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Residential Time of Use Energy Modelling

and Tariff Evaluation

Gideon Gope, Sunday A. Reju and Kalaluka Kanyimba

Abstract — Energy modelling for demand-side management

studies requires a very accurate description of the

characteristics of the end-use areas under examination,

including the correlation existing between consumption and

socio-economic and demographic characteristics of the users.

The paper presents a mixed strategy 2-player game model for

a residential energy consumption profile for winter and

summer seasons of the year using a dual-occupancy high-rise

(11-storey) building located within the Polytechnic of

Namibia, Windhoek. The optimum energy values and the

corresponding probabilities obtained from the model extend

the usual simple statistical analyses of minimum and

maximum energy values and their associated percentages.

The time-block and the week-day strategies depict critical

probabilistic values worth considering for decision purposes,

especially, the necessity and justification for a dual tariff

regime for the residential and workplace residents of the

building as against the existing institutional uniform energy

tariff policy. However, the critical energy game values being

proposed as the optimal tariff estimate parameters constitute a

unique result of the paper. Moreover, the morning and the

evening energy utility proportion values obtained from the

time series plots of the mean consumption for the two seasons

highly justify the efficiency of our game model optimal

solutions. The paper also presents regression model analysis

for the mean consumption profiles for the two seasons, as a

tool for the prediction of energy consumption at any given

time of use..

Key Words — Game theory, Time of use, Tariff evaluation.

I. INTRODUCTION

Energy consumption data for the case study 11-storey

residential complex (Poly-Heights) at the Polytechnic of

Namibia were collected for a period of one year. The time

logged data were collected at three distinct points in the

building, namely, on the incoming supply feeder to the

building, on the distribution board supplying three floors and

on the supply distribution board to a single (individual) unit

/flat.

1 This work was supported by the Polytechnic of Namibia under Grant No.

IRPC-POLY /2011/7030/476.

G. Gope is with the Department of Electrical Engineering, Polytechnic of

Namibia, Windhoek, Namibia (ggope@polytechnic.edu.na).

S. A. Reju was with the Regional Training and Research Institute for Open

and Distance Learning, National Open University of Nigeria, Lagos, Nigeria.

He is now with the Department of Mathematics and Statistics, Polytechnic of

Namibia, Namibia (sreju@polytechnic.edu.na).

K. Kanyimba is with the Department of Electrical Engineering,

Polytechnic of Namibia, Namibia (kkanyimba@polytechnic.edu.na).

A sampling time interval of 30 minutes was used so as to

make the energy consumption data fall within the same

metering block intervals as used by utilities in the region,

since the metering block interval used by utilities in the region

is a 30-minute time interval.

A shorter sampling time interval would result in a high

resolution data but this was not considered as critical for the

initial model development for demand-side management

studies and tariff evaluation.

The approach to energy model development is based on an

end-use approach. The method being applied during the

project is to derive the energy consumption or demand profile

of the consumers in the building from a probabilistic

aggregation of the demand profile of every flat in the building.

One of the reasons for the selection of Poly-Heights as the

case study complex is the close proximity of the individual

households to each other. This was aimed at minimizing the

cost of appliance surveys during the project phase.

The distinctive features of residential end-use justify the

introduction of a model having different probabilistic

functions and “behavioral” variables which allow for

reproduction of the customers'' behaviour in terms of

electricity usage. In this paper, a specific application of

mathematical game theory is explored for model extraction

and identification

II. BRIEF OVERVIEW OF GAME THEORY

Game theory as a branch of decision theory or Operational

Research (OR) is concerned with interdependent decisions.

The problems of interest involve multiple participants, each of

whom has individual objectives related to a common system

or shared resources. The theory basically describes the

analysis of competitive scenarios, hence the problems are

called games and the participants are called Players. Even for

strictly competitive games, the goal is simply to identify the

player’s optimal strategy.

In practice, many games are not strictly determined and

hence are in want of best pure strategies according to the

fundamental principles of game theory. Thus generally, for

some games, especially one that has no saddle points, it is best

to use a Mixed Strategy, wherein, instead of sticking to a

single pure strategy, a player chooses among the strategies at

random. A mixed strategy of a player in a strategic game is a

probability distribution over the player’s actions, that is, each

player chooses a probability distribution over his or her set of

actions rather than restricting the player to choosing a single

deterministic action.

Assigning a probability distribution over the set of

strategies of each player is expressed as follows.

Canadian Journal on Electrical and Electronics Engineering Vol. 3, No. 3, March 2012

115

Let probability that Player 1 will

use strategy (1)

Probability that Player 2 will

use strategy (2)

where and are the respective numbers of available

strategies.

The players’ plans of actions given by () and

() are known as the Mixed Strategies in contrast to

the original strategies known as the Pure Strategies. Though

there is no satisfactory measure of performance known for

evaluating mixed strategies, a very useful one is the Expected

Payoff, defined as below.

Using expressions (2.1) and the general payoff matrix, we

have, by definition

where is the payoff if Player 1 uses pure strategy and

Player 2 uses pure strategy .

Application of game theory in energy models is becoming

ubiquitous as seen in (Ferrero et al, 1998) and in (Neimane et

al, 2008). The latter specifically developed a cooperative game

model for the task of energy supply planning in the market

environment. The possibilities of forming coalitions between

companies competing within energy production, distribution

and sales were considered.

In our model, we consider a situation where time and week

days are taken as the competitive parameters, hence we define

the following:

Player 1 = Time Block

Player 2 = Week Days

Payoff = Energy value (from the daily 30-minute

time block readings)

Thus our general energy payoff matrix is given as follows.

Strategy

Week Days

Mon

Tue

...

Sun

Time

00.00

e

11

e

12

...

e

1n

00.30

e

21

e

22

...

e

2n

...

...

...

...

...

23.30

e

m1

e

m2

...

e

mn

where is the consumption energy value at time on day

Now for our model we define the following

- the week day strategies

- the daily time strategies (4)

- the energy consumption values

The following steps describe the solution technique in solving

the game model problem.

Step 1: Reduce the payoff matrix by dominance (optional, but

highly recommended).

Step 2: Convert to a payoff matrix with no negative entries

by adding a suitable fixed number to all the entries.

Step 3: Solve the associated standard linear programming

problem.

Step 4: Calculate the optimal strategies.

The expected value of the game is given by

where T is the Time strategy vector, E, the Energy payoff

matrix and D the Day strategy vector.

To solve a larger game, as in our model, we usually employ

a linear programming approach. The associated linear

programming problem for our model is given as follows:

subject to

Below are the energy consumption values (from the 48x7

energy payoff matrices), for the two seasons (showing total

energy readings in kWh for the first three and the last three

30-minute time blocks), however the matrix is truncated to

manage space.

TABLE 1

TRUNCATED TOTAL ENERGY DATA FOR

SUMMER DAYS (KWH)

Time

Mon

Tue

Wed

…

Sun

0.00 23.86878 23.82743 23.2569 … 24.15275

0.30 23.13838 22.34982 22.37926 … 22.82042

1.00 22.74585 22.14361 23.2265 … 22.40683

... … … … … …

22.30 29.5876 26.89516 29.0896 … 29.18996

23.00 27.00796 25.57397 25.7405 … 27.05877

23.30 24.9841 25.00659 25.4480 … 25.76252

In the absence of providing the complete actual (48-by-7

array) energy values for the summer season, Fig. 1 is the 3-

dimesional MAPLE surface plot representing the consumption

profile. For the winter season, Fig. 2 is also the 3-dimesional

surface plot representing the consumption profile.

(5)

(6)

(7)

(8)

Canadian Journal on Electrical and Electronics Engineering Vol. 3, No. 3, March 2012

116

Input Energy

Matrix Size

(48X7)

Input Energy

Matrix

Start Set System RHS = 1

MATLAB

Simplex

Routine

Output Mixed

Strategies

,

and Game

Value

Stop

C

C

Fig. 1. Summer Total Energy Consumption Surface Plot

Fig. 2. Winter Total Energy Consumption Surface Plot

III. GAME OPTIMAL SOLUTIONS AND REGRESSION

MODELLING

Game Model Optimal Solutions for Summer:

Implementing the solutions steps in the above condensed

flowchart algorithm (by running a MATLAB code) for the

(48x7) matrix system, we obtain the mixed strategy solution

with their respective probabilities and value of the game.

For the summer we have the following optimal mixed

strategies:

; ; ;

where

meaning that the probabilities for the three days of use,

namely Monday, Wednesday and Thursday are zero or they

are optimally insignificant.

For the same season we have the time strategy probabilities

as follows

; ; ;

with

Thus the (4x4) energy consumption matrix shown in Table

2 corresponding to the above strategies is hereby termed an

Energy Consumption Concentration Matrix, being the

associated dominated mixed strategy matrix with significant

energy consumption probabilities.

From Table 2, for the summer season, the expected value of

the game is given as follows.

TABLE 2

SUMMER ENERGY CONSUMPTION CONCENTRATION MATRIX

Days Tue Fri Sat Sun

Time Proba-

bilities 0.3195 0.0117 0.3409 0.3279

07.30 0.0695 39.65638 40.3427 34.2710 31.8429

09.30 0.1784 33.41002 31.6720 38.4210 33.92427

19.00 0.2106 36.83589 34.5814 34.2889 34.77837

20.00 0.5415 34.70487 36.0666 34.7353 36.33819

Game

Value

35.2666

Game Model Optimal Solutions for Winter:

For the winter season, we also have the following optimal

mixed strategies:

; ; ;

where

and ; ;

;

and

The Energy Consumption Concentration Matrix for the winter

is also presented below with the above significant energy

consumption probabilities.

TABLE 3

WINTER ENERGY CONSUMPTION CONCENTRATION MATRIX

Days Thu Fri Sat Sun

Time Proba-

bilities 0.0933

0.0786

0.6211 0.2070

09.00 0.1208 44.1326 42.7080 39.21589 39.3082

09.30 0.1460 38.4818 37.7524 39.86168 41.7991

10.00 0.2311 36.4835 38.6264 40.30111 41.0489

20.00 0.5021 41.0025 40.5711 40.02678 39.0969

Game

Value

39.9681

For the season, the expected value of the game is given by

Fig. 3. Energy Game Model Algorithm

Canadian Journal on Electrical and Electronics Engineering Vol. 3, No. 3, March 2012

117

The Regression Model Solutions:

The mean consumption time series plot for the summer

season and the regression curve are as follows (similar to a 2-

dimesional cutting plane through the surface plot in Fig. 1).

The best polynomial curve that fits the data for the mean

consumption during the two seasons is found to be of degree

15:

and the MATLAB code implementation of the Moore-Penrose

pseudo-inverse of the non-square array of the model generates

the following scatter plots and regression curves.

Fig. 4. Summer Mean Cons umption Time Series Scatter

Plot and Regression Curve

Fig. 5. Winter Mean Consumption Time Series Scatter Plot

and Regression Curve

Some predicted values using the regression curve for

summer are given below

TABLE 4

SUMMER ENERGY REGRESSION PREDICTION

Point on

The Curve

Time Actual

Energy

Predicted

Energy

Error

5 02.00 21.681531 22.57806 -0.8965

8 03.30 21.41427 20.64771 0.76656

12 05.30 26.294747 27.75580 -1.46105

34 16.30 28.86655 29.32644 -0.45996

40 19.30 35.97429 36.09450 -0.12020

Similarly, the best 15th degree polynomial helps to predict

the following winter energy values:

TABLE 5

WINTER ENERGY RE GRESSION PREDICTION

Point on

the Curve

Time Actual

Energy

Predicted

Energy

Error

5 02.00 23.80976 22.7216 1.08816

8 03.30 22.84293 23.6635 -0.82057

12 05.30 22.65505 22.1964 0.458652

34 16.30 30.11242 29.0495 1.062924

40 19.30 40.84608 40.3167 0.529381

Comparing the two seasonal means, we have the following

MATLAB plot.

Fig. 6. Summer and Winter Means Comparison Plots

A closer look at the above plots shows six (6) points of

intersection (times of use) where there are constant energy

consumption values for the two seasons. Selecting four of

these (for the early morning and late evening times), we have

05.00Hr = 23.1601748; 07.30Hr = 37.7529186;

17.30Hr = 31.7849060; 18.30Hr = 34.7047750

Thus the above constant consumptions for the year

obviously provides estimate for the average morning and

evening consumption proportions for the year, namely 0.31

and 0.69 of the energy demand for the morning and evening

peak times of use respectively. The calculations are done as

follows (using the time intervals):

Morning Average = (37.7529186 + 23.1601748)/6

= 10.152182

Evening Average = (31.7849060+ 34.7047750)/3

= 22.163227

Morning Proportion = 10.152182/32.315409

= 0.3141592 0.31

Evening Proportion = 22.163227/32.315409

= 0.6858408 0.69

Hence the above simple computations confirm a higher

energy utility demand in the evening more than in the morning

for the whole year as observed in the time series profile and

these specifically determine the numerical estimates of the

utility demand proportions. Further research is worth being

conducted for other buildings to investigate general estimates

of these proportions for a larger consumer population.

0 5 10 15 20 25 30 35 40 45 50

20

25

30

35

40

45

Summer Mean

Winter Mean

Canadian Journal on Electrical and Electronics Engineering Vol. 3, No. 3, March 2012

118

Interestingly, the morning and the evening proportion

estimates 0.31 and 0.69 respectively obtained above are very

close to the estimates of the morning and the evening

probabilities 0.2479 and 0.7521 (Table 6), obtained from our

game model solutions, to justify the latter and hence the

efficiency of our game model.

IV. MODEL INTERPRETATION AND ENER GY TARIFF POLICY

Tariff Policy:

The optimal solutions to our game model represented by

Tables 2 and 3 intuitively divide the consumers in our case

study building into what we would like to refer to as

(Morning) Workplace Consumers and (Morning and Evening)

Residential Consumers for energy consumption billing

purposes.

By the residential consumers, we mean the institution’s staff

members that use the flats as residence as against the

workplace consumers that use their flats as offices and so are

deemed to be out of the building during weekend days and

after the working hours.

TABLE 6

TIME OF USE ENERGY CONSUMPTION PROBABILITIES

Summer Winter

Time Probabilitie

s

Time Probabilities

07.30 0.0695 09.00 0.1208

09.30 0.1784 09.30 0.1460

19.00 0.2106 10.00 0.2311

20.00 0.5415 20.00 0.5021

The above time of use consumption probabilities show a

significant higher total probability of 0.7521 for the evening

consumers (assumed to be exclusively residential consumers)

as against just 0.2479 for the morning consumers (which yet

include the residential consumers), during the summer season.

The winter season equally shows a substantial consumption by

the residential (evening) consumers during the obvious

constant (20.00) hour of high consumption for the two

seasons.

Thus for a realistic tariff policy, the current uniform billing

for the two types of consumers obviously reveals an empirical

contradiction.

Utility Time and Day of Use Optimality:

The game model solutions summarised in Tables 6 provide

optimal time of use for the energy utility, and thus in turn

provide an optimal policy for the utility provider.

Peak and Off-Peak Demand Times:

• For summer, the solutions reveal a two-hour morning

optimal utility (or peak) demand between 07.30 and

09.30 and a one-hour evening optimal utility (peak)

demand between 19.00 and 20.00.

• However, for the winter, the morning optimal utility

(peak) demand is a one-hour duration between 09.00

and 10.00. The evening optimal utility (peak) demand

could also be conveniently taken to be a one-hour

duration between 19.30 and 20.30 with a mean hour

20.00.

• The time blocks outside the peak periods are the off-

peak times with zero probabilities.

Peak and Off-Peak Demand Days:

As seen in Table 7, for the optimal utility (or peak) days of

use, the weekend days (Friday to Sunday) remain constant for

the two seasons, with a switch between Tuesday in summer

and Thursday in winter. For the two seasons, Monday and

Wednesday remain non-optimal utility (or off-peak) days of

use.

TABLE 7

DAY OF USE ENERGY CON SUMPTION PROBABILITIES

Summer

Days Tue Fri Sat Sun

Probabilities 0.3195 0.0117 0.3409 0.3279

0.3195 0.6805

Winter

Days Thu Fri Sat Sun

Probabilities 0.0933 0.0786 0.6211 0.2070

0.0933 0.9067

Critical Time and Day of Use Energy Factor:

From Tables 2 and 3, the values of the game for the two

seasons are 35.2666 and 39.9682 for summer and winter

respectively. It is worth noting that these two energy values do

not belong to the data sets for each of the seasons but are

rather obtained from the game model solution algorithm. We

wish to locate the position of the values within the sequence of

energy values represented by the energy consumption

concentration (4 x 4) arrays shown in Tables 2 and 3.

For the two seasons, we have the following 17-term

sequential arrangement of energy values (in just two decimal

places and including the game values in brackets):

Summer

31.67; 31.84; 33.41; 33.92; 34.27; 34.29;

34.58; 34.70; 34.74; 34.78; [35.27]; 36.07;

36.34; 36.84; 38.42; 39.66; 40.34

Winter

36.48; 37.75; 38.48; 38.63; 39.10; 39.22;

39.31; 39.86; [39.97]; 40.03; 40.30; 40.57;

41.00; 41.05; 41.80; 42.71; 44.13

Interestingly, the game energy value for winter is the

median of the sequence. The adjacent energy values to the

game values for the seasons are indicated in square bracket

shaded cells in the following optimality Tables. This is for the

purpose of identifying the optimality sub-matrix enclosing the

game value, to investigate any possible pattern. For the two

seasons, the optimality array of energy values are consistently

bounded by the week-end (Friday and Sunday) energy values,

with evening limiting hour of 20.00; the winter optimality

array of values is uniquely the Saturday array, and the summer

optimality array of values is uniquely the Evening (19.00 –

20.00) double array.

Though the above analysis is simply of mathematical

interest, further research on additional case studies could

reveal a possible pattern that is worth further investigating as

an extensive modelling effort.

Canadian Journal on Electrical and Electronics Engineering Vol. 3, No. 3, March 2012

119

Optimal Energy Tariff Estimate Parameter:

With the location of the optimal energy game values within

the optimality array of energy values for the two seasons and

their further location within the optimality sub-arrays

identified above, there seems to be a strong justification for

suggesting the use of the game values as Optimal Tariff

Estimate Parameters for the seasons. Thus various

percentages of the parameter could be used for tariff

determination according to the probabilistic values obtained in

the optimal solution.

V. CONCLUSION AND SUGGESTIONS FOR FURTHER

RESEARCH

The optimal energy consumption values and the

corresponding probabilities obtained from the game model

extend the usual simple statistical analyses of minimum and

maximum energy values and their associated percentages. The

model solutions and results clearly suggest the classification

of the energy consumers in the building into two groups as

noted.

Moreover, the dominated time of use or peak periods during

the morning hours of the two seasons seems to shed some light

on the climatic (or seasonal) lifestyles of the consumers –

suggesting a possible later arrival of the workplace consumers

at the building during the winter as against an earlier arrival

during the summer.

From the results, it is crystal clear that residential

consumption dominates the energy usage which in turn

suggests an obvious and necessary separate energy tariffs for

the two groups. This is evident from the fact that the

dominated (optimal) energy consumption days are mostly

week-end days during the two seasons (which are “Days of

Use” that workplace consumers are mostly not present in the

building). However, the morning (09H00 – 10H00) hours of

Winter Thursdays and Fridays are equally significant in case

of using a common billing plan for both two groups of

consumers.

Proposing the energy game values as optimal estimate tariff

parameters necessitates some further research on the tariff

evaluation for the two seasons. Moreover, studying the

time/day of use patterns for other residential locations could

provide useful information on additional residential and

workplace lifestyle profiles of various residential areas. The

regression model approach that helps to predict energy

consumption for any time of use is quite significant as it

provides us with a tool to predict energy values that are not

among the actual readings.

REFERENCES

[1] Ferrero, R.W., Rivera, J. F., Shahidehpour, S. M. (1998), Application of

Games with Incomplete Information for Pricing Electricity in

Deregulated Power Pools, IEEE Transactions on Power Systems, Vol.

13, No. 1.

[2] Neimane, V., Sauhats, A., Inde, J., Vempers, G., Bockarjova, G (2008),

Using Cooperative Game Theory in Energy Supply Planning Tasks, 16th

PSCC, Glasgow, Scotland, July 14-18, 2008

[3] Frederick S. Hillier, Gerald J. Lieberman (1995) Introduction to

Operations Research, (6th edition) McGraw-Hill.

[4] Frederick S. Hillier, Gerald J. Lieberman (1995) Introduction to

Mathematical Programming, McGraw-Hill.

BIOGRAPHIES

Gideon Gope was born in Kadoma, Zimbabwe. H e

received his B.Sc. degree in electrical engineering and

M.Sc. degree in electrical power engineering from the

University of Zimbabwe in 1996 and 1999 respectively.

He is member of the Association of Energy Engineers

(A.E.E,USA) and is a certified energy auditor (CEA).He

is a former research scientist at the Scientific and

Industrial Research and Development Centre (SIRDC) i n

Zimbabwe and a former lecturer at the University of Zimbabwe (UZ, Harare,

Zimbabwe). He is currently lecturing courses i n power systems engineering

and power electronics application in power systems at the Polytechnic of

Namibia, Windhoek, Namibia. Hi s research interest is in power systems

modelling, energy management and energy modelling.

Sunday A. Reju born in Oboro, Nigeria, is a Professor

of Applied and Computational Mathemati cs. He

obtained his BSc (Hons) degree from University of Jos,

Nigeria, MSc degree (Applied Mathematics –

Continuum Mechanics) from University of Ibadan,

Nigeria and PhD degree (Computational Mathematics -

Optimization) from University of Ilorin, Nigeria. He

taught mathematics and computer science in a number of Universities in

Nigeria; supervised Master’s a nd doctoral theses in Computational

optimization; facilitated and coordinated capacity building programmes for

open and distance learning institutions and agencies supported by the

Commonwealth of Learning (Canada) in West African sub-region (2003-

2010). He is presently with the Polytechnic of Namibia, Namibia. He is a

member Mathematical Association of Nigeria, Nigerian Computer Society

and a senior member of International Association of Computer Science &

Information Technology (IACSIT). His scientific research interests are

mathematical modeling, optimal control and computational optimization

algorithms related to diffusion, fluid and wave problems. His other research

areas ar e open and distance learning modelling with special interest in

elearning and quality assurance.

Kalaluka Kanyimba was born in Mongu Zambia. He

obtained his Bachelor of Engineering degree in Electrical

Machines and Power Engineering at the University of

Zambia in 1989 and his Master of Engineering degree in

Power Systems at the Southeast University in P.R. China

in 1997. He is a member of the Institution of Engineering

and Technology (IET). H e i s currently a lecturer at the

Polytechnic of Namibia in the Department of Electrical

Engineering. His research interests include Control Engineering and

Automation, Electrical Machines & Drives, Energy Systems and Energy

Management & Modelling and Power Systems Protection. He has been

heavily involved in curriculum development at both diploma and degree

levels.

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