1
A multiple objective evolutionary approach for the design and selection of load control strategies A. Gomes, Member, IEEE, C. Henggeler Antunes and A. Gomes Martins, Member, IEEE
Abstract--Load management activities, even in scenarios characterized by an unbundled electricity market, maintain their potential attractiveness, not just due to operational issues but also because of potential economic benefits. However, multiple, incommensurate and conflicting objectives are at stake in the design and selection of load management actions. Evolutionary algorithms, working with a population of potential solutions, are well suited for such multiobjective optimization problems of combinatorial nature. Moreover, when applied to load management programs, they allow both for the design and the selection of control strategies. The combined use of this type of algorithms and adequate load models allows some of the concerns these actions may arise, such as the payback phenomenon, to be taken into account. In the proposed approach, the effects of load control strategies are computed at different demand aggregation levels. This capability and the explicit consideration of multiple objective functions in the mathematical model enable the proposed approach to be used in different possible scenarios related with power systems structure and by different entities. Index Terms— Load management, evolutionary algorithms, multiobjective optimization.
I. INTRODUCTION
T
he use of available demand-side resources by changing the regular working cycles of loads through the implementation of appropriate power curtailment actions have been done by some electric utilities in the last decades and with diverse goals. Some of them are related with operational benefits, such as increasing load factor, reducing peak power demand, reliability concerns or loss and costs reduction. Recently, this kind of programs have raised further attention mainly due to the economic interests related with the volatility and spikes of wholesale electricity prices and also because of reliability concerns (transmission congestion and generation shortfalls) [1] [2]. These programs, which include direct load control, interruptible power and voluntary load shedding, can This work was supported in part by the Foundation for Science and Technology (FCT) under Grant POCTI/ESE/38422/2001. The authors are with the Department of Electrical Engineering and Computers, University of Coimbra, Polo II 3030-290 Coimbra, Portugal and with INESC Coimbra, R. Antero de Quental 199, 3000-033 Coimbra, Portugal (email:
[email protected];
[email protected];
[email protected]).
be very attractive for a retailer dealing with volatile wholesale prices and fixed, over a certain time period, retail prices. The so-called payback effect and an eventual strong reduction in revenues are two examples of negative effects that can be originated by a Load Management (LM) program, meaning that a careful analysis must be done before the implementation of such actions to avoid undesirable consequences. The identification and selection of the power curtailment strategies under multiple, incommensurate and potentially conflicting objectives is a hard task to be carried out under a LM program framework. Some of the objectives that a Decision Maker (DM) may consider when dealing with this kind of problems are: minimize maximum power demand, maximize profits, minimize losses and minimize causing discomfort to the customers. The selection of adequate load shedding actions to be implemented over sets of loads, grouped according to some criteria, is therefore a Multiobjective (MO) optimization problem to be faced by LM programs. Evolutionary Algorithms (EAs) while search and optimization tools are well suited for complex and combinatorial MO problems since they work with a set of potential solutions (population) in each generation. The main aim of this study is to use EAs in the identification and selection of load shedding actions to be applied to the different groups of loads under control taking explicitly into account multiple axes of evaluation. Some related work is briefly referred to in section 2 and the multiobjective model to provide decision support in the design and selection of the load shedding strategies is presented in section 3. The characterization of the case study is made in section 4, and the main results are presented in section 5. II. RELATED WORK Several and different issues associated with the design and the implementation of LM programs are the subject of some works that can be found in the literature, with diverse approaches being proposed for each of them. For example, [3] and [4] deal with the decision part of the problem only, e.g. the selection of a control strategy from a pre-specified set. Other authors propose different ways to identify the set of potential solutions. [5] propose a method that first proceeds with the identification of control patterns, which are the off and on periods, for each group and then determines the optimal control strategy for the group by putting several periods together. [6] use a reduced model in an interactive (and
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iterative) way for the identification of potential control strategies. On the other hand, in some more traditional approaches this pre-identification stage is based on available knowledge from past experiences or from pilot programs carried out with the purpose of providing data for a suitable preevaluation study. Some diversity can also be found regarding the pursuit objectives. [7] propose a model aimed at reducing the spinning reserve, while the model presented in [8] aims at the evaluation of the impact of LM actions on the system reliability. However, the costs reduction or peak power minimization are the objectives generally used. For instance, these objectives can be found together or individually in [9], [10], [5], [11], among others. The maximization of the profit of utilities is the main objective reported in [12]. Besides these objectives, the eventual discomfort caused to customers resulting from the implementation of power curtailment actions is taken into consideration in the models reported in [6] and [13], once the acceptance or rejection of these activities can be influenced by the energy service quality. From both a utility and an electricity retailer point of view, a major concern LM may arise is the so-called payback effect, which consists in the increasing in peak power demand due to the power restoration of controlled loads. This phenomenon is related with thermostatic loads, which are commonly used in LM activities. In this type of loads, when a power curtailment occurs the demand is deferred in time and the diversified demand is changed in a way that may cause higher demand during the power restoration that it would be if no load shedding had occurred. This phenomenon must be carefully accounted for, otherwise peak demand reduction, which is one of the goals in most LM works, could not be attained. Some of the related works use a very simplified approach to the payback phenomenon. Based on diversified demand, [14] established that payback lasts for two hours only, and the demand in every fifteen minutes period is related with the amount of energy not consumed during the control period. These relationships have been identified from data collected in load research campaigns. [10] and [15] also use this approach. [11] also relates the payback with the diversified demand and control schedules, being the payback the load above the normal diversified demand. Other authors use physically based models in order to appropriately deal with the payback effect [6][16][17]. Two main conclusions can be drawn from this diversity of approaches. On one hand, the task dealing with the identification of the candidate solutions is a crucial step in a LM program and some more efforts must be put on this step, since the selection of a control strategy is done from that prespecified set. On the other hand, suitable models must be used, capable of dealing not only with the demand of controlled loads and the LM actions but also providing the necessary data for an adequate evaluation of the impacts to be carried out. In the approach presented in this paper an EA is used for the design and the selection of load control strategies to be
applied over several groups of loads. The loads to be considered for LM actions are air conditioners (ACs) and electric water heaters (EWHs) whose demand results from Monte Carlo simulations through the use of physically-based load models (PBLMs) that have been experimentally validated. These models reproduce the physical phenomena arising in these types of end use-loads, this way allowing for the identification of changes in the demand and the payback effect originated by LM actions. This is one of the main advantages of using PBLMs, once changes in individual power demand due to external load shedding actions can easily be accounted for both in individual and global demand. This ability to deal with load control actions, in the sense that they can easily reproduce the changed demand of controlled loads, makes PBLMs specially suited for such problems. The counterpart is the cpu time usually needed to simulate realistic situations. However, this is not an important disadvantage since in power systems one normally works with a day ahead (or more) forecasts. There are two main reasons why EA are suitable to deal with LM issues. Firstly, they are well suited for complex MO problems of combinatorial nature like this one involving the selection of a LM control action; secondly, due to their ability to deal with both design and selection of control strategies. In the combined use of an EA and the PBLMs for the identification and selection of load shedding actions the identification process proceeds as follows: the PBLMs reproduce the load demand according to some control strategy being simulated; the EA then evaluates the results and generates the next generation of control strategies, which are again simulated within PBLMs. This iterative process continues until some stop condition is reached. III. M ULTIOBJECTIVE MODEL The market structure volatility and the diversity found in electricity systems, ranging from a more traditional vertically integrated structure to a totally unbundled structure with different entities in each branch of activity, ask for a model able to deal with the different possible scenarios and with diverse goals. In the proposed approach, the effects of LM programs are evaluated at different demand aggregation levels that, together with the multiple objectives considered, make possible the use of this model in distinct market scenarios. The only effects that are evaluated are the ones related with changes in power and consumption levels. Such values are essential if a cost-benefit analysis or an environmental impact quantification is to be carried out. The objective functions considered in the proposed model are: • Minimization of peak power demand. A distribution utility (which owns and also manages the distribution grid) is generally interested in decreasing peak demand at, for example, sub-station and transformer stations levels due to capacity constraints, reliability concerns, or efficiency improvement through the reduction of losses. Power
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demand reduction may also be desirable due to costs associated with a specific demand level, where profits may substantially decrease because average wholesale prices are much higher than retail prices in a certain period of time. Peak reduction enables both the distribution utility and the retailer to have a better capability of continuously exploring the differences between purchasing and selling prices. In this paper, the reduction of power demand is computed at three different demand aggregation levels. In general, this accommodates a potential common interest for a distribution utility and a retailer selling electricity. In the case of a retailer, which buys electricity in the wholesale market and sells it at the retail level, economic interests may be the main purpose. In this case, the aggregate level can be the demand of all the customers and the less aggregate levels the demand imposed by a class or a specific sub-group of customers. The evaluation carried out at these different levels may also be used, once additional useful data is available, in the establishment of the “rewards” to be given to customers for their participation in such kind of activities. • Profit maximization. Profits in the selling of electricity are generally influenced by the amount of electricity sold and the time of day/season/year. In the presence of demand and wholesale price forecasts, the distribution utility/retailer can design adequate load shedding actions in order to maximize profits once retail prices are fixed. • Loss factor minimization. This objective function is related with both operational and economic goals. Losses are a function of some physical network parameters. In the absence of a detailed knowledge on the network characteristics that would allow evaluating the losses imposed at a specific power demand, an estimate is usually used through the loss factor. The loss factor is the ratio between the sum of losses and the maximum losses value. Since loss reduction is directly proportional to loss factor reduction, this has been considered in the model. • Minimize discomfort caused to customers. The electricity service provided by loads under control is changed, possibly postponed or even not provided at all, when load management actions are implemented. These changes can eventually cause some discomfort to customers that must be minimized, so that those actions become also attractive from the customers' point of view (with eventual reduction in their electricity bill) and/or at least not decrease their willingness to accept them. Discomfort is evaluated through two objective functions related with the time some state variable (controlled by loads) is over or under a pre-specified threshold level. The objective functions to be minimized are the maximum continuous time interval in which the situation has occurred and the total time it has occurred.
The mathematical formulation of the model is as follows. Notation: w – index for demand type at the less aggregate level i – index for time period under consideration j – index for group of loads k – index for load curtailment strategies N - number of intervals in which the load diagram is discretized aijk – power demand increase/decrease at aggregate level in time interval i, when strategy k is applied to group j d wijk – power demand increase/decrease at less aggregate level, type w, at interval i, when strategy k is applied to group j xj k – binary decision variable denoting whether load shedding strategy k is applied to group j Dwi – power at less aggregate level of type w at time interval i, without load curtailment A i – power at aggregate level at time interval i, without load curtailment g j k – profit when strategy k is applied to group j lj k – loss factor when strategy k is applied to group j tj k –total time in which temperature is beyond the threshold of discomfort when strategy k is applied to group j mj k –maximum interval of time, number of minutes, in which temperature is beyond the discomfort threshold when strategy k is applied to group j ei – energy profits at interval i p i – power profits at interval i Minimizing peak power at the more aggregate level corresponds to minimizing the maximum of Ai + a ijk x jk . By introducing the auxiliary variable v,
∑∑ j
k
the min-max problem can be transformed into: min v s.t. Ai +
∑ ∑a j
ijk
x jk − v ≤ 0.
k
In a similar way, for each power demand objective at less aggregate level (w=1 and w=2) one obtains min v1 ; s.t. D1i +
∑ ∑d j
1 ijk
x jk − v1 ≤ 0
k
and min v2 ; s.t. D2 i +
∑∑ d j
2 ijk
x jk − v 2 ≤ 0.
k
Profits are related with power and the amount of energy sold. Variations related with energy are given by aijk*ei *xjk and variations related with power are given by (Ai+aijk)*pi *xjk. These values are used on a daily basis and are evaluated according to variations on profits caused when strategy k is applied to group j,
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g jk = ∑ a ijk .ei .x jk + ∑ ( Ai + aijk ). pi .x jk . Thus the example, the DM may impose further constraints on the i
i
profit objective function is
max ∑∑ g jk x jk . j
k
1T Losses(t ).dt T ∫0 The loss factor, l, is given by l = , max [Losses(t )] where Losses(t ) is the loss function, max [Losses(t )] is the maximum losses value within the time interval considered. Usually, data about demand and losses are available in a l discrete way and therefore is given by
1 N ∑ Losses[i] N i =1 l= . max[Losses[i]] Using
l=
1 N
the
per
unit
system
representation
∑ Losses[i ] , the objective function is N
i =1
min
∑∑ l j
jk
x jk .
k
The minimization of discomfort is given by
min min
∑∑ t j
k
j
k
jk
∑∑ m
x jk , and
jk
x jk .
The mathematical model is thus
min v
acceptable values for the objective functions (reservation levels). EAs are unconstrained search tools and, in spite of the several approaches that can be implemented in order to deal with non-feasible solutions, in this work that task is carried out by the evaluation function. In the experiments already carried out this approach has shown to be sufficient to deal with infeasibility. IV. CHARACTERIZATION OF THE CASE STUDY As already referred to above the proposed approach can be used in different market structure scenarios. This is made possible by enabling changes of demand shape to be evaluated at different levels of aggregation and with different meanings according to the entity using the tool. In this work, a case study for a distribution utility based on real world data is presented. The utility faces some capacity constraints in one sub-station (SE) and two power transformers (PT1 and PT2). The objective of the study is to evaluate the use of LM actions aimed at reducing the maximum demand values in both aggregation levels and provide the utility with data needed for a thorough evaluation of this scenario. This means that changes in power and energy values produced by LM actions on the demand at SE, PT1 and PT2 must be computed. The SE feeds all the controlled loads through power transformer stations with no capacity constraints except those to be satisfied at PT1 and PT2. Typical demand at the sub -station (most aggregate level) and the two power transformers PT1 and PT2 (less aggregate power level) owned by the utility are shown in figure 1, for a typical summer workday.
min v1
min v2
Typical load diagram. SE - thick; PT1 - dotted; PT2 - thin
∑∑ j
min
k
∑∑ j
min
l jk x jk t jk x jk
k
m jk x jk
k
s.t.
Ai + ∑∑ aijk x jk − v ≤ 0 j
30
600
25
500 20 400 15 300 10
200
∑∑ j
35
700
100
5
0
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min
k
PT1, PT2 (kW)
j
800
SE (MW)
max ∑∑ g jk x jk
Fig. 1. Typical demand at SE, PT1 and PT2 (summer workday)
k
D1i + ∑ ∑ d 1 ijk x jk − v1 ≤ 0 j
k
j
k
D2 i + ∑ ∑ d 2 ijk x jk − v2 ≤ 0 x jk ∈ {0,1} ∀j , k . Other constraints can be incorporated into the model. For
Maximum peak demand values are 31632 kW, 602 kW and 707 kW at SE, PT1 and PT2, respectively. Table 1 shows some additional characteristics about the consumption and the number of consumers in each aggregation level. SE feeds consumers of residential, commercial and services types. PT1 feeds consumers of residential type only while PT2 feeds residential and commercial consumers.
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SE PT1 PT2
SE PT1 PT2
peak consumption nr of (kW) (kWh) consumers 31632 536432 25852 602 8731 626 707 8027 521 # consumers residential comercial services others 20408 2084 2361 999 626 0 0 0 247 260 0 14
Loads under control are grouped in such a way that all the loads belonging to one group are fed by the same power transformer. Each group contains one type of load only. Changes occurring in loads fed by PT1 (220 EWHs) are evaluated both at PT1 and SE demands, and similarly with changes in loads fed by PT2 (195 EWHs and 45 ACs). In table 2 some more characteristics about the controlled load are presented. The demand of all controlled groups of loads is shown in figure 2. The total number of loads evaluated at SE level is 1860 (1415 EWHs and 445 ACs).
period in the SE that occurs at about 12:00h. The controllable load at this time is 2174 kW (6.9% of the peak demand). Figures 3 and 4 show the global demand of PT1 and PT2 together with the demand of the controlled loads fed through each power transformer. In figure 3 a typical residential demand shape can be recognized. Despite the maximum demand value of the controlled loads fed through PT1 is 272 kW (at 08:45h), the available load for control at 20:45h, that is when maximum peak demand (602 kW) at PT1 occurs, is only 119 kW, which is 19.8 % of the peak value. The peak at PT2 (707 kW), occurs at 12:15h but two other periods of high demand values exist at 15:30h and 18:15h (see fig. 4). The controllable load during these periods is 292 kW (41.3%), 267 kW (39.2%) and 205 kW (29.8%), respectively. Total demand at PT1 (thin) and demand of loads under control (dotted) fed by PT1 700 600 500 400 kW
T ABLE 1 CHARACTERIZATION OF SE, PT1 AND P T 2
300 200 100
1000
10
500
5
0
0
23:00
21:00 22:00
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18:00 19:00
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9:00 10:00
15
7:00 8:00
20
1500
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25
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30
2500
Fig. 3. Demand at PT1 and demand of controllable loads fed by PT1 SE demand (MW)
35
3000
Total demand at PT2 (thin) and demand of loads under control (dotted) fed by PT2 800 700
Fig. 2 Global demand (SE) and demand of all loads under control
600 500 kW
0:00 1:00 2:00 3:00 4:00 5:00 6:00 7:00 8:00 9:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00
Total demand under control (kW)
3500
2:00 3:00
0
SE demand - thick; demand under control - dotted
400 300 200
Loads (ACs + EWHs) under control Number peak (kW) energy (kWh) total (SE) 1415 3000 30058 PT1 220 272 2358 PT2 195 348 3697 Air conditioners Number peak (kW) energy (kWh) total (SE) 445 2174 17036 PT1 0 0 0 PT2 45 36 278 Electric water heaters Number peak (kW) energy(kWh) total (SE) 970 1313 13025 PT1 220 272 2358 PT2 150 323 3419
The maximum demand under control is 3000 kW, occurring at 13:45h, which is not coincident with the maximum demand
100 0 0: 00 1: 00 2: 00 3: 00 4: 00 5: 00 6: 00 7: 00 8: 00 9: 0 10 0 :0 11 0 :0 12 0 :0 13 0 :0 14 0 :0 15 0 :0 16 0 :0 17 0 :0 18 0 :0 19 0 :0 20 0 :0 21 0 :0 22 0 :0 23 0 :0 0
T ABLE 2 CHARACTERIZATION O f SE, PT1 AND P T 2
Fig. 4. Demand at PT2 and demand of controllable loads fed by PT2
The efficacy of LM actions depends on the amount of controllable loads available and the number of groups under control. It seems reasonable that better results are obtained for a higher number of such groups. However, some economic and physical concerns may arise when this number increases too much. Loads under control have been arranged in 20 groups. Groups 1-4 are fed by PT1, groups 5-14 are fed by SE and the remaining 5 groups (16-20) are fed by PT2. The peak power (kW) and the consumption (kWh) of the 20 groups are displayed in tables 3 (EWHs) and 4 (ACs).
6 T ABLE 3 GROUPS OF EWHS ( GROUPS 1-4 FED THROUGH PT1; GROUPS 15, 16, 19 AND 20 FED THROUGH PT2) Group1 Group2 Group3 Group4 Group5 Peak power 96 57 67 66 150 Consumption 523 507 660 669 1109 Group6 Group7 Group8 Group9 Group10 Peak power 177 201 138 129 148 Consumption 1137 1076 1261 1271 1393 Group15 Group16 Group19 Group20 Peak power 46 95 122 114 Consumption 585 915 1179 740 T ABLE 4 GROUPS OF ACS
measure of their performance). The different steps of the evolution process are repeated until a stop condition is reached: a given number of generations is attained or the decision maker considers the results are satisfactory compromise solutions for a final decision to be based on. In each generation there are 20 potential solutions for the problem. One of the solutions computed by the EA is shown in figure 6. The horizontal axis represents the time on a daily basis and the power curtailment pattern (on/off periods) to be applied to each group of loads under control is represented in each window.
Group11 Group12 Group13 799 296 259 6133 2407 2087 Group14 Group17 Group18 Peak power 793 24 13 Consumption 6131 179 99 Peak power Consumption
The profits forecast used in the simulations have two terms. One is related with energy sold and the other one is related with power. This term is a fixed value per kW, which means that a higher demand originates higher values. In the periods of the day presenting higher power demand values, profits obtained with electricity sales are negative as shown in figure 5. The profit structure is not adequate for some scenarios in which the price of demand (kW) strongly depends on the values in such a way that higher demand values are penalized. Profits are computed on an annual basis. Profit forecast per kWh (thick); demand at SE (thin) 0.08
35
0.07
30
0.05
25
0.04
20 MW
Euro
0.06
0.03 0.02
15
0.01
10
0 5
-0.01
0 0:00 1:00 2:00 3:00 4:00 5:00 6:00 7:00 8:00 9:00 10:00 11:00 12:00 13:00 14:00 15:00 16:00 17:00 18:00 19:00 20:00 21:00 22:00 23:00
-0.02
Fig. 5. Profit forecast and global demand at SE
23:00
21:00 22:00
18:00 19:00 20:00
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15:00
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The EA is adequately parameterized before search starts, by incorporating specific characteristics of the problem [18] [19]. The first generation is a random population and the evolution process then proceeds in the following way: all the individuals are evaluated according to their performance in all objective functions, this task being carried out by using the PBLM according to the objectives being analyzed. The crossover and mutation steps follow, which allow the algorithm to proceed on the search space and thus to bring some new genetic material to the population. The next step is the selection of the fittest for being part of the next generation based on their fitness (a
0:00 1:00
V. RESULTS
Fig. 6. Power curtailments patterns to be implemented over the 20 groups of controlled loads (solution 1)
Table 5 displays the values obtained for each objective function with no LM actions and with the implementation of the power curtailment actions characterized in figure 6 (solution 1). This solution presents the best values for
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objective functions 2, 3 and 4, considering all 20 solutions computed by the EA. The load diagrams for these situations for PT1, PT2 and SE are displayed in figures 7, 8 and 9, respectively.
about 50 k Euro (year). With respect to the discomfort caused to customers, none of the thresholds (see section 3) related with this objective has been exceed, which is in accordance with the weak reduction in energy consumption.
T ABLE 5 OBJECTIVE FUNCTION VALUES Original Solution 1 31632 30901 602 562 0 0 707 579 0.54137 0.53518 7385 7335
Difference 731 40 128 0.00619 -50
% 2.31% 6.61% 18.08% 1.14% -0.68%
30 25 20 MW
Demand at SE (kW) Demand at PT1 (kW) Number of minutes Maximum Interval Demand at PT2 (KW) Loss factor Profits (k Euro)
Demand at SE: without(thin) and with (dotted) power curtailments 35
15 10 5
Demand at PT1: without(thin) and with (dotted) power curtailments 700 600 500
kW
400 300 200 100
23:00
21:00 22:00
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The payback effect is presented in the controlled demand at PT1 (fig. 7). The reduction in peak power demand at PT1 is 40 kW, which is 33.6% of the controllable load and 6.68% of the peak demand. The reduction in energy consumption is only about 0.27% (24 kWh).
Fig. 9. Demand at SE with and without power curtailments
The optimization process carried out resorting to EA results on several potential solutions to the problem. In the implemented EA, the DM may give the algorithm full control in the selection of the final compromise solution or he/she can choose it from the population of non-dominated solutions. In this case, let us suppose that the DM wants to know a solution that further improves the reduction of peak power demand at power transformer 2 (solution 2 presented in table 6). In fact this solution is the one that optimizes this objective function. Compared with solution 1, this non-dominated solution degrades the performance on the first 4 objective functions and improves the performance in the other objective functions.
Fig. 7. Demand at PT1 with and without power curtailments
The achieved reduction in maximum demand at PT2 is 128 kW, that is 43.8% of the load under control in the peak power demand period, and it represents 18.1% of maximum demand (fig. 8). The reduction in energy consumption is 4.35% (349 kWh). Demand at PT2: without(thin) and with (dotted) power curtailments
T ABLE 6 OBJECTIVE FUNCTIONS VALUE FOR A SECOND SO LUTION Demand at SE (kW) Demand at PT1 (kW) Number of minutes Maximum Interval Demand at PT2 (KW) Loss factor Profits (k Euro)
Original Solution 2 31632 31122 602 564 4916 18 707 561 0.54137 0.53468 7385 7351
Difference 510 38 146 0.00669 -34
% 1.61% 6.35% 20.62% 1.24% -0.46%
800 700 600
kW
500 400 300
If the degradation of the comfort objective functions is considered unacceptable then a compromise solution could be solution 3 (table 7), which strongly improves these objectives by slightly degrading objective functions 2, 5, 6 and 7.
200 100
0: 00 1: 00 2: 00 3: 00 4: 00 5: 00 6: 00 7: 00 8: 00 9: 0 10 0 :0 11 0 :0 12 0 :0 13 0 :0 14 0 :0 15 0 :0 16 0 :0 17 0 :0 18 0 :0 19 0 :0 20 0 :0 21 0 :0 22 0 :0 23 0 :0 0
0
Fig. 8. Demand at PT2 with and without power curtailments
At SE, the reduction of peak power (731 kW) is about 33.6% of controllable load and 2.31% of the peak power demand (fig. 9). The reduction in energy consumption is 0.71 % (909 kWh). The lower values obtained for the reduction in energy consumption explains the slight reduction in profits, which is
T ABLE 7 OBJECTIVE FUNCTIONS VALUE FOR A THIRD SOLUTION Demand at SE (kW) Demand at PT1 (kW) Number of minutes Maximum Interval Demand at PT2 (KW) Loss factor Profits (k Euro)
Original Solution 16 Difference 31632 30941 691 602 579 23 5 3 707 584 123 0.54137 0.53518 0.00619 7385 7337 -47
% 2.18% 3.87% 17.40% 1.14% -0.64%
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VI. CONCLUSION The identification of suitable load control strategies is a fundamental step in any LM program. The selection of control strategies that can be applied to the groups under control to be evaluated by multiple, incommensurate and conflicting objective functions is a complex combinatorial multiobjective problem. Its complexity is further increased by the diversity and even volatility found in power systems structure, since different entities pursuit different goals. Therefore, an adequate model to support the decision making process in the framework of LM activities should be able not only to deal with all the objectives at stake but also be useful for the different possible players in the power market. The multiple objective model presented in this paper aims at providing decision support by enabling DMs to rationalize the comparison between non-dominated solutions under the axis of evaluation: minimization of peak power demand at different demand aggregation levels, minimization of the discomfort caused to consumers, minimization of the loss factor, and maximization of the utility profits. The combined use of an EA and the PBLM makes the identification and the selection of the control strategies possible without the need of pre-specifying the LM actions. This is relevant since the starting point of the EA (the initial population) can be the population computed in the previous day, thus making possible to find satisfactory compromise solutions faster.
[10] Y.-Y. Hsu and C.-C. Su, "Dispatch of direct load control using dynamic programming,” IEEE Trans. on Power Systems, vol. 6, 3, pp. 1056-1061, Aug. 1991. [11] A. Cohen and C. Wang, "An optimization method for load management scheduling,” IEEE Trans. on Power Systems, vol. 3, 2, pp. 612-618, May 1988. [12] K.-H. Ng and G. B. Sheble, "Direct load control - A profit-based load management using linear programming,” IEEE Trans. on Power Systems, vol. 13, 2, pp. 688-695, May 1998. [13] B. Rautenbach and I. Lane, "The multi-objective controller: A novel approach to domestic hot water load control," IEEE Trans. on Power Systems, vol. 11, 4, pp. 1832-1837, Nov. 1996. [14] S. Lee and C. Wilkins, "A practical approach to appliance load control analysis: a water heater case study," IEEE Trans. on Power App. and Systems, vol. PAS-102, 4, pp. 1007-1013, Apr. 1983. [15] K. Bhattacharyya and M. Crow, "A fuzzy logic based approach to direct load control," IEEE Trans. on Power Systems, vol. 11, 2, pp. 708-714, May 1996. [16] P. Dolan, M. H. Nehrir and V. Gerez, "Development of a Monte Carlo based aggregate model for residential electric water heater loads," Electric Power Systems Research, vol. 36, pp. 29-35, 1996. [17] J. Laurent, G. Desaulniers, R. Malhamé and F. Soumis, "A column generation method for optimal load management via control of electric water heaters,” IEEE Trans. on Power Systems, vol. 10, 3, pp. 1389-1400, Aug. 1995. [18] A. Gomes, C. Antunes and A. Martins, “Parameter selection in a GA for multiobjective problems: a case study,” presented at the Workshop on Multiple Objective Metaheuristics, Paris, Nov. 2002. [19] A. Gomes, C. Antunes and A. Martins, “Adaptive mutation probability as an efficient tool to incorporate knowledge in an evolutionary algorithm for multiobjective electricity planning,” Research Report nº 7 2002, INESCC. http://www.inescc.pt/download/RR7_IC.pdf.
REFERENCES [1] G. Heffner and C. Goldman, "Demand responsive programs - an emerging resource for competitive electricity markets?,” Proceedings of the International Energy Program Evaluation Conference (2001 IEPEC), Aug. 2001. [2] E. Hirst and B. Kirby, "Retail-load participation in competitive wholesale electricity markets,” Edison Electric Institute, Jan. 2001. [3] R. Bhatnagar, S. Rahman, "Dispatch of direct load control for fuel cost minimization,” IEEE Trans. on Power Systems, vol. 1, 4, pp. 96-102, Nov. 1986. [4] C. Kurucz, D. Brandt and S. Sim, "A linear programming model for reducing system peak trough customer load control programs," IEEE Trans. on Power Systems, vol. 11, 4, pp. 1817-1824, Nov. 1996. [5] J. Chen, F. Lee, A. Breipohl and R. Adapa, "Scheduling direct load control to minimize system operation cost,” IEEE Trans. on Power Systems, vol. 10, 4, pp. 1994-2001, Nov. 1995. [6] H. Jorge, C. Antunes and A. Martins, “A multiple objective decision support model for the selection of remote load control strategies,” IEEE Trans. on Power Systems, vol. 15, 2, pp. 865-872, May 2000. [7] F. Lee and A. Breipohl, "Operational cost savings of direct load control," IEEE Trans. on Power App. and Systems, vol. PAS-103, 5, pp. 988-992, May 1984. [8] M. Fotuhi-Firuzabad and R. Billinton, "Impact of load management on composite system reliability evaluation short-term operating benefits,” IEEE Transactions on Power Systems, vol. 15, 2, pp. 858-864, May 2000. [9] D.-C. Wei and N. Chen, "Air conditioner direct load control by multi-pass dynamic programming," IEEE Trans. on Power Systems, vol. 10, 1, pp. 307-313, Feb. 1995.
Álvaro Gomes received his Master degree from the University of Coimbra in 1995. Currently, he is preparing his Ph.D. degree. He is a lecturer at the Department of Electrical Engineering and Computers, University of Coimbra and a researcher at INESC Coimbra. His research interests include demand-side management, load modeling and evolutionary algorithms. He is an IEEE member since 1992. C arlos Henggeler Antunes received his Ph.D. degree in Electrical Engineering (Optimization and Systems Theory) from the University of Coimbra in 1992. He is an associate professor at the Department of Electrical Engineering and Computers, University of Coimbra, and a researcher at INESC Coimbra. His research interests include multiple objective programming, decision support systems, and energy planning. António Gomes Martins received his Ph.D. degree in Electrical Engineering from the University of Coimbra in 1985. He is presently a full professor at the Department of Electrical Engineering and Computers at this University, where he is responsible for a R&D group on efficient use of energy resources. He is also a researcher at INESC Coimbra. His current research interests are energy planning, load modeling, demand-side management.
