- Email: [email protected]
Power outage planning
254
European Journal of Operational Research 49 (1990) 254-265 North-Holland
Power outage planning Mohammad Modarres Department of Industrial Engineering, Sharif University of Technology, Tehran, Iran Abstract: This paper is concerned with scheduling the planned electric power outage in a region where the expected demand for power exceeds the expected supply, during some interval of the year. To minimize the total economic loss of the power shortage, the best allocation of power outage among the consumers is determined. A hierarchical planning system is proposed to find the optimal solution with two levels of aggregation. The first step in this system is to determine the outage duration for each consumer in the shortage season, in order to minimize the total economic loss. In the second step, a disaggregate model determines when, and for how long, the power of a consumer will be cut off during a shortage season. Linear and mixed integer programming are used, respectively, to develop the aggregate as well as the disaggregate models. A heuristic algorithm is developed to solve the problem, practically. We also discuss some results of the model. Keywords: Developing countries, electric power, application of mathematical modelling, hierarchical planning system
1. Introduction The economy of many developing countries suffers severe damage, due to the shortage of the electric power. Not only is the industrial sector heavily dependent upon the reliable flow of the electricity, but even the agricultural sector, which still applies the traditional methods, needs the electric power for a variety of reasons. In these countries, the consumption of electricity is growing so fast, that supply falls short of satisfying the demand. Take Iran as a case. Although the annual growth rate of the installed capacity during the 1971-1981 period (12 percent) has passed the annual growth rate of per capita G N P (4.8 percent) as well as the population growth rate (3 percent), the shortage of electric power has been steadily growing. To justify such a phenomenon, one should cite the occurrence of the drastic changes in the life style in these countries, as well as the substitution of 'man' by 'machines'. There are many reasons to believe that the supply of the required amount of the electric power, in the short term, is difficult or even impossible. Power generation, transmission, and distribution, in general, is quite expensive. In fact, the main bulk of the required investment is in terms of foreign currencies, which the developing countries can not provide, easily. Even if they can provide the money to invest, it takes between five to ten years to design, purchase and install the equipment. Therefore, the power shortage is considered a fact of life in many developing countries. When the demand in the peak hours exceeds the supply, the power company has to cut the electricity of some consumers off. As a result, the consumers, especially those in the industrial sector experience considerable monetary loss. This paper introduces the problem of power shortages and determines the best way to distribute the planned outages among the consumers, to have the minimum loss. Clearly, the planning should cover all the sectors. However, in this paper, only the outage planning for the industrial firms will be considered. After the industrial outage planning, part of the power shortage will be compensated. Then, the outage Received March 1990 0377-2217/90/$03.50 © 1990 - ElsevierSciencePublishers B.V. (North-Holland)
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planning for the other sectors will be done, which is beyond the scope of this paper. However, it can be performed, using the same framework introduced for the industrial sector. The proposed mathematical model considers the outage planning at two different levels, aggregate and disaggregate. At the aggregate level, the model determines the length of the outage that should be assigned to each firm, while the detailed outage schedufing of each firm will be done at the disaggregate level. In the literature, not much can be found on this subject, since the developed countries do not face a problem of this nature. The reliability of the power supply in these countries is so high they almost do not have any power outage. A common measure of power reliability, called LOLP (Loss Of Load Probability), can illustrate the extent of power shortage in a country or a region. While LOLP in the USA is less than one day per ten years, many developing countries have power outages everyday, or at least in an interval of the year. Section 2 discusses the nature and classification of outage costs and establishes the fact that some costs depends on the outage duration, while some other costs vary as a function of the number of outages. In Section 3, the method of illustrating the supply and demand diagram will be discussed. In Section 4, the aggregate model, as well as the mathematical foundations of the properties of the optimal solutions will be presented. Section 5 contains some results from the application of the model. In Section 6, another mathematical model to disaggregate the results of the aggregate model and finally in Section 7 a heuristic algorithm to solve this model will be introduced.
2. Classification of outage costs Whenever the supply of electricity is less than the demand, the consumers lose in different ways. These economic losses are called 'outage costs'. A number of papers have investigated the nature of the outage costs and how to classify them. Many studies, in different countries have designed the suitable methodology to estimate the outage costs of the various sectors and have got some interesting results. To review the literature, see for example, Munasinghe (1979). Outage costs, in general, can be classified into two categories, direct and indirect costs. Any economic loss caused by a particular outage, say the production interruption, is called a direct cost. Indirect costs are not the results of any specific outage, but are consisting of all those losses a consumer affords, in order to decrease the expected costs when an outage occurs. One example of indirect costs is the investment and operation costs of a standby generator. A consumer will incur some indirect cost, when and only when he can realize some sort of savings in his direct costs. Therefore, provided the decision to pay extra indirect costs is reasonable, the sum of both types of costs will be less than the direct costs alone, when nothing is done to decrease the loss due to the outage. Based on this fact, in our model, only the direct cost is considered. We assume indirect costs can not be counted on, because they are the results of the individuals' actions to minimize their losses and following no prefabricated patterns. The direct costs also can be classified into two types. The first type, roughly, is a function of the number of the outages and independent of the outage length, especially, when the duration is not too short and exceeds a certain limit. It is clear that this limit depends on the nature of the product, manufacturing process and some other factors. This kind of cost can be thought of a set up cost. The second type, on the other hand, is a continuous function of the outage duration. This type of cost is caused by the production interruption, overtime payments to compensate the production losses and the cost of spoilage of the materials and products. This kind of cost, C(t), is not necessarily a linear function of the outage duration, t. However, in many manufacturing firms, it can be assumed that this kind of cost is the linear function of outage duration, t, if it is not too short. In our model, we assume the continuous function costs are linear and the set up costs are constant, regardless of how long the power is off. These assumptions can be justified, because a particular planned outages duration can not be very short, or the number of outages very large.
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3. Estimation of the power shortage Each year, the power shortage appears in specific months, which we call 'the shortage season'. For example, in Iran, the maximum shortage occurs in summers, especially June and July, when the demand for electricity increases rapidly, mostly because of high power consumption of the cooling systems. Since the outage planning is for the short term, usually, for one season only, the demand can be forecasted relatively precisely. The supply and demand for electricity can be shown by a graph, called Daily Load Duration Curve (DLDC). The Annual Load Duration Curve is a popular tool in the power generation planning. In our model, we are going to take advantage of the same idea, but for every day. In Figure 1, t represents the length of time of a day (in terms of hours) during which, the demand is at least D(t). We assume the supply level during a day is constant. As can be seen, during h hours (out of 24 hours), the demand exceeds the supply and expriencing some outages would be unavoidable. The area under the demand and over the supply curve, say E, represents the total daily shortage, in terms of megawatt hours. To compensate this shortage, the outage should be distributed among the industrial, residential, commercial and the other sectors. Since we are considering the industries only, a fraction of the total shortage, say Fr, should be distributed among the industrial firms. It is better to consider this fraction lower than the real fraction of the total energy consumed by those firms. This is so to decrease the economic and social losses of the industrial shutdowns. Although the patterns of DLDC in general can be changed on a daily basis, it can be assumed that in a shortage season there are only two patterns, working day and holiday patterns.
4. The aggregate outage planning models The aggregate level model determines how long, in a shortage season, the power of a firm should be cut off. Through analyzing the results of the disaggregate model, a detailed solution will be derived. However, as will be seen, even the result of the aggregate model alone provides a framework for decision making. In many regions, the planning can be done based on the results of the aggregate level model only.
Dmax
E
Supply curve
I
I I
g O3
I Dmin
I-
o
Demand curve
-+
t I I
I I I
I
I
t
h
Figure 1. Daily load duration curve
24
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In this section, we start with the presentation of the aggregate models with two different criteria. Then we show that these models can be converted into linear programming problems. The firms can be categorized into two different groups. The first consists of those firms which can not afford to be closed for a long time. For example, a food producer can not close more than three hours per day. The second group consists of the firms with no outage duration upper limit. The model considers all the industrial firms in a region. We use the following notation in the model: N Fu St Ct At T~
: : : : : :
T : Lt : : Xt :
Number of firms to consider in the region. Set of the firms with limited allowed outage duration. Total cost for firm i, of the first type (set up cost) per outage. Total cost of the second type for firm i, per hour. Megawatts used by firm i (energy consumption per hour). Total shortage of energy (megawatt hours) in the shortage season which should be distributed among the industrial firms, or the total season shortages times Fr (the fraction of the total shortage assigned to the industries, as defined previousely). Number of hours in a shortage season, during which demand exceeds supply. Upper bound for the allowed outage duration of firm i. The total length of the outage assigned to firm i. Number of outages assigned to firm i.
4.1. The first aggregate model (PI)
minimize
f ( x , y) = E[S~X, + CiY~]
(1)
i
subject to
~ A i Y ~> T~,
(2)
i
Yi-LiX~.<<,O, i~Fu,
(3)
Y~
(4)
i = 1 . . . . . N,
k~i
0< ~
i = l . . . . . N,
X,> 0: integers,
i = 1 . . . . . N.
(5) (6)
Constraint (4) compares the outages length of each firm with the others, where q >/1. It prevents the big gaps between the outage lengths of the firms. In a special case, where q---1, all firms will have equal outage lengths.
4.2. The second aggregate model (P2) Substituting the constraint (4) with the following one will result in the second model:
Y ~ < ~ [ q / ( N - 1 ) ] E X k,
q>~l,
i = 1 . . . . . N.
(7)
k¢i
In this model, the outage length of any firm is compared with the average of the others (excluding its own).
The following lemma and theorem provide the properties of the optimal solutions of the first and second models. Since the lemma is trivial, the proof has been omitted.
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Lemma 1. In both models the optimal values of the integer decision variables Xi, i = 1 . . . . . N, are as follows: Xi=
1 [Yii/Li]
ifi~Fu, otherwise,
(8)
where, for any real value a, we define [a] as an integer value, such that
(9)
[a]-l
Using Lemma 1, the integer decision variables X,, i = 1,..., N, can be omitted from the model. Therefore, the first part of the objective function as well as the constraints (3) and (6) will be deleted. Now, we see that the first and second aggregate models have been changed into the following models of (P3) and (P4), respectively. 4.3. Problem (P3): A new version of the first aggregate model
minimize
(10)
g ( y ) = E C1Yi i
subject to
E A i Y i ~ re,
(11)
i
Y~
i = l ..... N,
(12)
i=l,...,N,
(13)
O<~ Y i < T ,
where Wi=maxYk, k=~i
i = 1 ..... N.
(14)
4.4. Problem (P4)." A new version of the second aggregate model
This model is the same as (P3), except that the constraint (12) should be substituted by the following constraint: Y ~ < . [ q / ( N - 1 ) ] ~] Yk,
i = l . . . . . N.
(15)
Problem (P4) is a linear programming problem and can be solved by the simplex method easily (especially by the upper bound variation). To solve problem (P3), the result of Theorem 1 will be used. Before stating this theorem, we assume that the following relation holds: [C,/AI] <~[C2/A2] <~ . . . <<.[CN/AN].
(16)
By rearranging the decision variables, the relation (16) is always true. Theorem 1. Assuming the relation (16) is satisfied, then for any q >1 1, problem (P3) is equivalent to the following linear programming problem (P5): (P5)
minimize
(17)
g(y)=ECiY~ i
subject to
(18)
Y~AiYi >1 T~, i
(19)
Y1 <~qY2, O<~Y,~T,
i=l,...,N.
(20)
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Proof. Suppose that the optimal solution of (P5) consists of the vector
.....
L]
(21)
We claim that the following relation holds: Y1 >/Y2 >/ "'" >/ YN-
(22)
If this is not true, then for example, Y~ < Y2, while,
[Cl/A1] [C2/A2].
(23)
Now, consider a new solution Y'=[YI+u,
Y z - ( A 1 / A 2 ) u,
Y3 . . . . .
YN],
(24)
where
0 <~u <~n u n { ( a 2 / A i ) Y 2, T } .
(25)
It is clear that Y' is a feasible solution and also that g ( Y ' ) < g(Y). This contradicts the optimality of Y. Thus, (22) must hold.The proof will be complete if we show that Y is also an optimal solution of (P3), or if we show that constraint (12) holds. From (14) and (22), we have W1 = Y2, IV,,= Y1,
(26) i = 2 . . . . . N.
(27)
From (26), constraint (12) holds for i = 1, and from (22) and (27), it also holds for i >t 2; thus the proof is complete. []
5. Some results of the aggregate model An important result of the aggregate model comes from Lemma 1. That is, the number of outages assigned to a firm should be as small as possible. In other words, according to the properties of the optimal solution, if the power of a firm should be off for Y~ hours, then two cases may occur. When there is no upper limit for the length of the outage, then turn the power of this firm off just once and for ~ hours, in the shortage season. However, if for some technical reasons, a firm can not afford to be closed for more than L i hours, then each outage length of this firm should be exactly L i hours, in order to minimize the number of shortages. The above policy is different from what is applied in lran, these days. In the shortage season, the power company cuts the total power of an area off for about one to two hours a day. Adopting our suggested policy not only does bring the total losses down, but the firm can take some preventive steps to decrease its losses. For example, if a firm can not have power for two weeks, then it can either close the plant in that period and consider it as the annual vacation or operate the plant partially, using a standby generator, and make annual overhaul maintenance. Before the shortage season, the power company can determine the length of the outage assigned to all firms and hold discussion with them to come up with an agreement about the timing of the outage. One result of this agreement is the possibility of changing the working hours of a firm, for example shifting from day-hours to night-hours. The model also determines the shadow costs, a basis for the discount rate that the power company can consider for the firms that are willing to change their working hours.
6. The disaggregate model The aggregate model determines how long a firm must be without electric power during the shortage season. It can not be used, however, to specify the cut off time for each firm's power. If there are only a
M. Modarres /Power outage planning
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Dmax
C
I I
>.
I I
I
Dmin
I I I
24 Figure 2. Real time supply and d e m a n d curve
few firms in a region, the outage scheduling can be performed on the basis of trial and error in such a way that at any time, the total power consumption of shut down firms approximately equals the fraction of the power shortage which has been assigned to the industrial firms. However, if there are too many firms in a region, a supporting system for outage scheduling will be needed. Therefore, we introduce another mathematical model to disaggregate the result of the aggregate level, i.e. to determine the starting and finishing times of all outages given to the firms. Before presenting the disaggregate model, it is worthwhile to consider Figure 2 as the graph of supply and demand plotted against time. To see the difference between this graph and D L D C (Figure 1), one must note that in this graph, time is the real or clock time. In fact, in Figure 2 the shortage (or surplus) of power is displayed for any moment during a day. The major obstacle in outage scheduling is the lack of flexibility in matching the saved energy (due to the outages) with the shortages, at any time. This is illustrated in Figure 3. Every day, the shortage of power starts at time tl and lasts up to t s. The power company has to cut the power of some firms off during that interval. Suppose that the outage is assigned to just one firm, say firm i, whose power consumption is A i megawatts. Then, in that interval the demand curve shifts down, as much as Ai. In Figure 3, D.C.(l) represents the total demand for all firms, while D.C.(2) represents the total demand of all firms excluding firm i in the shortage interval. AA', BB', CC' or D D ' are equivalent to A~, the power consumption for firm i. As can be seen at time t 2, the original shortage is N N ' megawatts, but we have cut A, megawatts off, which is more than N N ' . This means we have closed a firm down and saved energy more than we needed. On the other hand, it is not possible to gradually decrease the firm's demand for power in order to make it fit to the shortage pattern of any moment. Therefore, in the intervals tl - t 3 as well a s t 4 - t 5 , the saving has been more than necessary. The areas of triangles AA'B and C D ' D represent the extra amount of energy which is saved due to the outage assigned to firm i. Based on the above mentioned obstacle, the objective function of the disaggregate model is to minimize the total amount of energy which has been saved unnecessarily. Recalling from Section 4, some firms can have a limited allowed outage duration, while the others do not have such a limit. From Lemma 1, any firm of the second type c a n only have one outage during the shortage season. Furthermore, the length of the outage is determined by the aggregate model. Therefore, the only thing which must be determined at the disaggregate level is the starting time of the outage. For the firms with a limited allowed outage duration, the model should determine the dates and the starting times for the outages. From Lemma 1, it is known that the length of each outage is exactly equivalent to the allowed limit. In this model, we assume that the hourly supply and demand for a shortage season is known. For simplicity, we assume there is only one Daily Load Duration Curve pattern in the shortage season.
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M. Modarres / Poweroutageplanning
"o
Supply curve
g
I tl •
t2
t3
I
L
t4
t5
24
Figure 3. Effectof an outage on the demand curve However, there is no loss of generality if there are more than one pattern. In that case, some minor modifications will be necessary. The following notation is used in this model. : Number of days in the shortage season. : Number of hours per day during which outage planning must be performed. : Number of firms to consider in the region. Set of the firms with limited allowed outage duration. G: T : Number of hours in a shortage season, during which demand exceeds supply. L i : Upper bound for the allowed outage duration of firm i. : The total length of the outage assigned to firm i. Wjk : Amount of unnecessary saving of energy in terms of megawatts that exceeds te difference between the demand and the supply, during the j-th hour of the k-th day of the season. : Starting time of the outage for the firm i (measured from the beginning of the season, considering the shortage times only) if the firm has no outage duration limit. : Starting time of the outage for firm i on day k if the firm has limited outage duration. ~ijk " Z e r o - o n e decision variable which is 1 when the power of firm i is off during the j-th hour of the k-th day. Bjk " Amount of power shortage in the j-th hour of the k-th day of the season which is to be compensated by the industries. O
t N
6.1. The disaggregate model (P6)
minimize
Y'- E Wjk k j
(28)
subject to
EA,v
j,-Bj,,
j=l
..... t,
k = 1 . . . . . D,
(29)
i
}-". Y'~ V,.jk >~ ~ , k
j
i = 1 . . . . . N,
(30)
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262
Vijk < ~ l - [ ( U ~ - k t + t - j ) / T
],
Vijk<~l-[(kt-t+j-Ui-Yi)/T
V~jk<~l-[(j-U~k-Li)/t], U,.<~T,
iEFu,
j=l
j=l
j = l .... ,t,
iEFu,
j=l
. . . . . t,
0 or 1,
k = l . . . . . D,
k = l ..... D,
. . . . . t,
k = l . . . . . D,
k - - 1 . . . . . D,
(31) (32) (33) (34)
(36)
. . . . . t,
(37)
i ~ F~,
U,k>~0: integers, =
i ~ F u,
k = l . . . . . D,
(35)
U, >/0: integers,
Vij k
j = l ..... t,
if~Fu,
U~k<~t+l, Wjk>~0,
],
i~Fu,
Viik ~ < l - [ ( U , k - j ) / t ] ,
iq~Fu,
i ~ F u,
i = 1 . . . . . N,
(38) k = l ..... D, j = 1 . . . . . t,
(39) k = 1 . . . . . D.
(40)
The constraints (31) and (32) guarantee that the firm i, with no limited outage duration, will be out of power only from the moment U~ for as long as Y~ hours, i.e. the value of V,jk is 1 only when the current time is between U, (the starting time of the outage) and U~+ Y~ (the finishing time of the outage). In order to investigate this claim, we can consider three different cases: (a) When the current time is before the starting time of the outage, i.e. (k - 1)t + j < U,; then from (31), V~jk < 1. On the other hand, since V,jk is a binary variable, it has to be 0. Therefore, no outage will be given to firm i before U,. (b) When the current time is after the finishing time of the outage, i.e. (k - 1)t + j > U, + Y~; then from (32), Viik < 1, or we conclude that the value of this variable is 0. (c) When the current time is between the starting and the finishing time of the outage, i.e. U, < (k - 1)t + j < U/+ Y/; then from (31) and (32) we can conclude that V,7k can be either 0 or 1. However, since the length of the outage must be at least Y/ hours (from the constraint (30)), the value of this variable must be 1. The constraints (33) and (34) do the same thing, but for the firms with the limited allowed outage duration.
7. A heuristic algorithm for the disaggregate model Since the application of the disaggregate model usually leads to a very large mixed integer programming, neither the branch and bound nor any of the decomposition methods can be used to find a solution. Instead, in this section we present a heuristic algorithm to solve this problem. As mentioned in Section 6, the disaggregate model tries to fit the demand of the firm which is planned to close down to the shortage pattern. Therefore, a firm with higher consumption and longer outage duration is more difficult to fit in this pattern. Based on this obstacle, the algorithm is developed. Through the following theorem, we will show that the best result can be obtained when the total amount of energy which has been saved due to the closing down of the industrial firms is the same for every day of the shortage season. A larger deviation from the uniform outage planning will result in a worse objective function of the disaggregate model. Clearly, it is almost impossible to have a schedule with uniform daily power shortages. However, there is a great flexibility in power scheduling for the residential and commercial sections and it is possible to cut any amount of power at any time. The reason is that the city can be divided in different regions in many ways, and a short outage duration can be conveniently assigned for these regions. Therefore, the power scheduling for these areas can be used as a fine tune up to get a near optimal solution.
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7.1. Assumptions of the algorithm (1) The real time demand curve (Figure 2) does not change during the shortage season, however, it may shift up or down. In a shortage season, there usually is no major fluctuations in the demand of industrial firms. The only possible change in the demand curve is due to the power consumption of the cooling (or heating) system which is a function of the outside temperatures and causes the curve to shift up or down. Therefore, this assumption is quite justified. (On holidays the structure of demand is quite different, but since there usually is no power shortage on these days the outage planning is not necessary.) (2) The length of planned power outage for any firm is a multiple of days and not a fraction of it. These assumptions hold for any firm not belonging to the set of F u firms, as defined in Section 6, because it is neither practical nor economical for a firm to operate only a few hours a day. These assumptions, however, do not hold for the firms belonging to F u. Such firms will be discussed later. Theorem 2. Under the above assumptions, (a) comparing any two feasible solutions of the disaggregate model, the one with smaller value of the
following term is a better solution: D
E (Oj) 2' j=l
(41)
where Oj is the total amount of planned power outage on the j-th day; (b) if the daily planned power outages is identical, then it is the optimal solution. Proof. Suppose the total saving on the i-th day of the season is equal to Oj; then referring to Figure 3, the demand curve shifts down as much as AA' or CC', both of which are equal to Oj. Based on the definition of the disaggregate model of Section 6, the objective function is equal to the amount of unnecessary saving due to planned outages, or the area of BAA' and C D D ' . (In some cases where there exists more than two triangles, the proof is still valid.) It can be assumed with good acceptable accurance that B A N and C D D ' 1 PB + CD'). Then, from the property of triangles are triangles, so the sum of their area is equal to 7Oj(A and also from the first assumption that the daily demand curve is the same, it can be concluded, regardless of the amount of Oj, that A'B/AA' = r
and
C D ' / D D ' = r'.
Therefore, the objective function related to the j-th day is equal to ½(r + r ' ) times squre of Oj, and the total objective function for the shortage season is as follows:
D ½ ( r + r ' ) Y'~ (Oj) 2.
(42)
j=l Since ½(r + r ' ) is a constant, the proof of the first part of the theorem is complete. The second part of the theorem is proved by resorting to the fact that if the sum of finite number of variables is constant, then the sum of their squres will attain its minimum when the variables are equal. []
7.2. Outline of the algorithm First, we divide the firms into two different groups. The first group includes all the firms without and the second group consists of the ones with the limited outage duration. The next step is to rearrange the firms within each group. In the first group, the firms are ranked in descending order of their power consumption, i.e. A 1 > A 2 > A 3 > • • •. In the second group, the firms are rearranged by the length of their single outage, i.e. by L i as defined in Section 6. Therefore, in this group
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the firm number 1 has the longest single outage duration. In case of ties, the firms with longer outage duration, and in the second group the firm with the greater A i will be picked first. The algorithm consists of two stages. In the first stage, the outage scheduling of the firms belonging to the first group will be determined. In this stage, the firms will be scheduled one by one, by their ranks, as defined above. In the second stage the same procedure will be done for the firms of the second group. Any firm that must be scheduled in the first stage has only one planned outage in the shortage season which lasts Y, hours or Yi/t days. We round off Y~/t to the closest integer and call it D~. Therefore, D~ is the outage length of firm i in terms of days. We start with the first firm of the first group. This firm will be assigned an outage which starts on the first day and will last for D a days. Meanwhile, we set Oj equal to A 1, for j = 1 . . . . . D 1. Then, on the next day, the power of the second firm will be cut off for D 2 days, and for this period again, we set Oj = A 2. This process will be repeated for firm number 3, then 4 and so on, respectively, until the remaining days of the season is less than the required outage length of the next firm. Then, the remaining firms will be scheduled in the best possible intervals where the total outage distribution along the season has less deviation from the ideal leveled demand reduction, as stated in Theorem 2. In general, once k - 1 firms are scheduled, at the k-th iteration, the k-th firm should be scheduled, i.e. the starting day of this firm's outage should be determined. Since the outage length of this firm (D k days) has already been determined by the aggregate model, the finishing day of the outage assigned to this firm can be calculated easily. Now suppose that the starting day of the outage assigned to the k-th firm coincides with the i-th day. Hence, the current amount of Oj will be increased as much as A k megawatts, for j -- i, i + 1 . . . . . i + D k. By Theorem 2, to get the best result, we should determine the starting day of the outage, say i*, that minimizes the following term, for i = 1 . . . . . ( D - D k + 1): i+Dk--1 E ( O t + A k ) 2" l=i
(43)
This can be done easily by inspection. Once a firm's outage is scheduled, the value of Oj for the corresponding interval will be updated. Once the scheduling for all firms of the first group is completed, the shortage pattern for the whole season will be modified. By subtracting the value of final Oj from the original Bjk (hourly shortage, as defined in Section 6), its current value will be obtained, for j = 1 . . . . . t and k = 1 . . . . . D. A table will be set up, in which updated data, including the value of Bjk for each hour of the season as well as the shortage length and maximum daily Bjk are displayed. In the second stage, the outage scheduling for the firms belonging to F u will be carried on. The number of outages assigned to each firm can be more than one, but the length of each outage will not exceed t (the shortage length of each day). In the second stage, the first firm of F u will be picked, first. This firm must have as many as )(1 outages, each of which with a duration of L 1. This firm will be given outages on the )(1 days of the season with the highest Bjk, provided the shortage length of such days is at least equal to L v Since in the aggregate model, only one set up cost per day has been considered, the outages of this firm should be scheduled either at the beginning or at the end of a working day, but not in the middle. Then the shortage pattern as well as the above mentioned table will be modified accordingly. This procedure will go on for the second, third, . . . firms until all the firms have been scheduled.
7.3. Implementing the algorithm A computer program is written to find the best interval for each firm's outage duration and to keep track of the value of Oj for each day in the first stage, and also to update the value of Bjk for each hour of the season after the first and the second stage. This program determines the days on which a firm should have an outage in the second stage of the algorithm. A real example with seven firms was solved by hand in order to make sure that the algorithm can be implemented even without using a computer. This makes
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the algorithm more applicable in the cities where using the computer is difficult for many reasons, including the lack of computer experts. In addition, since the people in charge can understand and implement it easily, no resistance against it can be expected. In larger areas, where computers services are available, it is not wise to schedule by hand. After finishing outage scheduling for all industrial firms and the plotting the new shortage patterns, one can see that the modified shortage patterns for different days are no longer similar. However, this is not a worrisome situation because later on, as mentioned before, one can compensate the extra shortage by the outages allocated to the residentials, commercials and the other sectors. The planning in these sectors, contrary to the industrial sector is quite flexible, as described before. In these sectors, it is possible to cut almost any amount of power at any time.
8. Summa~ and conclusion In this paper, the issue of electric power shortage in the developing countries as well as the method of power outage planning, in order to minimize the cost of power shortage, was discussed. The importance of the problem arises from the fact that the cost of production interruption due to the power shortage is significant. Two mathematical models have been developed to determine the outage length assigned to each firm, and to schedule the power outage for the firms during the season. Based on the results of the mathematical models, a policy has been suggested which is simple and practical and is quite different from what is now applied in Iran. The number of outages assigned to each firm should be as few as possible, usually once per season, although its duration may be the total outage length assigned to that firm. An algorithm has been developed to schedule outage intervals for the firms. The algorithm is simple and comprehensive, especially for the managers in small towns or remote areas in which the computer facilities are not at hand easily. The algorithm can even be implemented by hand or by a programmable calculator. However, it was shown that the algorithm can produce a solution which is not mathematically far from the solution of the model.
Acknowledgement It is a great pleasure to thank Professor Bruce L. Miller of the University of California, Los Angeles, and Professor Hashem Mahlooji of Sharif University of Technology for reviewing this paper and for their valuable comments.
References Central Bank of Iran (1984), "Annual economic report of Islamic Republic of Iran", Tehran. Modarres, M. (1986), "A general framework of power planning when demand exceeds supply", Journal of Barname Va Towse'e (in Persian) Tehran. Office of Power Planning, Ministry of Energy (1985), "Statistics of Iranian electric power", Tehran. Munasinghe, M. (1979), The Economics of Power System Reliability and Planning, The Johns Hopkins University Press, Baltimore, MD.
European Journal of Operational Research 49 (1990) 254-265 North-Holland
Power outage planning Mohammad Modarres Department of Industrial Engineering, Sharif University of Technology, Tehran, Iran Abstract: This paper is concerned with scheduling the planned electric power outage in a region where the expected demand for power exceeds the expected supply, during some interval of the year. To minimize the total economic loss of the power shortage, the best allocation of power outage among the consumers is determined. A hierarchical planning system is proposed to find the optimal solution with two levels of aggregation. The first step in this system is to determine the outage duration for each consumer in the shortage season, in order to minimize the total economic loss. In the second step, a disaggregate model determines when, and for how long, the power of a consumer will be cut off during a shortage season. Linear and mixed integer programming are used, respectively, to develop the aggregate as well as the disaggregate models. A heuristic algorithm is developed to solve the problem, practically. We also discuss some results of the model. Keywords: Developing countries, electric power, application of mathematical modelling, hierarchical planning system
1. Introduction The economy of many developing countries suffers severe damage, due to the shortage of the electric power. Not only is the industrial sector heavily dependent upon the reliable flow of the electricity, but even the agricultural sector, which still applies the traditional methods, needs the electric power for a variety of reasons. In these countries, the consumption of electricity is growing so fast, that supply falls short of satisfying the demand. Take Iran as a case. Although the annual growth rate of the installed capacity during the 1971-1981 period (12 percent) has passed the annual growth rate of per capita G N P (4.8 percent) as well as the population growth rate (3 percent), the shortage of electric power has been steadily growing. To justify such a phenomenon, one should cite the occurrence of the drastic changes in the life style in these countries, as well as the substitution of 'man' by 'machines'. There are many reasons to believe that the supply of the required amount of the electric power, in the short term, is difficult or even impossible. Power generation, transmission, and distribution, in general, is quite expensive. In fact, the main bulk of the required investment is in terms of foreign currencies, which the developing countries can not provide, easily. Even if they can provide the money to invest, it takes between five to ten years to design, purchase and install the equipment. Therefore, the power shortage is considered a fact of life in many developing countries. When the demand in the peak hours exceeds the supply, the power company has to cut the electricity of some consumers off. As a result, the consumers, especially those in the industrial sector experience considerable monetary loss. This paper introduces the problem of power shortages and determines the best way to distribute the planned outages among the consumers, to have the minimum loss. Clearly, the planning should cover all the sectors. However, in this paper, only the outage planning for the industrial firms will be considered. After the industrial outage planning, part of the power shortage will be compensated. Then, the outage Received March 1990 0377-2217/90/$03.50 © 1990 - ElsevierSciencePublishers B.V. (North-Holland)
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planning for the other sectors will be done, which is beyond the scope of this paper. However, it can be performed, using the same framework introduced for the industrial sector. The proposed mathematical model considers the outage planning at two different levels, aggregate and disaggregate. At the aggregate level, the model determines the length of the outage that should be assigned to each firm, while the detailed outage schedufing of each firm will be done at the disaggregate level. In the literature, not much can be found on this subject, since the developed countries do not face a problem of this nature. The reliability of the power supply in these countries is so high they almost do not have any power outage. A common measure of power reliability, called LOLP (Loss Of Load Probability), can illustrate the extent of power shortage in a country or a region. While LOLP in the USA is less than one day per ten years, many developing countries have power outages everyday, or at least in an interval of the year. Section 2 discusses the nature and classification of outage costs and establishes the fact that some costs depends on the outage duration, while some other costs vary as a function of the number of outages. In Section 3, the method of illustrating the supply and demand diagram will be discussed. In Section 4, the aggregate model, as well as the mathematical foundations of the properties of the optimal solutions will be presented. Section 5 contains some results from the application of the model. In Section 6, another mathematical model to disaggregate the results of the aggregate model and finally in Section 7 a heuristic algorithm to solve this model will be introduced.
2. Classification of outage costs Whenever the supply of electricity is less than the demand, the consumers lose in different ways. These economic losses are called 'outage costs'. A number of papers have investigated the nature of the outage costs and how to classify them. Many studies, in different countries have designed the suitable methodology to estimate the outage costs of the various sectors and have got some interesting results. To review the literature, see for example, Munasinghe (1979). Outage costs, in general, can be classified into two categories, direct and indirect costs. Any economic loss caused by a particular outage, say the production interruption, is called a direct cost. Indirect costs are not the results of any specific outage, but are consisting of all those losses a consumer affords, in order to decrease the expected costs when an outage occurs. One example of indirect costs is the investment and operation costs of a standby generator. A consumer will incur some indirect cost, when and only when he can realize some sort of savings in his direct costs. Therefore, provided the decision to pay extra indirect costs is reasonable, the sum of both types of costs will be less than the direct costs alone, when nothing is done to decrease the loss due to the outage. Based on this fact, in our model, only the direct cost is considered. We assume indirect costs can not be counted on, because they are the results of the individuals' actions to minimize their losses and following no prefabricated patterns. The direct costs also can be classified into two types. The first type, roughly, is a function of the number of the outages and independent of the outage length, especially, when the duration is not too short and exceeds a certain limit. It is clear that this limit depends on the nature of the product, manufacturing process and some other factors. This kind of cost can be thought of a set up cost. The second type, on the other hand, is a continuous function of the outage duration. This type of cost is caused by the production interruption, overtime payments to compensate the production losses and the cost of spoilage of the materials and products. This kind of cost, C(t), is not necessarily a linear function of the outage duration, t. However, in many manufacturing firms, it can be assumed that this kind of cost is the linear function of outage duration, t, if it is not too short. In our model, we assume the continuous function costs are linear and the set up costs are constant, regardless of how long the power is off. These assumptions can be justified, because a particular planned outages duration can not be very short, or the number of outages very large.
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3. Estimation of the power shortage Each year, the power shortage appears in specific months, which we call 'the shortage season'. For example, in Iran, the maximum shortage occurs in summers, especially June and July, when the demand for electricity increases rapidly, mostly because of high power consumption of the cooling systems. Since the outage planning is for the short term, usually, for one season only, the demand can be forecasted relatively precisely. The supply and demand for electricity can be shown by a graph, called Daily Load Duration Curve (DLDC). The Annual Load Duration Curve is a popular tool in the power generation planning. In our model, we are going to take advantage of the same idea, but for every day. In Figure 1, t represents the length of time of a day (in terms of hours) during which, the demand is at least D(t). We assume the supply level during a day is constant. As can be seen, during h hours (out of 24 hours), the demand exceeds the supply and expriencing some outages would be unavoidable. The area under the demand and over the supply curve, say E, represents the total daily shortage, in terms of megawatt hours. To compensate this shortage, the outage should be distributed among the industrial, residential, commercial and the other sectors. Since we are considering the industries only, a fraction of the total shortage, say Fr, should be distributed among the industrial firms. It is better to consider this fraction lower than the real fraction of the total energy consumed by those firms. This is so to decrease the economic and social losses of the industrial shutdowns. Although the patterns of DLDC in general can be changed on a daily basis, it can be assumed that in a shortage season there are only two patterns, working day and holiday patterns.
4. The aggregate outage planning models The aggregate level model determines how long, in a shortage season, the power of a firm should be cut off. Through analyzing the results of the disaggregate model, a detailed solution will be derived. However, as will be seen, even the result of the aggregate model alone provides a framework for decision making. In many regions, the planning can be done based on the results of the aggregate level model only.
Dmax
E
Supply curve
I
I I
g O3
I Dmin
I-
o
Demand curve
-+
t I I
I I I
I
I
t
h
Figure 1. Daily load duration curve
24
M. Modarres / Power outage planning
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In this section, we start with the presentation of the aggregate models with two different criteria. Then we show that these models can be converted into linear programming problems. The firms can be categorized into two different groups. The first consists of those firms which can not afford to be closed for a long time. For example, a food producer can not close more than three hours per day. The second group consists of the firms with no outage duration upper limit. The model considers all the industrial firms in a region. We use the following notation in the model: N Fu St Ct At T~
: : : : : :
T : Lt : : Xt :
Number of firms to consider in the region. Set of the firms with limited allowed outage duration. Total cost for firm i, of the first type (set up cost) per outage. Total cost of the second type for firm i, per hour. Megawatts used by firm i (energy consumption per hour). Total shortage of energy (megawatt hours) in the shortage season which should be distributed among the industrial firms, or the total season shortages times Fr (the fraction of the total shortage assigned to the industries, as defined previousely). Number of hours in a shortage season, during which demand exceeds supply. Upper bound for the allowed outage duration of firm i. The total length of the outage assigned to firm i. Number of outages assigned to firm i.
4.1. The first aggregate model (PI)
minimize
f ( x , y) = E[S~X, + CiY~]
(1)
i
subject to
~ A i Y ~> T~,
(2)
i
Yi-LiX~.<<,O, i~Fu,
(3)
Y~
(4)
i = 1 . . . . . N,
k~i
0< ~
i = l . . . . . N,
X,> 0: integers,
i = 1 . . . . . N.
(5) (6)
Constraint (4) compares the outages length of each firm with the others, where q >/1. It prevents the big gaps between the outage lengths of the firms. In a special case, where q---1, all firms will have equal outage lengths.
4.2. The second aggregate model (P2) Substituting the constraint (4) with the following one will result in the second model:
Y ~ < ~ [ q / ( N - 1 ) ] E X k,
q>~l,
i = 1 . . . . . N.
(7)
k¢i
In this model, the outage length of any firm is compared with the average of the others (excluding its own).
The following lemma and theorem provide the properties of the optimal solutions of the first and second models. Since the lemma is trivial, the proof has been omitted.
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Lemma 1. In both models the optimal values of the integer decision variables Xi, i = 1 . . . . . N, are as follows: Xi=
1 [Yii/Li]
ifi~Fu, otherwise,
(8)
where, for any real value a, we define [a] as an integer value, such that
(9)
[a]-l
Using Lemma 1, the integer decision variables X,, i = 1,..., N, can be omitted from the model. Therefore, the first part of the objective function as well as the constraints (3) and (6) will be deleted. Now, we see that the first and second aggregate models have been changed into the following models of (P3) and (P4), respectively. 4.3. Problem (P3): A new version of the first aggregate model
minimize
(10)
g ( y ) = E C1Yi i
subject to
E A i Y i ~ re,
(11)
i
Y~
i = l ..... N,
(12)
i=l,...,N,
(13)
O<~ Y i < T ,
where Wi=maxYk, k=~i
i = 1 ..... N.
(14)
4.4. Problem (P4)." A new version of the second aggregate model
This model is the same as (P3), except that the constraint (12) should be substituted by the following constraint: Y ~ < . [ q / ( N - 1 ) ] ~] Yk,
i = l . . . . . N.
(15)
Problem (P4) is a linear programming problem and can be solved by the simplex method easily (especially by the upper bound variation). To solve problem (P3), the result of Theorem 1 will be used. Before stating this theorem, we assume that the following relation holds: [C,/AI] <~[C2/A2] <~ . . . <<.[CN/AN].
(16)
By rearranging the decision variables, the relation (16) is always true. Theorem 1. Assuming the relation (16) is satisfied, then for any q >1 1, problem (P3) is equivalent to the following linear programming problem (P5): (P5)
minimize
(17)
g(y)=ECiY~ i
subject to
(18)
Y~AiYi >1 T~, i
(19)
Y1 <~qY2, O<~Y,~T,
i=l,...,N.
(20)
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259
Proof. Suppose that the optimal solution of (P5) consists of the vector
.....
L]
(21)
We claim that the following relation holds: Y1 >/Y2 >/ "'" >/ YN-
(22)
If this is not true, then for example, Y~ < Y2, while,
[Cl/A1] [C2/A2].
(23)
Now, consider a new solution Y'=[YI+u,
Y z - ( A 1 / A 2 ) u,
Y3 . . . . .
YN],
(24)
where
0 <~u <~n u n { ( a 2 / A i ) Y 2, T } .
(25)
It is clear that Y' is a feasible solution and also that g ( Y ' ) < g(Y). This contradicts the optimality of Y. Thus, (22) must hold.The proof will be complete if we show that Y is also an optimal solution of (P3), or if we show that constraint (12) holds. From (14) and (22), we have W1 = Y2, IV,,= Y1,
(26) i = 2 . . . . . N.
(27)
From (26), constraint (12) holds for i = 1, and from (22) and (27), it also holds for i >t 2; thus the proof is complete. []
5. Some results of the aggregate model An important result of the aggregate model comes from Lemma 1. That is, the number of outages assigned to a firm should be as small as possible. In other words, according to the properties of the optimal solution, if the power of a firm should be off for Y~ hours, then two cases may occur. When there is no upper limit for the length of the outage, then turn the power of this firm off just once and for ~ hours, in the shortage season. However, if for some technical reasons, a firm can not afford to be closed for more than L i hours, then each outage length of this firm should be exactly L i hours, in order to minimize the number of shortages. The above policy is different from what is applied in lran, these days. In the shortage season, the power company cuts the total power of an area off for about one to two hours a day. Adopting our suggested policy not only does bring the total losses down, but the firm can take some preventive steps to decrease its losses. For example, if a firm can not have power for two weeks, then it can either close the plant in that period and consider it as the annual vacation or operate the plant partially, using a standby generator, and make annual overhaul maintenance. Before the shortage season, the power company can determine the length of the outage assigned to all firms and hold discussion with them to come up with an agreement about the timing of the outage. One result of this agreement is the possibility of changing the working hours of a firm, for example shifting from day-hours to night-hours. The model also determines the shadow costs, a basis for the discount rate that the power company can consider for the firms that are willing to change their working hours.
6. The disaggregate model The aggregate model determines how long a firm must be without electric power during the shortage season. It can not be used, however, to specify the cut off time for each firm's power. If there are only a
M. Modarres /Power outage planning
260
Dmax
C
I I
>.
I I
I
Dmin
I I I
24 Figure 2. Real time supply and d e m a n d curve
few firms in a region, the outage scheduling can be performed on the basis of trial and error in such a way that at any time, the total power consumption of shut down firms approximately equals the fraction of the power shortage which has been assigned to the industrial firms. However, if there are too many firms in a region, a supporting system for outage scheduling will be needed. Therefore, we introduce another mathematical model to disaggregate the result of the aggregate level, i.e. to determine the starting and finishing times of all outages given to the firms. Before presenting the disaggregate model, it is worthwhile to consider Figure 2 as the graph of supply and demand plotted against time. To see the difference between this graph and D L D C (Figure 1), one must note that in this graph, time is the real or clock time. In fact, in Figure 2 the shortage (or surplus) of power is displayed for any moment during a day. The major obstacle in outage scheduling is the lack of flexibility in matching the saved energy (due to the outages) with the shortages, at any time. This is illustrated in Figure 3. Every day, the shortage of power starts at time tl and lasts up to t s. The power company has to cut the power of some firms off during that interval. Suppose that the outage is assigned to just one firm, say firm i, whose power consumption is A i megawatts. Then, in that interval the demand curve shifts down, as much as Ai. In Figure 3, D.C.(l) represents the total demand for all firms, while D.C.(2) represents the total demand of all firms excluding firm i in the shortage interval. AA', BB', CC' or D D ' are equivalent to A~, the power consumption for firm i. As can be seen at time t 2, the original shortage is N N ' megawatts, but we have cut A, megawatts off, which is more than N N ' . This means we have closed a firm down and saved energy more than we needed. On the other hand, it is not possible to gradually decrease the firm's demand for power in order to make it fit to the shortage pattern of any moment. Therefore, in the intervals tl - t 3 as well a s t 4 - t 5 , the saving has been more than necessary. The areas of triangles AA'B and C D ' D represent the extra amount of energy which is saved due to the outage assigned to firm i. Based on the above mentioned obstacle, the objective function of the disaggregate model is to minimize the total amount of energy which has been saved unnecessarily. Recalling from Section 4, some firms can have a limited allowed outage duration, while the others do not have such a limit. From Lemma 1, any firm of the second type c a n only have one outage during the shortage season. Furthermore, the length of the outage is determined by the aggregate model. Therefore, the only thing which must be determined at the disaggregate level is the starting time of the outage. For the firms with a limited allowed outage duration, the model should determine the dates and the starting times for the outages. From Lemma 1, it is known that the length of each outage is exactly equivalent to the allowed limit. In this model, we assume that the hourly supply and demand for a shortage season is known. For simplicity, we assume there is only one Daily Load Duration Curve pattern in the shortage season.
261
M. Modarres / Poweroutageplanning
"o
Supply curve
g
I tl •
t2
t3
I
L
t4
t5
24
Figure 3. Effectof an outage on the demand curve However, there is no loss of generality if there are more than one pattern. In that case, some minor modifications will be necessary. The following notation is used in this model. : Number of days in the shortage season. : Number of hours per day during which outage planning must be performed. : Number of firms to consider in the region. Set of the firms with limited allowed outage duration. G: T : Number of hours in a shortage season, during which demand exceeds supply. L i : Upper bound for the allowed outage duration of firm i. : The total length of the outage assigned to firm i. Wjk : Amount of unnecessary saving of energy in terms of megawatts that exceeds te difference between the demand and the supply, during the j-th hour of the k-th day of the season. : Starting time of the outage for the firm i (measured from the beginning of the season, considering the shortage times only) if the firm has no outage duration limit. : Starting time of the outage for firm i on day k if the firm has limited outage duration. ~ijk " Z e r o - o n e decision variable which is 1 when the power of firm i is off during the j-th hour of the k-th day. Bjk " Amount of power shortage in the j-th hour of the k-th day of the season which is to be compensated by the industries. O
t N
6.1. The disaggregate model (P6)
minimize
Y'- E Wjk k j
(28)
subject to
EA,v
j,-Bj,,
j=l
..... t,
k = 1 . . . . . D,
(29)
i
}-". Y'~ V,.jk >~ ~ , k
j
i = 1 . . . . . N,
(30)
M. Modarres / Power outage planning
262
Vijk < ~ l - [ ( U ~ - k t + t - j ) / T
],
Vijk<~l-[(kt-t+j-Ui-Yi)/T
V~jk<~l-[(j-U~k-Li)/t], U,.<~T,
iEFu,
j=l
j=l
j = l .... ,t,
iEFu,
j=l
. . . . . t,
0 or 1,
k = l . . . . . D,
k = l ..... D,
. . . . . t,
k = l . . . . . D,
k - - 1 . . . . . D,
(31) (32) (33) (34)
(36)
. . . . . t,
(37)
i ~ F~,
U,k>~0: integers, =
i ~ F u,
k = l . . . . . D,
(35)
U, >/0: integers,
Vij k
j = l ..... t,
if~Fu,
U~k<~t+l, Wjk>~0,
],
i~Fu,
Viik ~ < l - [ ( U , k - j ) / t ] ,
iq~Fu,
i ~ F u,
i = 1 . . . . . N,
(38) k = l ..... D, j = 1 . . . . . t,
(39) k = 1 . . . . . D.
(40)
The constraints (31) and (32) guarantee that the firm i, with no limited outage duration, will be out of power only from the moment U~ for as long as Y~ hours, i.e. the value of V,jk is 1 only when the current time is between U, (the starting time of the outage) and U~+ Y~ (the finishing time of the outage). In order to investigate this claim, we can consider three different cases: (a) When the current time is before the starting time of the outage, i.e. (k - 1)t + j < U,; then from (31), V~jk < 1. On the other hand, since V,jk is a binary variable, it has to be 0. Therefore, no outage will be given to firm i before U,. (b) When the current time is after the finishing time of the outage, i.e. (k - 1)t + j > U, + Y~; then from (32), Viik < 1, or we conclude that the value of this variable is 0. (c) When the current time is between the starting and the finishing time of the outage, i.e. U, < (k - 1)t + j < U/+ Y/; then from (31) and (32) we can conclude that V,7k can be either 0 or 1. However, since the length of the outage must be at least Y/ hours (from the constraint (30)), the value of this variable must be 1. The constraints (33) and (34) do the same thing, but for the firms with the limited allowed outage duration.
7. A heuristic algorithm for the disaggregate model Since the application of the disaggregate model usually leads to a very large mixed integer programming, neither the branch and bound nor any of the decomposition methods can be used to find a solution. Instead, in this section we present a heuristic algorithm to solve this problem. As mentioned in Section 6, the disaggregate model tries to fit the demand of the firm which is planned to close down to the shortage pattern. Therefore, a firm with higher consumption and longer outage duration is more difficult to fit in this pattern. Based on this obstacle, the algorithm is developed. Through the following theorem, we will show that the best result can be obtained when the total amount of energy which has been saved due to the closing down of the industrial firms is the same for every day of the shortage season. A larger deviation from the uniform outage planning will result in a worse objective function of the disaggregate model. Clearly, it is almost impossible to have a schedule with uniform daily power shortages. However, there is a great flexibility in power scheduling for the residential and commercial sections and it is possible to cut any amount of power at any time. The reason is that the city can be divided in different regions in many ways, and a short outage duration can be conveniently assigned for these regions. Therefore, the power scheduling for these areas can be used as a fine tune up to get a near optimal solution.
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7.1. Assumptions of the algorithm (1) The real time demand curve (Figure 2) does not change during the shortage season, however, it may shift up or down. In a shortage season, there usually is no major fluctuations in the demand of industrial firms. The only possible change in the demand curve is due to the power consumption of the cooling (or heating) system which is a function of the outside temperatures and causes the curve to shift up or down. Therefore, this assumption is quite justified. (On holidays the structure of demand is quite different, but since there usually is no power shortage on these days the outage planning is not necessary.) (2) The length of planned power outage for any firm is a multiple of days and not a fraction of it. These assumptions hold for any firm not belonging to the set of F u firms, as defined in Section 6, because it is neither practical nor economical for a firm to operate only a few hours a day. These assumptions, however, do not hold for the firms belonging to F u. Such firms will be discussed later. Theorem 2. Under the above assumptions, (a) comparing any two feasible solutions of the disaggregate model, the one with smaller value of the
following term is a better solution: D
E (Oj) 2' j=l
(41)
where Oj is the total amount of planned power outage on the j-th day; (b) if the daily planned power outages is identical, then it is the optimal solution. Proof. Suppose the total saving on the i-th day of the season is equal to Oj; then referring to Figure 3, the demand curve shifts down as much as AA' or CC', both of which are equal to Oj. Based on the definition of the disaggregate model of Section 6, the objective function is equal to the amount of unnecessary saving due to planned outages, or the area of BAA' and C D D ' . (In some cases where there exists more than two triangles, the proof is still valid.) It can be assumed with good acceptable accurance that B A N and C D D ' 1 PB + CD'). Then, from the property of triangles are triangles, so the sum of their area is equal to 7Oj(A and also from the first assumption that the daily demand curve is the same, it can be concluded, regardless of the amount of Oj, that A'B/AA' = r
and
C D ' / D D ' = r'.
Therefore, the objective function related to the j-th day is equal to ½(r + r ' ) times squre of Oj, and the total objective function for the shortage season is as follows:
D ½ ( r + r ' ) Y'~ (Oj) 2.
(42)
j=l Since ½(r + r ' ) is a constant, the proof of the first part of the theorem is complete. The second part of the theorem is proved by resorting to the fact that if the sum of finite number of variables is constant, then the sum of their squres will attain its minimum when the variables are equal. []
7.2. Outline of the algorithm First, we divide the firms into two different groups. The first group includes all the firms without and the second group consists of the ones with the limited outage duration. The next step is to rearrange the firms within each group. In the first group, the firms are ranked in descending order of their power consumption, i.e. A 1 > A 2 > A 3 > • • •. In the second group, the firms are rearranged by the length of their single outage, i.e. by L i as defined in Section 6. Therefore, in this group
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the firm number 1 has the longest single outage duration. In case of ties, the firms with longer outage duration, and in the second group the firm with the greater A i will be picked first. The algorithm consists of two stages. In the first stage, the outage scheduling of the firms belonging to the first group will be determined. In this stage, the firms will be scheduled one by one, by their ranks, as defined above. In the second stage the same procedure will be done for the firms of the second group. Any firm that must be scheduled in the first stage has only one planned outage in the shortage season which lasts Y, hours or Yi/t days. We round off Y~/t to the closest integer and call it D~. Therefore, D~ is the outage length of firm i in terms of days. We start with the first firm of the first group. This firm will be assigned an outage which starts on the first day and will last for D a days. Meanwhile, we set Oj equal to A 1, for j = 1 . . . . . D 1. Then, on the next day, the power of the second firm will be cut off for D 2 days, and for this period again, we set Oj = A 2. This process will be repeated for firm number 3, then 4 and so on, respectively, until the remaining days of the season is less than the required outage length of the next firm. Then, the remaining firms will be scheduled in the best possible intervals where the total outage distribution along the season has less deviation from the ideal leveled demand reduction, as stated in Theorem 2. In general, once k - 1 firms are scheduled, at the k-th iteration, the k-th firm should be scheduled, i.e. the starting day of this firm's outage should be determined. Since the outage length of this firm (D k days) has already been determined by the aggregate model, the finishing day of the outage assigned to this firm can be calculated easily. Now suppose that the starting day of the outage assigned to the k-th firm coincides with the i-th day. Hence, the current amount of Oj will be increased as much as A k megawatts, for j -- i, i + 1 . . . . . i + D k. By Theorem 2, to get the best result, we should determine the starting day of the outage, say i*, that minimizes the following term, for i = 1 . . . . . ( D - D k + 1): i+Dk--1 E ( O t + A k ) 2" l=i
(43)
This can be done easily by inspection. Once a firm's outage is scheduled, the value of Oj for the corresponding interval will be updated. Once the scheduling for all firms of the first group is completed, the shortage pattern for the whole season will be modified. By subtracting the value of final Oj from the original Bjk (hourly shortage, as defined in Section 6), its current value will be obtained, for j = 1 . . . . . t and k = 1 . . . . . D. A table will be set up, in which updated data, including the value of Bjk for each hour of the season as well as the shortage length and maximum daily Bjk are displayed. In the second stage, the outage scheduling for the firms belonging to F u will be carried on. The number of outages assigned to each firm can be more than one, but the length of each outage will not exceed t (the shortage length of each day). In the second stage, the first firm of F u will be picked, first. This firm must have as many as )(1 outages, each of which with a duration of L 1. This firm will be given outages on the )(1 days of the season with the highest Bjk, provided the shortage length of such days is at least equal to L v Since in the aggregate model, only one set up cost per day has been considered, the outages of this firm should be scheduled either at the beginning or at the end of a working day, but not in the middle. Then the shortage pattern as well as the above mentioned table will be modified accordingly. This procedure will go on for the second, third, . . . firms until all the firms have been scheduled.
7.3. Implementing the algorithm A computer program is written to find the best interval for each firm's outage duration and to keep track of the value of Oj for each day in the first stage, and also to update the value of Bjk for each hour of the season after the first and the second stage. This program determines the days on which a firm should have an outage in the second stage of the algorithm. A real example with seven firms was solved by hand in order to make sure that the algorithm can be implemented even without using a computer. This makes
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the algorithm more applicable in the cities where using the computer is difficult for many reasons, including the lack of computer experts. In addition, since the people in charge can understand and implement it easily, no resistance against it can be expected. In larger areas, where computers services are available, it is not wise to schedule by hand. After finishing outage scheduling for all industrial firms and the plotting the new shortage patterns, one can see that the modified shortage patterns for different days are no longer similar. However, this is not a worrisome situation because later on, as mentioned before, one can compensate the extra shortage by the outages allocated to the residentials, commercials and the other sectors. The planning in these sectors, contrary to the industrial sector is quite flexible, as described before. In these sectors, it is possible to cut almost any amount of power at any time.
8. Summa~ and conclusion In this paper, the issue of electric power shortage in the developing countries as well as the method of power outage planning, in order to minimize the cost of power shortage, was discussed. The importance of the problem arises from the fact that the cost of production interruption due to the power shortage is significant. Two mathematical models have been developed to determine the outage length assigned to each firm, and to schedule the power outage for the firms during the season. Based on the results of the mathematical models, a policy has been suggested which is simple and practical and is quite different from what is now applied in Iran. The number of outages assigned to each firm should be as few as possible, usually once per season, although its duration may be the total outage length assigned to that firm. An algorithm has been developed to schedule outage intervals for the firms. The algorithm is simple and comprehensive, especially for the managers in small towns or remote areas in which the computer facilities are not at hand easily. The algorithm can even be implemented by hand or by a programmable calculator. However, it was shown that the algorithm can produce a solution which is not mathematically far from the solution of the model.
Acknowledgement It is a great pleasure to thank Professor Bruce L. Miller of the University of California, Los Angeles, and Professor Hashem Mahlooji of Sharif University of Technology for reviewing this paper and for their valuable comments.
References Central Bank of Iran (1984), "Annual economic report of Islamic Republic of Iran", Tehran. Modarres, M. (1986), "A general framework of power planning when demand exceeds supply", Journal of Barname Va Towse'e (in Persian) Tehran. Office of Power Planning, Ministry of Energy (1985), "Statistics of Iranian electric power", Tehran. Munasinghe, M. (1979), The Economics of Power System Reliability and Planning, The Johns Hopkins University Press, Baltimore, MD.