Optimal step-by-step restoration of distribution systems during excessive loads due to cold load pickup

ELEOTfllO POI,iJEfl

IW;, ELSEVIER

Electric Power Systems Research 32 (1995) 121-128

Optimal step-by-step restoration of distribution systems during excessive loads due to cold load pickup Canbolat U ak, Anil Pahwa Department ~)1"Electrical and Computer Engineering, Kansas State University, 261 Durland Hall, Manhattan, KS 66506-5105, USA Received 30 September 1994

Abstract

Restoration of a distribution system following extended outages may cause problems if the substation transformer capacity cannot supply the initial load. Excessive initial load is a result of loss of diversity in the distribution system. Dividing the system into sections and restoring these sections step by step is one of the solutions to overcome excessive load due to cold load pickup. In step-by-step restoration, the order of restoration of sections becomes an important factor in increasing the reliability and decreasing the average customer interruption duration. In this paper, step-by-step restoration of a distribution system to utilize transformer capacity to the maximum extent and to minimize the customer interruption duration is studied.

Keywords: Distribution network restoration; Cold load pickup; Customer interruptions; Reliability evaluation

I. Introduction Thermostatically controlled devices such as air conditioners, heaters, and heat pumps provide the largest contribution to the total load in a typical house. During normal conditions, diversity among individual loads on the residential feeder is present, and therefore the aggregated load of a number of houses is less than the connected load. But, after an outage in a distribution system, some or all thermostatically controlled devices will be O N as soon as the power is restored. If an outage involves a large number of customers and is of long duration, it may result in excessive load during restoration. Restoring power to a distribution system under such conditions is called cold load pickup (CLPU). C L P U appeared first in the 1940s as a problem because of high inrush currents that last a few seconds and prevent the circuit from being reenergized after extended outages. Using very inverse characteristic relays or sectionalizing the distribution system were two of the solutions used by engineers to overcome this problem. Enduring current, which is the result of loss of diversity, did not demand attention in the 1940s and 1950s. This was partly because the current was not as high as inrush and motor starting currents. Also, long duration of enduring current did not force the thermal 0378-7796/95/$09.50 ~; 1995 Elsevier Science S.A. All rights reserved SSDI 0378-7796(94)00902-G

limits on distribution equipment because of large margins between substation capacity and system load. Large margins were necessary for high system reliability since the distribution systems were in their infancy and support from other substations was either very limited or did not exist. Although no problem existed at that time with the enduring component of restoration, Ramsaur mentioned in 1952 [1] that enduring current might limit the amount of load that could be picked up at once, and suggested that a sectionalizing scheme should provide a solution to this problem. Since then, penetration of thermostatically controlled devices such as air conditioners, water heaters, heat pumps, etc. has resulted in some investigation of C L P U enduring currents. Starting from the late 1970s, more literature has appeared on prediction and analysis of C L P U behavior. In 1979, McDonald et al. [2] studied electrically heated homes to predict the magnitude and duration of the peak demand following a power outage in cold weather. Their work was experimental and one of the first attempts to predict the magnitude and duration of the peak demand as a function of outside temperature and outage duration. The same model was used, with slight modifications, by Miller et al. [3]. However, the results could only be applied to systems with a similar type of load. Therefore, a more general description of the load was needed. Chong and

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Debs [4] gave a physically based load model for individual loads as a function of weather and human use patterns. Their methodology consisted of two basic steps: (i) modeling of the individual electrical loads, and (ii) aggregation of these loads to find the total demand. Following that, physically based load models which make use of stochastic theory have been investigated in detail by many authors [5-9]. Mortensen and Haggerty [10] present and discuss five mathematical models that have been studied in the literature. Aggregated load behavior, when a large number of customers are considered, can be determined based on these models either by use of numerical techniques to solve partial differential equations derived from individual load models or by Monte Carlo simulation based on the stochastic difference equation given in Ref. [10]. On the other hand, instead of stochastic and detailed models, simplified models are used to find the effect of CLPU loads in substation and distribution transformers. Because the thermal response of a transformer to a load is slow, simplified models are sufficient to analyze loading capabilities. Wilde [11] investigated the effects of CLPU at the substation transformer by using a CLPU model in which postoutage load is constant for some time and then decreases linearly from undiversifted load to diversified load. Aubin et al. did a similar study using a piecewise linear CLPU model to find the overloading capability of distribution transformers [12]. Quick restoration of distribution systems during excessive system load conditions caused by extended outages is the main goal of this paper. For restoration of distribution systems under such conditions, when system load is higher than the system capacity, one solution is to sectionalize the system. Therefore, the load behavior of sections and the restoration sequence of these sections play an important role in the restoration procedure. Determination of restoration times is important because operation engineers need to know when the sections should be restored. Also, system reliability calculations and customer satisfaction are clearly correlated to interruption durations. In the following sections, first we present the CLPU load model used in this paper. Then, based on this model and the maximum capacity of the transformer, we give a numerical procedure to find the restoration times. Since the restoration times are sequence dependent, customer interruption duration is minimized by using a procedure called the adjacent pairwise interchange method. This work is an extension of the work done by the authors of this paper in Ref. [13].

2. CLPU model A difficult aspect of this work is selection of a suitable model to represent the dynamics of the load for

LoadS i (t)

SDi

. . . . . . . . . . . . . .

I',

~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

t~

Time

t

Fig. I. C L P U model of the aggregated load of each section in a distribution system.

the enduring portion of CLPU. The model should be mathematically simple and yet it must account for the behavior of aggregated load as closely as possible. Analytical models could change from a simple straightline model to a more complicated high-order polynomial or a sigmoid type function. High-order polynomial and sigmoid type functions have the potential to make the problem very complicated and intractable. At the same time, simple straight-line models do not accurately represent the load behavior. Only extended outages will be considered because they are the most severe case in distribution systems. After extended outages, diversity is completely lost and the load is the highest at the beginning of restoration. Generally, it is considered that the diversity is completely lost if the outage lasts more than half an hour. Higher than normal loads may be expected for shorter outage durations but their effect on the distribution system may not be as important as that of extended outages. Therefore, the aggregated load model does not need to account for the behavior of the partial loss of load diversity in the system. Actual restoration data adequate to justify the models used for CLPU are extremely rare. Generally, utilities record average demand over 15 minutes and this resolution is too low to verify CLPU models. In Ref. [ 11], two sets of high-resolution current readings in the same circuit during CLPU restoration are given. Load change from undiversified to a diversified level in the circuit as shown in that paper may be closely represented by an exponential function. However, an exponential function model does not take into account the duration of undiversified load. It assumes that the diversity starts just after restoration. Therefore, to include the duration of undiversified load, a delayed exponential function, as shown in Fig. 1, is used to model the CLPU behavior of the aggregated loads in distribution systems. A delayed exponential model for

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cold load pickup of thermostatically controlled devices has previously been suggested by Lang et al. [14]. A mathematical expression for the delayed exponential model can be written as S,(t) = {SD~ + (Sug - SD~) e x p [ - ~(t -- t~)]}u(t -- t~)

+ St;,[ 1 - u(t - ti)]u(t - T~)

(1)

where S~(t) is the load of the ith section, SDi is the diversified load, Su~ is the undiversified load, and ~g is the rate of decay of the load on the ith section; u(t) is the unit step function, T~ is the restoration time, and t , - T, is the undiversified load duration on the ith section.

3. Problem formulation

It is assumed that the main feeders have remotely controlled sectionalizing switches which can be opened and closed to supply power to the sections. The sectionalizing switches do not need to have ratings high enough to interrupt feeder currents because restoration starts after all the sections lose power. We assume that the total substation transformer capacity is not sufficient to switch on all the loads simultaneously because of the undiversified load on the system resulting from an outage of long duration. In general, complete loss of diversity mainly depends on both the outage duration and the outside temperature. For example, on a very hot summer day, short outage durations may result in a complete loss of diversity because temperatures inside the houses will increase quickly to reach the ambient temperature, resulting in fast diversity loss in the circuit. Supplying power to loads during C L P U should be done while considering the transformer loading capacity. If the loading capacity is exceeded, the sections need to be restored in a sequence. Sections with priority loads such as hospitals, fire stations, and police stations must be supplied with power as soon as possible. Priority loads are included in the restoration procedure as priority constraints. Also, if a section that needs to be supplied power is at the end of the feeder, then all sections in that feeder must be supplied power first. Thus, selection of a section to restore power may also require energizing the upstream sections in the feeder. These types of constraints are called the precedence constraints. It is assumed that there are n sections which can be supplied power from a substation transformer. The maximum capacity of the transformer will be represented by SMr. The aggregated load behavior of each section is given by Eq. (1). The problem is to restore all n sections without violating the transformer loading constraint SMT. This can be written as

~ S H ( t ) ~< SMT

t ~>0

(2)

where [i] will be defined in Section 4. Note that the total undiversified load of n sections is higher than the transformer loading capacity. Therefore, restoration of some of the sections will be delayed to a point where the difference between SMT and the total load is larger than or equal to the undiversified load of the section to be restored.

4. Restoration times

The time when a section is restored is called the restoration time of that section. Restoration times of the sections will depend on the restoration order. Each section will be represented by a number such as 1, 2 . . . . . n, and its restoration order will be enclosed within brackets. For example, if the ith section is the first section to be restored in the restoration sequence, then the first element of the index will be i and, because it is restored first, it is given the section number [1]. There could be m sections which can be restored in the first step of restoration without violating Eq. (2). In this case, the restoration times of sections [1], [2] . . . . . [m] will be equal to To, which is the time when restoration started. In other words, the first m sections are restored simultaneously, T[q = T[21 . . . . . TL,,,1 = T0. The variable m will change depending on the undiversified load of the sections, the loading capacity SMT, and the priority and precedence constraints. The remaining n - m sections will be restored in steps at times T[,,,+ q, TI.... 21. . . . . TEn1. The restoration times of the sections that are restored step by step will obey the inequality Tii I < Ttg+q for i = m + 1. . . . . n - 1. In other words, two sections are not restored simultaneously when a step-by-step procedure is followed. The transformer load as a function of time with respect to a restoration order can be expressed using Eq. (1) of the C L P U model for n sections: S(t) = ~

SH(t)

i=1

= ,-,Y" ({SDH +

-- SoE, )

x exp[ - ~te](t - t t n ) ] }u(t -t[il ) K

+ Sutj]u(t - TH)[1 - u(t - tH)])

(3)

This is a general equation based on the restoration time Tt, 1. In the restoration procedure, section [k] should be restored as soon as possible, that is, when SMT -- S(T[k1) = Sulk] is satisfied. Therefore, the restoration time of section [k] can be found, such that

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C. Uvak, A. Pahwa /Electric Power Systems Research 32 (1995) 121 128

k l( SU[k] ---- SMT - - ~ , i=l

x exp[ - 0¢(T[k] -- t[i ])] } U(T[k] -- t[/l) + SuHu(T[k] - TH)[1 - u(T[k] -- tH)])

(4)

Since TEk] > T [ i ] for i = 1 , 2 , 3 , . . . , k - 1 and k > m , the corresponding unit step function u(TEk] -T]i]) is equal to one. This is true because restoration is done step by step and simultaneous restoration of two sections is not allowed except in the first step. The restoration time of a section can be written analytically if the rates of load decays in all the sections are the same [13]. In general, some or all sections may have different rates of load decay. Therefore, it is not possible to find a closed-form solution for the restoration time Tfk1. However, this problem can be solved with various root finding iterative techniques. Here, a well-known iterative technique based on Newton's method is used to find the restoration times of sections during C L P U in a distribution system. Newton's method, also called the N e w t o n - R a p h s o n method, requires the evaluation of both the function and the derivative at arbitrary points. It converges quadratically for well-behaved functions, that is, the number of significant digits approximately doubles with each step near the root. In this problem, a transformer or substation total load S ( t ) is a summation of exponential functions. Thus S ( t ) is a well-behaved function and does not have multiple roots. Therefore, the N e w t o n Raphson method is suitable for finding the restoration times of the sections. The restoration time for section [k] is found by solving the equation f ( t ) = SMT - -

Sulk]

--

S(t) = 0

(5)

The time t which satisfies Eq. (5) will be the restoration time of section [k]. To apply Newton's method, we need the derivative o f f ( t ) which is equivalent to the derivative of the load: f'(t)

df(t) dt

dS(t) dt

(6)

The transformer load S ( t ) when k - 1 sections are restored prior to section [k] is the same as Eq. (3) except n is replaced by k - 1. Note that the transformer load S(t) is not a continuous function at t~,-1 and TE~1. But the derivative of S ( t ) exists in piecewise continuous regions. The time derivative of the transformer load S ( t ) in the continuous region can be written as S'(t)

dS(t) dt -

~- l ~ %](Sutq - Sotq)

Iteration steps for the solution are then

Told

5. System and customer interruption duration indices Service reliability indices have been defined by the IEEE and they can be found in Refs. [15,16]. Two of these service reliability indices are: • system average interruption duration index SAIDI -

sum of customer interruption durations total number of customers served

• customer average interruption duration index CAIDI -

sum of customer interruption durations total number of customers interrupted

SAIDI and CAIDI are very sensitive to the duration and number of outages. Both SAIDI and CAIDI could be improved with remotely controlled switches. Also, service reliability levels could be improved by careful selection of the restoration sequence of the sections during CLPU. An optimal sequence should reduce SAIDI and CAIDI. Severe weather conditions resulting in widespread outages will cause the system to have high values of these indices. Therefore, faster restoration of distribution systems will play a very important role in system reliability. Estimation of improvement in the interruption duration is important to evaluate the economic effectiveness of the proposed methodology. However, economic evaluation is beyond the scope of this paper. Let us assume that the outage duration is given by T o u t and that the sections could be restored in a sequence at the end of this outage with To = 0. Then the sum of customer interruption durations (SCID) for section [i] will be SCIDfil = CEil(T[i] + Tout)

(9)

for i = 1, 2 . . . . . n. The value Cv] gives the number of customers on section [i]. The total customer interruption duration (TCID) for n sections can be written based on SCID[i] as

i=l

(7)

(8)

Iteration is stopped when lithew - to~dl[ ~< e, where ~" is a small number. When this stopping condition is satisfied, T[~w gives the restoration time T[kl of section [k]. Note that only the restoration times of n - m sections need to be calculated. The remaining m sections are restored in the first step of the restoration procedure according to the restoration sequence.

T C I D = ~" SCID[i] = ~, C[i](T[i j + Tout)

i=1

× exp[ - %1(t - t[q)]u(t - t[i])

old

SMT -- Sulk] -- S(T[k 1) T[k] ~-"--[k] -~, old S (T~I) new

{SD[i] ~- (Su[i] -- SD[i] )

(10)

i=1

Therefore, the system average interruption duration index for the restoration because of this outage is

C. Ufak, A. Pahwa /Electric Power Systems Research 32 (1995) 121 128

&

S(t)

L CMTH + To.,) SAIDI = i=l

125

(11)

n

C,. + ~ C[i] i-I

~7_~C;i] is the total number of customers that are interrupted and Cr is the number of remaining customers in the system. The total number of customers in the system is considered constant for all the outages. If all the customers are out of power, such as in a severe storm, then C, will be equal to zero. Similarly, the customer average interruption duration index can be calculated as T[k-2]

n

Z q,1TH CAIDI - ' : ~

+ Tou,

T[k-l]

T[u

Fig. 2. The restoration sequence o f two adjacent sections, a and b.

(12)

L C[,] i

1

The first term on the right-hand side is a function of the section restoration times. Since Tout and Cti] are constants, improvement in both SAIDI and CAIDI is possible with selection of a proper sequence of sections for restoration. From Eqs. (11) and (12) it can easily be seen that the restoration sequence which minimizes CAIDI will also minimize SAIDI. In these equations, the number of customers and the outage time Tout are constant values. Therefore, to find a restoration sequence that optimizes CAIDI and SAIDI, we need to solve the minimization problem min L Clq TH

(13)

i-I

A search for the minimum is achieved with a procedure called the adjacent pairwise interchange method.

6. Adjacent pairwise interchange method Sequencing problems occur whenever there is a choice of order in which a number of tasks can be performed. Mainly, there are two approaches to solve sequencing p r o b l e m s - algebraic sequencing methods and Monte Carlo sampling [17]. Algebraic methods include the adjacent pairwise interchange method, branch-and-bound algorithms, and various mathematical programming procedures, such as linear and dynamic programming. Monte Carlo sampling can be described as a selection of the best of a large number of randomly generated sequences. Almost any sequencing problem can be approached by the Monte Carlo method. However, algebraic sequencing methods are probably more powerful because an optimal sequence can be reached by step-by-step procedures. The drawback is that there are no general algebraic methods that are applicable to large numbers of sequencing prob-

lems. Storage requirements or computing-time constraints could also be the limiting factors. The restoration procedure during C L P U is similar to the single-machine scheduling problem where scheduling times are sequence dependent [18,19]. In a distribution system with n sections, there are n! different permutations. Thus, the determination of an optimal sequence for an objective function, in general, is a combinatorial problem and intractable for large n. The adjacent pairwise interchange method (APIM) is simple and well suited for search of an optimal solution for the restoration procedure where restoration times are sequence dependent. In APIM, a sequence is sought for which all adjacent pairwise interchanges lead to an increase in the objective function [18]. For example, we seek a restoration order of sections which will minimize the total interruption duration; that is, we find the order [1], [2] . . . . . [n] where TCID is the minimum. Now, let us assume that an initial restoration sequence is selected. As shown in Fig. 2, the improvement in total customer interruption duration by interchanging two adjacent sections (a and b) can be investigated. We define the total customer interruption durations TCID~ and TCID2 for two different orders, such as TCID~ for restoration order a - , b and TCID2 for restoration order b ~ a . If TCID~ < TCID2, then the restoration order a--, b gives a smaller total customer interruption duration. This procedure is applied between all pairs of adjacent sections in a restoration sequence until there is no reduction in the total customer interruption duration. However, this method does not guarantee a global minimum solution. In general, APIM is known to result in a local optimum but not necessarily in a global optimum [20]. Therefore, one can only hope that the resulting local optimum is a good solution relative to the global optimum. There are various ways to increase the likelihood of obtaining a good solution. For example, increasing the number of local optima being evalu-

C. Ufak, A. Pahwa ~Electric Power Systems Research 32 (1995) 121 128

126

--U53i. . . . . . . . . . .

:

SECTION 1

SECTION 4

Initial Sequence --U:3

SECTION 7

<3

O

% ~

O

SECTION 2

SECTION 5

SECTION 8

O

k

O

LTJ

©

SECTION 3

SECTION 6

SECTION 9

SUBSTATION SECTION 10 E:2

swap(i, i-1)

O

SECTION I1 O

SECTION 12

circuit Breaker

O

Three-phase Sectionalizer

[]

Tie switch

Fig. 4. One-line diagram of a distribution system. swap(i, i-l) I Y

N

Table t CLPU parameters of each section

i=i-1

~-

Y@N i

Found optimum]

I

Section no.

St;, (p.u.)

SD, (p.u.)

kti (min)

1/~i (min)

C,

1 2 3 4 5 6 7 8 9 10 11 12 Total

0.1333 0.2000 0.3000 0.1667 0.1667 0.2667 0.2000 0.2000 0.2333 O.3000 0.2167 0.1833 2.5667

0.0667 0.0833 0.1000 0.0500 0.0667 0.1000 0.0667 0.0833 0.0667 0.1000 0.0667 0.0667 0.9168

30 20 10 30 25 30 30 30 30 15 20 15

120 160 120 120 140 120 120 100 100 200 150 120

600 700 900 300 550 1100 400 350 800 700 600 500 7500

Fig. 3. Flowchart for the adjacent pairwise interchange method. Sba~e = 30 MVA.

ated or choosing a good initial sequence may give a better solution. It can also be shown that both the initial sequence and the APIM switching strategy may affect the resulting local optimum solution. The APIM strategy used in this paper starts from a predetermined or random initial sequence. The interchange of pairs begins at the last position in the sequence and proceeds back to the front; that is, the procedure moves from section [n] to section [1]. When a pair of sections to be interchanged is found, the swap is made, and the search strategy continues towards the beginning of the sequence. If at least one interchange is made then the procedure starts over from the last position in the sequence. The APIM flowchart is shown in Fig. 3.

7. Numerical example For illustration, we will consider a distribution system with twelve sections. The distribution system consists of four feeders each with three sections separated by two three-phase sectionalizer switches, as shown in Fig. 4. There are also tie switches which connect sections to neighboring feeders. The tie switches are added

to the system to increase the number of feasible solutions. The total number of customers in the distribution system is 7500. The number of customers and the C L P U parameters of each section are given in Table 1. In general, the maximum short-term loading capacity of a substation transformer changes depending on the load duration and load shape [21]. But, for simplicity, we will consider that the maximum transformer capacity SMv is equal to 1.667 p.u. for the selected conditions. Both the preoutage and postoutage transformer loads for this example are 0.91 p.u. and the total undiversified load of the distribution system is 2.5667 p.u. Since the total undiversified load of the system exceeds the substation transformer capacity, step-by-step restoration is required to return the system to normal operating conditions. The precedence constraints for the system are considered in the APIM algorithm. For example, section 2 can be energized if either section 1, section 5, or section 3 has been energized. Any restoration order which does not account for the precedence constraints is discarded from the search process. Feeder current, thermal limits, and voltage drop limits are not considered in this example.

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C. U~'ak, A. Pahwa /Electric Power Systems Research 32 (1995) 121 128

Table 2 The results of APIM for different initial sequences and the global minimum and maximum sequences No.

Initial sequence

Final sequence

Average CID (rain/customer)

1 2 10-3.9.4 8 5 7 6 11 12 11 3 6 9-4.2 10 7 12 5 1 8 8 10 1 3 5 6 4 2 11-9.--12-7 10 1 2 9 3 8 5 6 %4-12-11 Descending order of undiv, loads Global minimum Global maximum

[4,1,10,2,3,5,6,11} 12 9 7-8 {4, 10,5,7, 1,8,6,9} 2 3 11 12 {1,10,2,11,4,7,31 12-8-5 6 9 {1,2,10,5,3,6,9} 11-12 7 4 8 {10,11,7,8,12,5,6,1} 9 2-3-4 {1,2,10,11,3,6,91-5-12 7 4 8 {4,5,10,7,8,2,11} 3 12 9 6 1

34.59 40.64 49.90 28.17 40.13 27.40 57.60

The optimal restoration orders and the customer interruption durations for five initial sequences are shown in Table 2. F o u r of these sequences are chosen randomly. The fifth initial sequence is arranged in descending order of undiversified loads. The best result is obtained by the fourth initial sequence with 28.17 minutes of average customer interruption duration. This value is very close to the global minimum average customer interruption duration of 27.4 minutes. F o r the best result, sections 1, 2, 10, 11, 3, 6, and 9 should be restored at the beginning of restoration and sections 5, 12, 7, 4, and 8 should be restored step by step without exceeding SMT. This is shown as {1, 2, 10, 11, 3, 6, 9 } 5-12-7-4 8 in Table 2. In this example, using the descending order of undiversified loads as the initial sequence does not result in a good solution. However, using different initial sequences increases the likelihood of obtaining a better solution. The global m a x i m u m and minimum values of the customer interruption duration are found to be comparable to those found by the A P I M algorithm. As one might expect, searching for all the combinations of sequences m a y take a long time for a large number of sections and m a y not be necessary, as demonstrated here, since the solution obtained using the A P I M algorithm is very close to the global solution. In the worst-case scenario, the average customer interruption duration per customer will be 57.6 minutes. The solution of the A P I M algorithm gives a restoration sequence which reduces the average customer interruption duration to 28.17 minutes per customer. This is an improvement of about half an hour per customer. For a large number of customers, the restoration sequence will be very important for increasing the reliability and for decreasing the interruption costs to the customers.

distribution system. The C L P U characteristics of aggregated load in each section m a y allow restoration of the distribution system step by step when such a condition occurs. In step-by-step restoration, the sequence of sections should be selected in such a way that the average customer interruption duration is minimized. In this research, the minimum average customer interruption duration is found by using the adjacent pairwise interchange method. This method does not guarantee a global o p t i m u m but it provides a good and fast solution to the problem. This study shows that considerable improvement in step-by-step restoration is feasible with a proper procedure. The improvement depends on cold load pickup parameters, transformer thermal characteristics, loading capacities, number of sections, and other constraints such as the priority and precedence. Since cold load pickup is one of the most severe conditions that a distribution system experiences, restoration capabilities of the system and procedures to return it to a normal condition as fast as possible will benefit not only the operation engineers but also the design engineers. However, a good knowledge of load behavior during cold load pickup is important for the field implementation of this methodology. In the future, availability of high-resolution data from the field during an actual restoration m a y validate or improve the cold load pickup model used.

Acknowledgements

This research was supported by the National Science Foundation G r a n t No. ECS-9311257.

References 8. C o n c l u s i o n s

The restoration procedure becomes an important element when the load exceeds the substation transformer capacity because of the loss of diversity in a

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