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