plenty of works concentrate on load prediction. Most ofthese researches focuses on the load curve itself, using ma-chine learning ways to build forecasts. Usually, those worksput an approach directly to modeling without illustrating thereason. In this paper, a bottom-up method is developed tosimulate the behavior of household appliances. Paper [2]-[5]show Markov model have a good effect on load predictioncorrectly displaying dynamic transition regularity. Insteadof static probability distribution, this paper tends to use aMarkov model to forecast the possible transition direction.

1C. SHAO is student in master degree, University of Nantes, 44000Nantes, France chuanyong.shao at eseo.fr

2E. CHAUVEAU is with ESEO Tech, ESEO engeeniring school, 49000Angers, France and reseacher at IREENA, University of Nantes, 44600Saint-Nazaire, France eric.chauveau at eseo.fr

Fig. 1. Refrigerator construction scheme, with major power consumingparts. Rectangle up left represent the cold enclosure of the refrigerator.

Paper [6] and [7] introduce a method to decompose the stateof an household appliance. Considering the protection to theclients’ privacy, the idea is to find a common model for ap-pliances in a non-intrusive way. Fist of all, the decompositionof the appliance create sub power consumption units. Then aMarkov model is applied to convenient sub-system analysis.This model involves in the variation of the states of sub-systems along the time, then simulate the possible averagepower consumption within a certain time period.

II. METHOD

According to former studies [8] and [9], major part ofhousehold power consumption comes form freezing or heat-ing devices. Under this circumstances, the refrigerator hasa quite complicated operations states, and is usually top-ranked in high energy consuming household appliances. Asa result, the chosen experiment subject in this article is therefrigerator.

Three main power consumption units are considered insidethe refrigerator: The electrical Board, the light bulb andthe compressor as presented in figure 2. Inspired by paper[7], these three parts can be analyzed using two statesMarkov model respectively. For each part, there are onlytwo complementary states: On or Off. As long as we get thetransition probability between the two states as well as theinitial states, we can infer following states over time stepafter step.

All of these three units are supposed to consume constantpower during running time. If the prediction of the behaviorof these units is obtained, the power consumption can thenbe determined.

Fig. 2. The assumed total power consumption of a refrigerator relevant todescription in figure 1. Any power value is relevant to considered refrigeratormodel.

For Markov like sub-parts, the analysis based on Markovprobable model starts from the inner construction of thetime sequence. It was proposed by a russian mathematicianAndrey Andreyevich Markov. Markov model is applied tosolve questions such as:• If the system is under the state i, then from now

on, what is the probability of the state being j, aftertransited by n steps.

• In a market demand prediction problem, if there is acompany occupy part of the sale route currently, thenafter n steps, how much property of the market occupiedby this company?

Thus, Markov model can be used to show the variation ofa process. Then it comes to the problem of Markov process:In case of the state at time t0 is give, the state at time t wheret > t0 only depends on the state t0 and has no relationshipwith the states before this moment.

Definition 1: Assuming that {X (t) , t ∈ T} is a randomprocess, if for any n numbers of time t, t1 < t2 < ... <tn−1 < tn, (n > 3) , with the condition that X(ti) = Xi,i = 1, 2, ..., n − 1 , the distribution function of X(t) is oneof the same as the one of Xn when X(tn−1) = Xn−1, i =1, 2, ..., n−1 happens. Then {X (t) , t ∈ T} is called Markovprocess. That is:

p {X (tn) ≤ Xn | X(t1) = X1, ..., X(tn−1) = Xn−1} (1)

Markov chain is the simplest Markov process. The time tand state X of a Markov chain are both discrete.

Definition 2: For a random sequence {Xn, n = 0, 1, 2...}in the discrete space I , at anytime n, and any statei0, i1, ..., in−1, given i, j:

p {X − n+ 1 = j | Xn = i, ...,X0 = i0} =p {Xn+1 = j | Xn = i} (2)

Then random sequence {Xn, n = 0, 1, 2, ...} is calledMarkov chain.

Definition 3 (Single-step Transition Probability):We suppose that {Xn, n = 0, 1, 2, ...} is a Markov chain.

We denote the single-step transition probability at time n asthe following conditional probability:

pij = {Xn+1 = j | Xn = i} (3)

where: pij(n) ≥ 0, i, j ∈ I and∑

j∈I pi,j(n) = 1, i ∈ I .In this paper, all of the power consumption units can be

represented by two-states circumstances like in figure 3, withA and B representing the states, and ti→j the transition fromstate i to state j. Each state is associated with a given value1 and 0 representing the states respectively.

Fig. 3. A two discrete states process (A and B) with any possible transitionbetween each of them ti→j .

III. RESULTS

A. Refrigerator example

The research is based on historical data of a refrigeratorplaced in ESEO engineering school in Angers, France. Thisdevice can only be accessed by the staff in ESEO (excludingthe students). The recorded data contains the date, the real-time power consumption of the whole refrigerator, the statesof the door, and the ambient temperature at every samplingtime. The sampling time step is 1 second. The research’smodels of each convenient sub-part consider user influenceas randomness. Figure 4 shows the total load of the fridgeunder a period of running time demonstrating the assumptionof the power consuming is right.

Fig. 4. The measured total power consumption [W] (top) and door opening[binary] (bottom) of an operating fridge. Recording sample is about 30minutes long.

1) The Controller Board: The controller board is theeasiest to analyze inside the refrigerator. It always worksto control the operation units in the fridge. So there is onlyone On states for the controller board.

2) The Light Bulb: The light bulb doesn’t work all thetime. It’s switched on when the door of the fridge is open,and off when the door is close. Considering the light bulbby its own, its state working state is driven by randomness.It is more dependent on the behavior of the users. So it canbe seen as a probable event. In this paper the states of thebulb are only On or Off.

3) The compressor: The operation principle of the com-pressor involves the process of thermal exchange. The con-densing system of a fridge is composed of (see figure 1) acompressor, and a refrigerant pipe (internal and external ofthe refrigerator). Under order of a temperature sensor insidethe fridge, if the temperature inside the fridge is higher thanthe set level, then the compressor starts to work. The essentialof the compressor is a pump driven by a motor. Due to aparticular refrigerant gas saved inside the pipe, when thecompressor is working, the pump constricts and enhance thepressure initiating the condensing cycle. Then the gas whichis becoming hot because of the pressure will be sent to theexternal heat exchanger meeting the cooler air. This resultsin changing of the gas to liquid phase. The compressor goeson working and then the liquid is pushed into the internalcoil through a capillary. At this time, the temperature insidethe fridge is higher than the liquid, which will lead the liquidabsorbing the heat inside the fridge to convert back to gas.That is the principle of the condensing cycle system.

Considering the compressor by its own, its working stateis due to:• measured temperature inside the refrigerator;• ambient temperature of the room;• and refrigerator users habits.

So, as the same as the light bulb, there are only two possiblestates (On or Off ) for the compressor.

B. Recording

A specific equipment show in figure 5 is use to log dataduring a given period, from seconds to many months, with1 second sample time.

Recorded data are (see example in table III):• current date and time stamp;• real power consumption of the whole refrigerator;• door state (open or close);• ambient temperature.

C. Analyzing

Measurements in figure 4 clearly shows:• a power profile due to light bulb relevant to door

opening and closing;• a power profile due to compressor;• a constant power due to electronic controller board.

Using recorded power variations from in figure 4 and sub-system decomposition, power profile of each sub-part of therefrigerator are built independently step by step:

Fig. 5. Data recording equipment containing a power analyzer (bottomcenter) with two adapted current clamps (top right), a processor board(bottom right) with door Hall effect sensor connection, and a USB memorystick. Door Hall effect sensor and ambient temperature sensor are notrepresented.

1) Light bulb power consumption refers to door open-ing/closing and power consumption is relevant to bulbspecifications.

2) Extract supposed bulb power consumption profile, thensubtract to total power profile.

3) Remaining power profile shows two complementarypower states. Lower one for standby mode only dueto controller board, higher one due to cold compressorpower adding.

4) Power variation is supposed to be due only to compres-sor. Power variation value is quite constant over hours,days, and weeks, and quantity is relevant to inductionmotor compressor specification.

5) Subtract this last power power profile to remainingpower to find controller board power.

Controller board power consumption is very small, lessthan 1W in our case, compared to compressor and light bulbso that it can be ignored. Finally, the total power consumptionis supposed to be composed of only two main parts: the lightbulb and the compressor.

D. Modeling

As shown in figure 4, door state is a two state randomvariable, and so it is for the compressor power profile. So,for each light bulb and compressor, a simple 2 states Markovchain is considered.

Power consumption profiles from previous analysis foreach sub-part of the refrigerator can be computed to calculatetransition matrix over a given recording period. This canbe done by simply counting any transition over the givenconsidered time period.

E. Simulate

The simulation presented is developed with Matlab TM.Chosen data range is between 19.11.2018 and 25.11.2018within a full working week. For the whole simulation

process, every day’s prediction is completed by the sametransition matrix created by the first day of the week, hereon Monday.

According to previously described method and modeling,the algorithm is:

1) Extract the sub-profile of the bulb and its power2) Obtain the states of the bulb directly from door state3) Separate out the states of the compressor according to

the sub-profile of the total power consumption whichis higher than 75W .

4) Compute the transition matrix for each sub-system.5) Ensure the initial states of each sub-system in the

simulation keep in path with the reality.6) Calculate the total simulated power.According to the simulation, the transition matrix over the

considered week is:(0.9995 0.00050.1340 0.8660

)Let’s consider detailed simulated results within the first

day as an example to show the transition of the states ofsub-systems.

Fig. 6. State of the bulb from measurement (Top) and Simulation (bottom)over a day.

Fig. 7. State of the compressor from measurement (Top) and Simulation(bottom) over a day

Fig. 8. Total power comparison between measurement (Top) and simulation(bottom) over a day.

IV. DISCUSSIONFor each considered sub-part of the refrigerator:• Bulb: Referring to the measurement it is obvious that in

a real case, the door is opened between 8:00 and 9:00 inthe morning, 11:00 and 13:00 in the mid-day. However,the simulator presents a sequence distributing randomlyalong the time during a day.

• Compressor: In theory, the compressor operates quiteregularly, but the simulation presents a more irregularprofile.

• Controller board: The power consumption of the elec-trical board is smaller than 0.02W and can be ignoredcompared to the bulb and compressor consumptions.

Simulation results shows good global representativenessof the model refer to measurements as shown in tables IVand III. However, because influence of the environment isneglected here, the simulated results does not always meetreality. This specifically embodied in the following aspects:• The detailed simulation of the states is not convenient

with the real distribution of the states for each sub-system.

• The accuracy of the simulation is not good enoughcompared to the measurement. If there is a large amountof devices gathering together, the simulated load maynot be good to put into use.

• This model doesn’t fit to each day within one week, dueto the effect of important events. The operation statusof the working days is different from that in weekends.This is because nobody works during weekends orholidays.

Sub-system decomposition gives a custozimable modelthat can be derived to any similar appliance with knownspecifications (light bulb power, cold compressor power) ordeduced by simple analysis of non intrusive measurements.

The primary simulation demonstrates that it is impossibleto use one unchangeable transition matrix to predict all ofthe daily power consumption (see table I. Then anothersimulation is taken to observe the average power using thetransition matrix of each day respectively (see IV.

TABLE IAVERAGE POWER OF THE REFRIGERATOR DURING ONE WEEK

Meas. [W] 16.16 15.64 18.34 18.93 15.57 14.47 15.03Simu. [W] 14.77 12.33 15.64 16.74 14.56 15.93 15.93

TABLE IISIMULATED AVERAGE POWER OF EACH DAY RESPECTIVELY

Meas.[W] 16.16 15.64 18.34 18.93 15.57 14.47 15.06Simu.[W] 14.77 14.91 18.86 20.33 15.20 13.93 10.22

Table I shows that in weekends, the simulated results ofthe mean power of the device is higher that the reality.In fact, on Saturday and Sunday, because the institutionis not accessible, the door is never opened. Theoretically,the simulated power should be similar to the real power.There is no possibility for the fridge to consume more powerin the simulation. Conversely, table IV gives the averagepower simulated through the transition matrix of each dayrespectively. This model is more accurate than the formerone. This indicates that the transition matrix changes fromday to day. Although the difference among working days isnot so obvious, but it will still contribute to the differenceof the results. More significantly, there is a big importancebetween working days and week-ends. It illustrates thatpeople’s activity do effect the behavior of the appliances.Besides, the ambient temperature will influence the workingstate of the heating system too. These factors must be takeninto account to get a more accurate model.

V. CONCLUSION

According to the analysis and discussion we can draw thefollowing conclusions.

1) Markov chains can be used to predict the behavior ofthe power consumption of the household devices witha good accuracy on average power.

2) Sub-system decomposition is relevant to reach a goodaccuracy, and can be put into practice using non-intrusive measurements.

3) Because there are a lot of objectively influencingfactors, so the Markov model in this paper is notaccurate enough to put into use. Especially if there is aload profile consisted of a large number of householdsub-profile.

4) Even if the average power is similar between mea-surements and simulation, the modeled transition stateprocess differs a lot from the measurement. To get abetter simulator, a different Markov model should becreated for different time sections.

5) Taking the environmental factors into account, workcan be developed to make this model more adaptable.Environment factors such as the number of the users,or the ambient temperature can be then considered.

6) Besides, important events influence a lot the resultstoo. The Markov models built for working days andfor holidays or weekends should be separated.

APPENDIX

TABLE IIIRECORDINGS SAMPLE

Date Time Power Bulb Amb. temp.YYYYMMDD HH:MM:SS W Binary oC

...20181119 08:07:33 0.0185 0 22.820181119 08:07:34 0.0171 0 22.820181119 08:07:35 0.0177 0 22.820181119 08:07:36 0.0176 1 22.820181119 08:07:37 0.0175 1 22.820181119 08:07:38 31.0700 1 22.820181119 08:07:39 31.0400 1 22.820181119 08:07:40 31.0100 1 22.820181119 08:07:41 31.0200 1 22.820181119 08:07:42 31.0100 1 22.820181119 08:07:45 31 0 22.820181119 08:07:46 31.0100 0 22.820181119 08:07:47 0.0172 0 22.820181119 08:07:48 0.0167 0 22.820181119 08:07:49 0.0173 0 22.820181119 08:07:50 0.0169 0 22.820181119 08:07:51 0,0175 0 22.720181119 08:07:52 0.0171 0 22.820181119 08:07:55 0.0184 0 22.820181119 08:07:56 0.0184 0 22.820181119 08:07:57 0.0168 0 22.720181119 08:07:58 0.0181 0 22.8

...

ACKNOWLEDGMENT

The authors wishes to express their appreciation to ThierryPommeau (ESEO Tech), who helped them by developingthe measurement setup, and then make this research workfeasible.

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