Black-start decision-making with interval representations of uncertain factors

Electrical Power and Energy Systems 79 (2016) 34–41

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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes

Black-start decision-making with interval representations of uncertain factors Hong Wang a, Zhenzhi Lin b, Fushuan Wen c,⇑,1, Gerard Ledwich d, Yusheng Xue e, Yuzhong Zhou a, Yuchun Huang f a

Electric Power Research Institute, China Southern Power Grid, Guangzhou 510080, China School of Electrical Engineering, Zhejiang University, Hangzhou 310027, China Institut Teknologi Brunei, Bandar Seri Begawan BE1410, Brunei Darussalam d School of Electrical Engineering and Computer Science, Queensland University of Technology, Brisbane, Queensland 4001, Australia e State Grid Electric Power Research Institute, Nanjing 210003, China f Guangzhou Power Supply Bureau Co., Ltd., Guangzhou 510310, China b c

a r t i c l e

i n f o

Article history: Received 22 February 2014 Received in revised form 26 November 2015 Accepted 4 December 2015

Keywords: Black-start Decision-making Interval values Goal programming Risk attitude factor

a b s t r a c t Finding an optimal black-start scheme plays an important role in speeding up the restoration procedure of a power system after a blackout or a local outage. In a practical black-start decision-making procedure, uncertainties are inevitable. Some factors such as index values and indexes’ weights can be better described as uncertain interval values. So far, the black-start decision-making problem with uncertainties has not yet been systematically investigated. Given this background, a new approach for black-start decision-making based on interval values is developed. First, a decision-making matrix with interval values is normalized by using the error propagation theory. Then, a linear goal programming model is developed to seek the ideal vector of index weights and the evaluation values of all candidate blackstart schemes can be obtained. A risk attitude factor based method is presented to sort the schemes. Finally, a sample example is served for demonstrating the essential feature of the proposed method, and comparisons with three existing methods are also carried out. Simulation and comparison results show that the proposed method could not only take different kinds of uncertainties into account, but also overcome several shortcomings of the existing methods. Ó 2015 Elsevier Ltd. All rights reserved.

Introduction In recent years, several large-area blackouts happened all over the world [1,2], such as the power failure in US and Canada on August 14th in 2003, the interconnected power gird blackout in Western Europe on November 4th in 2006, and the recent blackout in India on July 30th in 2012. Although the self-healing capacity of a power system can be enhanced by the development of smart grids, it is not possible to avoid large-area blackouts completely. Therefore, it is necessary to study the issues associated with power system restoration after a complete blackout or a local outage, so as to implement rapid, intelligent and efficient power recovery [3–6]. The power system restoration process after a blackout can be divided into three stages: black-start, network reconfiguration ⇑ Corresponding author at: Department of Electrical and Electronic Engineering, Institut Teknologi Brunei, Bandar Seri Begawan BE1410, Brunei Darussalam. E-mail address: [email protected] (F. Wen). 1 Taking leave from School of Electrical Engineering, Zhejiang University, Hangzhou 310027, China. http://dx.doi.org/10.1016/j.ijepes.2015.12.033 0142-0615/Ó 2015 Elsevier Ltd. All rights reserved.

and load restoration. Among these three stages, the black-start stage is defined as the one in which the black-start units, after a large blackout, supply power to the units that cannot start independently without the help of other systems or units [7]. Therefore, black-start represents the first restoration stage after a blackout, and finding the optimal black-start scheme is one of the key issues having impacts on the restoration speed of the power system concerned. The black-start decision support system is an important module of the future decision support system in a smart grid environment, with an objective of speeding up the restoration procedure after a blackout or local outage. So far, much research work has been done in developing blackstart decision-making and system restoration methods. An expert system based decision-making for power system restoration is proposed in [8,9]. In [10], an optimal restoration approach based on the Wide Area Measurement System (WAMS) is presented. In [11], a hierarchical case-based reasoning method is presented for determining the black-start schemes. In [12], the vague set is first employed to deal with the black-start decision-making problem

H. Wang et al. / Electrical Power and Energy Systems 79 (2016) 34–41

with interacting attributes, and then a concept of the vague-valued fuzzy measure presented and a mathematical model for black-start decision-making developed. In [13], a new service restoration system for distribution systems is described, and more practical solutions can be obtained through the multiple criteria fuzzy evaluation for the candidate schemes. In [14], a real-time restoration decision support system is presented and a detailed design framework of the system introduced. The proposed system is combined with the dispatcher training simulation system of EdF (Electricite de France) in France, and is demonstrated by realtime data. In [15], a black-start decision-making method based on an intuitionistic fuzzy set and further the intuitionistic fuzzy Choquet integral operator are presented. In [16], a novel method using the entropy weight-based decision-making theory is developed to evaluate and optimize the black-start schemes. It can be seen that in the existing research literature the index values of alternative black-start schemes and the weights of the indexes for evaluating black-start schemes are mostly treated as deterministic. However, some of index values and the indexes’ weights are uncertain in practical black-start decision-making, and are better represented by interval ranges. To the best of our knowledge, the black-start decision-making problem considering uncertain quantities described as interval values has not yet been investigated thoroughly. In addition, the weights are acquired from the subjective experience of experts, or calculated based on the objective entropy weight theory in which the authority of each expert in the weight determination is not well reflected. As a result, ideal indexes’ weights should be determined by a combination of the subjective and objective methods, so as to make the final decision-making result more reasonable. Given this background, a new approach for black-start decisionmaking based on interval values is first developed. The method is able to deal with uncertain quantities described by interval values, which can better reflect actual black-start decision-making procedures. First, some theories of interval values and error propagation are introduced. A black-start decision-making method based on interval values is then proposed. Finally, a case study is employed to demonstrate the new method and to compare the proposed method with some existing methods. The theories of interval values and error propagation Definition of an interval value ~ where A closed interval [bL, bU] is denoted as an interval value b, bL 2 R, bU 2 R, bL 6 bU. A matrix (or vector) with interval elements is denoted as an interval value matrix (or vector). Taking the error ~ can also be described distribution into account, the interval value b ~ ¼ b  Db, with the interval midpoint defined as b ¼ ðb þ b Þ=2 as b and the limit error as Db = (bU  bL)/2 . L

U

A sorting method for interval values Interval values cannot be sorted by traditional sorting methods because there are intersections among them. Therefore, a risk attitude factor based method is introduced to sort interval values [17]. ~ ¼ ½bL ; bU , an interval mapping function For an interval value b

~ can be introduced to transform an interval value into an ue ðbÞ actual value, and described as follows.

~ ¼ ue ðbÞ

U

L

b þb U L þ eðb  b Þ 2

ð1Þ

where e is the risk attitude factor (|e|60.5), and represents the attitude or degree that a decision-maker is willing to take risks. It can

35

~ can map an interval value, which is be seen from Eq. (1) that ue ðbÞ represented by its midpoint, width and risk attitude factor, into an actual value, and then the interval values can be sorted easily. The error propagation theory In the well-established measurement theory, the error of an indirect measurement can be obtained through the error function of a direct measurement, and this is the so-called error propagation [18], which is described as follows. Suppose that a group of direct measurement values is as follows: u1, u2, . . ., um; an indirect measurement value is v; and there exists a continuous and differentiable function: v = f(u1, u2, . . ., um). In addition, suppose that the random errors of u1, u2, . . ., um are du1 ; du2 ; . . . ; dum , respectively; the random error of v is dv; and the random errors dui are independent. Thus, the random error propagation relationship can be expressed as

d2v ¼

2 m  X @f i¼1

@ui

d2ui

ð2Þ

A black-start decision-making method based on interval values Power system restoration is a complex process with many uncertain factors. For example, the start-up power required by each unit is affected by the operating state of the power system concerned and may not be simply described by an accurate value. Given this background, an interval value is more suitable for describing it with its uncertainties taken into account. Thus, the decision-making method based on interval values is useful for optimizing candidate black-start schemes in practical black-start decision-making, since different kinds of uncertain information can be taken into account in this method. Black-start decision-making process Generally speaking, the framework of a decision-support system for black-start consists of three functional modules in practice, i.e. formation, verification and optimization of black-start schemes. First, in the formation module of initial black-start schemes, the power system topology database is first searched automatically by the breadth first search philosophy according to the distribution of local black-start power. All the awaiting start-up power plants and possible black-start paths are determined, thus all the possible black-start schemes are generated automatically and an initial scheme database formed. Secondly, in the verification module of feasible black-start schemes, relevant simulation and analysis software packages need to be employed to compute and verify all the initially determined black-start schemes, on aspects including unit self-excitation checking, no-load switching overvoltage checking, power frequency overvoltage checking, frequency checking, and low frequency oscillation checking. The technical feasibility of each scheme is such checked, and feasible schemes are picked out as the candidate ones. Finally, in the optimization module of candidate black-start schemes, an optimal black-start scheme will be selected from the candidate schemes by using one of black-start decision-making methods. The selection of the optimal blackstart scheme is of great significance and plays an important role in the black-start decision-support system [19]. In this paper, the third black-start functional module is mainly focused. Thus, a new black-start decision-making method based on interval values is presented for optimizing black-start schemes with uncertain information.

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H. Wang et al. / Electrical Power and Energy Systems 79 (2016) 34–41

Selected evaluation indexes In practice, the black-start process can be impacted by many factors. Since there are many indicators (or indexes) relevant to the evaluation of the black-start schemes, the dispatchers need to find out those indicators (or indexes) that best reflect the merits or shortcomings of various schemes systematically. The overall objective of the power system restoration is to restore as much load as possible in a shortest period. Therefore, the restoration time and available generation capacity should be selected as evaluation criteria with priority. The former reflects the rapidity of the unit start-up, while the latter represents the generation capacity that generating units can supply to the system. Therefore, based on the above two criteria, the following five indicators (or indexes) are chosen for evaluating the black-start schemes [15]. (1) The rating capacity of each generating unit The rating capacity of each unit determines how much power the unit can provide to the system when it is started up. If the units with large rating capacities are started in the first stage of the system restoration, they can supply more power to the system, and this is very helpful for restarting more units and restoring more loads. (2) The state of each unit The state of each unit is determined by the cylinder temperature of its turbine. Based on the temperature, the states of units can be divided into five types, including ultracold, cold, warm, hot, and ultrahot, which can be represented by 1, 3, 5, 7, 9, respectively. Among the five states, the units in the ultrahot state need least time to start up while the units in the ultracold require longest time to start up on the contrary. The scale value of each unit’s state can be determined according to the decreasing curve of the cylinder temperature of its turbine when the unit is shutdown. (3) The ramping ratio of each unit The ramping ratios of various generating units can be quite different. In the black-start stage of the system restoration, important loads need to be restored with priority. Obviously, the units with higher ramping ratios should be started up first so as to supply power to the system loads more quickly. (4) The start-up power required by each unit In the initial restoration stage, only a few units can act as blackstart units, and the power generated by them is limited. Thus, the units requiring less start-up power should be started up first.

The black-start decision-making matrix with interval values and its normalization In a practical black-start decision-making procedure, there are always some uncertain factors. Some uncertain factors can be appropriately described by interval values. The interval values of the indexes can be determined according to the operating conditions of the power system concerned, such as the units’ states and the start-up power required; while those of the weights can be given by operators or domain experts based on their knowledge and practical experience. For a specific black-start decision-making problem, suppose that there are n candidate black-start schemes and m selected indexes, M = {1, 2, . . ., m}, N = {1, 2, . . ., n}. Define an index set C = {c1, c2, . . ., cm}, and it can be divided into the benefit-type index set P and cost-type index set Q, P \ Q = £, P [ Q = C; the set of candidate black-start schemes is represented by A = {a1, a2, . . ., an}; the weights of the indexes are described by a weight vector ~ 1; x ~ 2; . . . ; x ~ m T , and x ~ j ¼ ½xLj ; xUj  is an interval number ~ ¼ ½x x

which represents the weight of index cj (j e M), 0 6 xLj 6 xUj 6 1; the black-start decision-making matrix with n-alternative black-start schemes and m-evaluation indexes is described by ~ ¼ ð~ ~ij is the interval value of the jth index related X xij Þnm , where x to the ith scheme, and ~ xij ¼ ½xLij ; xUij , while xUij and xLij are respectively ~ij . In addition, the interval the upper and lower bounds of x number ~ xij can also be expressed as ~ xij ¼ xij  Dxij , with the interval midpoint as xij ¼ ðxLij þ xUij Þ=2, and the limit error as

Dxij ¼ ðxUij  xLij Þ=2. In the black-start decision-making problem, the selected evaluation indexes in the benefit-type set P and cost-type set Q cannot be compared directly, since different indexes may have different units and dimensions. Therefore, in order to compare different index values, it is necessary to normalize the decision-making ~ ¼ ð~ matrix X xij Þnm into the normalized decision-making matrix ~ ~ Y ¼ ðyij Þnm before starting to optimize the black-start decision~ij ¼ ½yLij ; yUij  is the normalized form of making procedure, where y ~ xij , 0 6 yLij 6 yUij 6 1. ~ are As the parameters ~ xij in the decision-making matrix X expressed as interval numbers, the conventional normalization methods will not be applicable. Thus, a normalization method based on the combination of the weighted sum model and the error propagation theory could serve for this purpose. In order to obtain the normalized black-start decision-making ~ the first step is to determine the interval midpoint matrix matrix Y, Y ¼ ðyij Þnm , where yij is the interval midpoint of the interval value ~ij . The weighted sum model [20] is employed to normalize xij as yij , y as detailed below:

(5) The number of switching operation In the black-start process, the units are supplied power by some black-start units through the restoration paths. The restoration time for each path depends on the number of switching operation in substations and lines along the path significantly. The number of switching operation also has some impacts on the system such as power frequency overvoltage and switching overvoltage. Of the selected five indexes above, the first three can be viewed as benefit-type indexes and the last two as cost-type indexes. The selected indexes may be different for various power systems depending on the specific characteristic of each system. However, the presented black-start decision-making method here is applicable to different systems.

yij ¼

8 xij > > > Pn <

x l¼1 lj

cj 2 P

1

xij > > > : Pn

1 l¼1 xlj

cj 2 Q

ð3Þ

In the black-start decision-making problem, the limit errors of x1j ; x2j ; . . . ; xnj are represented as Dx1j ; Dx2j ; . . . ; Dxnj , respectively, and the limit errors of y1j ; y2j ; . . . ; ynj as Dy1j ; Dy2j ; . . . ; Dynj , respectively. Furthermore, there exists a differentiable function yij ¼ f ðxij Þ expressed by Eq. (3). Thus, the error propagation theory expressed by Eq. (2) can be employed to calculate the limit error ~ij and matrix DY = (Dyij)nm, where Dyij is the limit error of y described as follows.

37

H. Wang et al. / Electrical Power and Energy Systems 79 (2016) 34–41

8 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P 2 P > n n > 2 2 2 > Dxij x þxij Dxkj > l¼1;l–i lj k¼1;k–i > > > cj 2 P Pn 2 > > < x l¼1 lj r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Dyij ¼  Dx2 D x2 Pn 2 Pn > > > x6ij 1xij l¼1 x1lj þx12 k¼1;k–i x4kj > > ij ij kj > cj 2 Q > P 2 > > n 1 :

~i ¼ ½d ; d  for the ith black-start scheme can be determined value d i i as follows. L

ð4Þ

U

8 < dL ¼ Pm yL x i j¼1 ij j : dU ¼ Pm yU x i j¼1 ij j

ð7Þ

l¼1 xlj

~ij ¼ yij  Dyij can be obtained given that yij and Dyij are Thus, y ~ ¼ ðy ~ij Þnm can be obtained. already known, and finally Y A goal programming based comprehensive evaluation method As the black-start decision-making matrix is described by using ~ ¼ ½d ; d  (i = 1, 2, . . ., n) for interval values, the evaluation value d i i i the ith candidate black-start scheme will be expressed by interval ~ , the values as well. In order to determine the evaluation value d L

U

i

weights of the indexes need to be first determined. In most existing black-start decision-making methods, the weights are directly given by the experts, which are subjective and uncertain. An entropy-weight based method is presented to determine the weights in [16], but cannot deal with heuristic experience or knowledge, which is very helpful in the black-start decisionmaking. Given this background, a linear goal programming model is presented in this work to seek the ideal weight vector from the interval ranges of index weights given by the experts, and can avoid some disadvantages of the existing methods and achieve more reasonable decision-making results. ~ has two endpoints dL and dU , Each interval evaluation value d i

i

i

and it is necessary to employ the same weight vector in determining the two endpoints. The interval values are employed to describe different kinds of uncertain information in the blackstart decision-making process. So a larger range of an interval value means more uncertain information involved. Given this background, a multi-objective optimization model [21] is applied to seek the ideal weight vector x ¼ ½x1 ; x2 ; . . . ; xm T .

8 P L L  > min di ¼ m > j¼1 yij xj > > > < max dU ¼ Pm yU x i j¼1 ij j > > s:t: xLj 6 xj 6 xUj > > > : Pm  j¼1 xj ¼ 1

i ¼ 1; 2; . . . ; n i ¼ 1; 2; . . . ; n j ¼ 1; 2; . . . ; m

ð5Þ

The optimization model described by Eq. (5) represents a multiobjective optimization problem. The extreme points of the objecP Pm U U L U L L tive functions di and di are m j¼1 yij xj and j¼1 yij xj , respectively. Thus, the optimization model described by Eq. (5) can be transformed into the following linear goal programming model [22]:

8 P P > min z ¼ ni¼1 pi uþi þ ni¼1 qi ui > > P Pm L L > þ L  > > s:t: m i ¼ 1; 2; . . . ; n > j¼1 yij xj  ui ¼ j¼1 yij xj > > Pm U  Pm U U >  < i ¼ 1; 2; . . . ; n j¼1 yij xj þ ui ¼ j¼1 yij xj > xLj 6 xj 6 xUj j ¼ 1; 2; . . . ; m > > > > > Pm x ¼ 1 > > j¼1 j > > : þ  ui ; ui P 0 i ¼ 1; 2; . . . ; n

A sorting method for the black-start schemes based on the risk attitude factor ~1 ; d ~2 ; . . . ; d ~n ) are described by Since the evaluation values (i.e. d interval numbers, intersections may exist among them. Thus, they cannot be sorted by traditional sorting methods. A risk attitude factor based method for sorting the interval values is employed here to sort the black-start schemes with interval evaluation values, and can deal with subjective intentions of the power system operators or experts. In this way, both subjective and objective factors encountered in the black-start decision-making process can be addressed. ~ of scheme a as determined For the interval evaluation value d i

i

in Section ‘A goal programming based comprehensive evaluation ~ Þ is introduced to repmethod’, an interval mapping function u ðd e

i

resent the risk attitude of the black-start decision-makers. Based ~i Þ can be obtained as on Eq. (1), u ðd e

ue ðd~i Þ ¼

U

L

di þ di U L þ eðdi  di Þ 2

ð8Þ

where e is the risk attitude factor (|e| 6 0.5), which represents the attitude/degree that the power system operators or experts are willing to take risks. The evaluation results of the black-start schemes have direct impacts on the power system restoration procedure. The risk attitude of the power system operators/experts can generally be classified as conservative (pessimistic), neutral, and adventurous (optimistic) types, and the range of e can be assigned as 0.5 6 e < 0, e = 0, and 0 < e 6 0.5, respectively for these three kinds of risk attitudes. The risk attitude factor can be determined by the power system operators or experts. Considering the high reliability requirement of power supply and the rapidity requirement of the restoration process, it is expected that the blackstart decision-making experts would not like to take high risks. Therefore, they can generally be deemed as the conservative type. Hence, the risk attitude factor e in the black-start decision-making could be assigned as 0.5 6 e < 0. ~ Þ of For the purpose of comparisons, the final sorting value Uðd i

ith scheme can be determined by the interval mapping function u ðd~ Þ, as described below. e

ð6Þ

 where uþ i and ui (i = 1, 2, . . ., n) are the positive and negative deviation variables, respectively; pi and qi (i = 1, 2, . . ., n) are the coefficients of the objective function. If the power system operators or experts do not have any preference on the specific black-start schemes, pi and qi can both be assigned as 1. The ideal weight vector x⁄ can be obtained by solving the presented goal programming model, and then the interval evaluation

i

ue ðd~i Þ ði ¼ 1; 2; . . . ; nÞ ~ k¼1 ue ðdk Þ

~Þ¼ Uðd i Pn

ð9Þ

~ (i = 1, 2, . . ., n) can be sorted Finally, the interval number d i

~ Þ of each black-start according to the final sorting value Uðd i ~ scheme. The larger Uðd Þ is, the higher ranking the interval number i

will be, and hence the better the black-start scheme will be. Thus, ~ Þ equals all the black-start schemes can be sorted in this way. If Uðd i

~ Þ, their interval midpoints d and d can be compared: if to Uðd j i j ~ is equivalent to d ~ ; if d > d , the di ¼ dj , the interval number d i j i j ~ ~ interval number d ranks higher than d . i

j

38

H. Wang et al. / Electrical Power and Energy Systems 79 (2016) 34–41

LCP

BHP XNP

ZC

SJ LC

JC ZHC

ST

CHS

BJ

GT

ZX JLP SG YT

CH

JH

XH TX XT

JT

LH RH

GH

TPP

TP

XTP GZP LM

LY XCP

TH BS

WXM

YCP HC

TC BT

GBP

RB

HD

CS

HPBP KY

FC

MSP FS

HPAP NC

HYCP

ZJP HG

HYBP

GQ

YF

MZABP LHS HDZ DY

PY

SHJ

NSP

XZ

LJP

STP

MZCP

LHSP Plant

Substation

Line

Fig. 1. A simplified actual power system for case studies.

The procedure of the black-start decision-making method based on interval values Table 1 The candidate black-start schemes to be evaluated. Scheme No.

The black-start paths

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 a21

XNP ? BJ ? GT ? JC ? BHP XNP ? BJ ? LY ? XCP XNP ? BJ ? LY ? GZP XNP ? BJ ? CH ? SG ? TP ? TPP XNP ? ZC ? LC ? LCP XNP ? ZC ? LC ? SJ ? ST ? JLP XNP ? ZC ? XT ? XH ? XTP XNP ? ZC ? KY ? GBP XNP ? ZC ? KY ? GQ ? MZABP XNP ? ZC ? KY ? GQ ? MZCP XNP ? ZC ? KY ? HYBP XNP ? ZC ? KY ? HYCP XNP ? ZC ? KY ? HPAP XNP ? ZC ? TX ? BS ? HPBP XNP ? ZC ? TX ? YCP XNP ? ZC ? TX ? CS ? XZ ? STP XNP ? ZC ? TX ? TC ? PY ? LHS ? LHSP XNP ? ZC ? TX ? TC ? PY ? ZJP XNP ? ZC ? TX ? TC ? PY ? SHJ ? LJP XNP ? ZC ? TX ? TC ? PY ? DY ? HDZ ? NSP XNP ? ZC ? TX ? TC ? PY ? DY ? YF ? MSP

In summary, the proposed black-start decision-making method based on interval values can deal with different kinds of uncertainties in the actual black-start decision-making process. The integral black-start decision-making steps using the new black-start decision-making method are summarized as follows: (1) The set of candidate black-start schemes is determined after relevant searching, simulation and verification, and can be represented as A = {a1, a2, . . ., an}. An optimal black-start scheme will be selected from them based on the proposed decision-making method. (2) According to the experience of experts and the system operating condition, the evaluation index set C = {c1, c2, . . ., cm} that can best reflect the merits or shortcomings of blackstart schemes is determined. The weight vector of all m ~ 1; x ~ 2; . . . ; x ~ m T is given based on the knowl~ ¼ ½x indexes x edge and practical experience of black-start experts. ~ ¼ ð~ (3) The black-start decision-making matrix X xij Þnm is determined based on the system operating condition, and ~ xij ¼ ½xLij ; xUij  represents the interval value of the jth index related to the ith scheme.

39

H. Wang et al. / Electrical Power and Energy Systems 79 (2016) 34–41 Table 2 Index values on interval numbers used to evaluate the black-start schemes. Scheme No.

Plant

Rating capacity (MW)

Unit state

Ramping ratio (MW/h)

Startup power (MW)

Number of switching operation

[0.36 0.44] [0.387 0.473] [3.15 3.85] [0.09 0.11] [0.045 0.055] [0.18 0.22] [0.135 0.165] [0.135 0.165] [0.047 0.057] [0.63 0.77] [2.7 3.3] [7.2 8.8] [6.3 7.7] [8.1 9.9] [3.6 4.4] [0.72 0.88] [0.09 0.11] [13.5 16.5] [0.18 0.22] [0.225 0.275] [0.81 0.99]

4

a1

BHP

25.0

[2 3]

a2

XCP

192.0

[4 5]

a3

GZP

255.0

[4 5]

a4

TPP

51.0

[1 2]

a5

LCP

25.0

[1 2]

[10.125 12.375] [69.12 84.48] [91.8 112.2] [22.5 27.5] [9 11]

a6

JLP

88.0

[4 5]

[18 22]

a7

XTP

60.0

[2 3]

a8

GBP

47.5

[2 3]

a9

MZABP

38.0

[6 7]

a10

MZCP

138.5

[8 9]

a11

HYBP

150.0

[6 7]

[13.5 16.5] [21.375 26.125] [10.305 12.595] [37.35 45.65] [27 33]

a12

HYCP

420.0

[8 9]

a13

HPAP

500.0

[4 5]

a14

HPBP

600.0

[4 5]

a15

YCP

100.0

[8 9]

a16

STP

55.0

[8 9]

a17

LHSP

107.5

[8 9]

a18

ZJP

600.0

[7 8]

[12.375 15.125] [29.025 35.475] [81 99]

a19

LJP

84.0

[1 2]

[18 22]

a20

NSP

115.5

[4 5]

a21

MSP

103.0

[2 3]

[36.45 44.55] [13.905 16.995]

[75.6 92.4] [90 110] [108 132] [36 44]

3 3 5 3 5 4 3 4 4 3 3 3 4 3 5 6 5 6 7 7

Table 3 The interval values of indexes’ weights given by black-start decision-making experts. Indexes

c1

c2

c3

c4

c5

Weights

[0.18 0.22]

[0.27 0.33]

[0.09 0.11]

[0.13 0.17]

[0.225 0.275]

~ ¼ ð~ (4) The initial black-start decision-making matrix X xij Þnm is ~ ~ij Þnm by Eqs. (3) and (4), based on the normalized into Y ¼ ðy weighted sum model and error propagation theory. (5) The linear goal programming model expressed by Eq. (6) is established to find the ideal weight vector

x ¼ ðx1 ; x2 ; . . . ; xm ÞT based on the given weight vector ~ 1; x ~ 2; . . . ; x ~ m T . ~ ¼ ½x with interval values x

Table 4 The evaluation values, sorting values and ranking results of all candidate black-start schemes. Scheme No.

~ Evaluation value d i

~Þ Sorting value Uðd i

Ranking order

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 a21

[0.0229 [0.0476 [0.0504 [0.0313 [0.0489 [0.0347 [0.0310 [0.0349 [0.0608 [0.0490 [0.0441 [0.0687 [0.0617 [0.0651 [0.0488 [0.0396 [0.0554 [0.0691 [0.0236 [0.0343 [0.0206

0.0246 0.0504 0.0533 0.0338 0.0525 0.0370 0.0333 0.0373 0.0648 0.0517 0.0463 0.0722 0.0650 0.0687 0.0514 0.0418 0.0589 0.0727 0.0255 0.0367 0.0221

20 11 7 17 8 15 18 14 5 9 12 2 4 3 10 13 6 1 19 16 21

0.0271] 0.0532] 0.0558] 0.0381] 0.0577] 0.0399] 0.0366] 0.0407] 0.0695] 0.0536] 0.0478] 0.0738] 0.0671] 0.0709] 0.0529] 0.0435] 0.0624] 0.0743] 0.0288] 0.0397] 0.0244]

~1 Þ; Uðd ~2 Þ; . . . ; Uðd ~n Þ) for all black(9) The sorting values (i.e. Uðd start schemes are determined by Eq. (9) based on the risk attitude factor e given by power system operators or power experts. Then, the sorting result can be obtained and the optimal black-start scheme determined. Case studies To demonstrate the feasibility and effectiveness of the developed black-start decision-making method based on interval values, a simplified power system as shown in Fig. 1, representing a part of the Guangdong power system in south China, is employed for case studies. Suppose that the black-start unit is placed in Plant XNP of the system and has already restarted after a blackout. The power system topology database is first searched automatically by the breadth-first search philosophy according to the location of the black-start unit, and all possible black-start schemes are generated automatically. Then, relevant simulation and analysis software packages are employed to compute and verify all the initially determined black-start schemes, on aspects including unit selfexcitation checking, no-load switching overvoltage checking, power frequency overvoltage checking, frequency checking, and low frequency oscillation checking. In this way, 21 technically feasible schemes are attained and listed in Table 1. The black-start unit located in Plant XNP can provide the start-up power supply to 21 non-black-start units. The 21 candidate black-start schemes are represented by A = {a1, a2, . . ., a21}; the set of evaluation indexes selected is represented by C = {c1, c2,. . ., c5}, including the rating capacity of each unit, the unit state, the ramping ratio of each unit, the start-up power required, and the number of switching operations along the restoration paths. More detailed data are listed in Table 2. The interval values of indexes’ weights given by black-start decision-making experts based on their knowledge and experience are listed in Table 3. The risk attitude factor e is specified to be 0.3 in this case.

~1 ; d ~2 ; . . . ; d ~n ) for all can(6) The interval evaluation values (i.e. d ~ didate black-start schemes can be determined by x⁄ and Y

The interval decision-making matrix and its normalization

based on Eq. (7). (7) The risk attitude factor e is specified by power experts or power system operators. ~ Þ of the ith black-start (8) The interval mapping function ue ðd i scheme is determined by Eq. (8).

It can be seen from Table 2 that the values of indexes c1 and c5 are deterministic, but they can be regarded as special interval numbers with equal upper and lower bounds. The interval midpoints of them are constant and limit errors are 0. Thus, the values of indexes c1 and c5 can be handled as same as the interval numbers.

40

H. Wang et al. / Electrical Power and Energy Systems 79 (2016) 34–41

Table 5 The impacts of various risk attitude factors on the sorting results of candidate black-start schemes. No.

~ Evaluation value d i

e = 0.5 Sorting value

Ranking order

Sorting value

Ranking order

Sorting value

Ranking order

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11 a12 a13 a14 a15 a16 a17 a18 a19 a20 a21

[0.0229 [0.0476 [0.0504 [0.0313 [0.0489 [0.0347 [0.0310 [0.0349 [0.0608 [0.0490 [0.0441 [0.0687 [0.0617 [0.0651 [0.0488 [0.0396 [0.0554 [0.0691 [0.0236 [0.0343 [0.0206

0.0243 0.0505 0.0535 0.0332 0.0519 0.0368 0.0329 0.0370 0.0646 0.0520 0.0466 0.0729 0.0655 0.0691 0.0518 0.0420 0.0588 0.0734 0.0250 0.0364 0.0218

20 11 7 17 9 15 18 14 5 8 12 2 4 3 10 13 6 1 19 16 21

0.0246 0.0504 0.0533 0.0338 0.0525 0.0370 0.0333 0.0373 0.0648 0.0517 0.0463 0.0722 0.0650 0.0687 0.0514 0.0418 0.0589 0.0727 0.0255 0.0367 0.0221

20 11 7 17 8 15 18 14 5 9 12 2 4 3 10 13 6 1 19 16 21

0.0249 0.0504 0.0532 0.0344 0.0530 0.0372 0.0336 0.0376 0.0650 0.0514 0.0460 0.0716 0.0646 0.0682 0.0510 0.0417 0.0589 0.0720 0.0260 0.0369 0.0224

20 11 7 17 8 15 18 14 4 9 12 2 5 3 10 13 6 1 19 16 21

0.0271] 0.0532] 0.0558] 0.0381] 0.0577] 0.0399] 0.0366] 0.0407] 0.0695] 0.0536] 0.0478] 0.0738] 0.0671] 0.0709] 0.0529] 0.0435] 0.0624] 0.0743] 0.0288] 0.0397] 0.0244]

e = 0.3

First, the interval midpoint xij and limit error Dxij of ~ xij can be ~ ¼ ð~ computed by the interval decision-making matrix X xij Þ215 as determined based on the power system operating state at that ~ ¼ ð~ time. Then, the decision-making matrix X xij Þ215 can be normalized based on the error propagation theory. In this example, c1, c2 and c3 are benefit-type indexes, while c4 and c5 are costtype ones. The normalized midpoint matrix Y ¼ ðyij Þ215 and limit error matrix DY ¼ ðDyij Þ215 can be determined by Eqs. (3) and (4), ~ respectively. Afterward, the normalized decision-making matrix Y can be obtained from Y and DY. Determination of the ideal weight vector based on the goal programming model The calculation of the evaluation values of all candidate black-start schemes is based on the known weights of all indexes. Since the weights are given as interval values in this case, then the linear goal programming model is applied to obtain the ideal weight vector x ¼ ðx1 ; x2 ; . . . ; x5 ÞT . Set pi and qi (i = 1, 2, . . ., 21) as 1 with no preference, and the linear goal programming model based on Eq. (6) is built and the optimization toolbox in MATLAB is employed to solve. The ideal weight vector such obtained is x⁄ = [0.18, 0.33, 0.11, 0.155, 0.225]T. Then, the interval evaluation ~1 ; d ~2 ; . . . ; d ~21 ) for all the black-start schemes can be values (i.e. d obtained based on Eq. (7), and are shown in Table 4. The risk attitude factor based method to sort black-start schemes

e = 0.1

Hence, when the black-start decision-making method based on interval values is employed, the 18th scheme is the most preferred one. Discussions Considering the importance of power system security, power system operators or experts can be generally deemed as the conservative type. Thus, the risk attitude factor e in the black-start decision-making can be assigned as 0.5 6 e < 0. However, the sorting results of interval values of candidate black-start schemes depend on the risk attitude factor to a certain extent. As a result, the impacts of different values of e on the sorting results or the sensitivity of the sorting results to the values of e should be discussed. In the above case studies, the risk attitude factor e is specified as 0.3. Now, e is specified as 0.5 and 0.1 for examining the impacts of different values of e on the sorting results. According to Eqs. (8) and (9), the sensitivity analysis of the sorting results to different values of the risk attitude factor is shown in Table 5. It can be seen from Table 5 that different values of the risk attitude factor do not have significant impacts on the sorting results of candidate black-start schemes. The reason might be that the inter~1 ; d ~2 ; . . . ; d ~21 are not close with each other val evaluation values d and the intersections between interval values are not large in this test case. However, the ranking order of some candidate black-start schemes changes with the risk attitude factor. The ranking of the 5th scheme and that of the 9th scheme change with the variations of e. The similarity of these two schemes is that both of them have ~ . Therefore, it can be a large width of interval evaluation values d i

It can be seen from Table 4 that there is a high degree of intersection among the evaluation values in the candidate black-start ~2 and d ~5 . The risk attitude factor e is specified schemes, such as d as 0.3. Then, the sorting values of all the candidate black-start ~1 Þ; Uðd ~2 Þ; . . . ; Uðd ~21 Þ) can be obtained, and the final schemes (i.e. Uðd

black-start decision-making results are shown in Table 4. It can be seen from Table 4 that the final preferred sequence of the candidate black-start schemes is

a18 > a12 > a14 > a13 > a9 > a17 > a3 > a5 > a10 > a15 > a2 > a11 > a16 > a8 > a6 > a20 > a4 > a7 > a19 > a1 > a21

concluded that the evaluation values with a larger interval width are more likely to be affected by the change of the risk attitude factor. On the whole, the sensitive of the final sorting results to the risk attitude factor is relatively small in the test case. Moreover, in order to demonstrate the validity of the proposed black-start decision-making approach based on interval values, the sorting results of the same instance using three existing black-start decision-making methods are carried out for comparisons. Both the black-start decision-making method based on traditional subjective weights and the method based on entropy weights in [16] are chosen to compare the impacts of the determination of indexes’ weights on the sorting results. The vague set based method

H. Wang et al. / Electrical Power and Energy Systems 79 (2016) 34–41

Table 6 Sorting results of different black-start decision-making methods for comparisons. Ranking order

Method based on interval values

Method based on traditional subjective weights

Method based on entropy weights

Method based on vague set theory

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

a18 a12 a14 a13 a9 a17 a3 a5 a10 a15 a2 a11 a16 a8 a6 a20 a4 a7 a19 a1 a21

a18 a13 a12 a14 a3 a10 a17 a9 a2 a11 a8 a6 a16 a5 a15 a4 a7 a20 a19 a21 a1

a18 a14 a12 a13 a3 a10 a2 a15 a11 a17 a16 a9 a20 a6 a8 a7 a21 a1 a19 a4 a5

a3 a12 a13 a18 a14 a9 a2 a17 a10 a16 a4 a5 a6 a11 a15 a7 a8 a20 a1 a21 a19

proposed in [12] is also chosen to compare the effectiveness of different black-start decision-making methods. Comparisons of the sorting results using different decision-making methods are carried out and the results are shown in Table 6. It can be seen from Table 6 that the sorting results are not identical but follow the same trend. The schemes 18, 12, 14 and 13 are generally considered to be better schemes while the schemes 21, 1, 19 and 7 are considered to be poor ones. Different black-start decision-making methods may focus on different aspects in the black-start decision-making process, so it is inevitable that these approaches cannot completely reach an identical ranking order. However, if the sorting results of one approach are significantly different from the others, then this approach cannot be considered as a good one. Furthermore, it can also be concluded that the proposed black-start decision-making approach based on interval values is effective and feasible. The decision-making method for black-start schemes based on traditional subjective weights mainly emphasizes the experience and knowledge of power experts, while the method based on entropy weights emphasizes the importance of objective data in actual power systems. The proposed decision-making method based on interval values represents a combination of subjective weights and objective weights, and the results such obtained are more reliable. The vague set theory based method can take the interactions among various indexes into account, while the proposed method based on interval values can take different kinds of uncertainties in the black-start process into account. As a result, the proposed black-start decision-making method based on interval values extends the methodologies associated with black-start decision-making and improves the reliability of black-start decision-making results. Concluding remarks In this work, the black-start decision-making matrix and indexes’ weights are described as interval values for the first time. A new approach for black-start decision-making based on interval values is presented to deal with different kinds of uncertainties in black-start process. First, the decision-making matrix with interval values is normalized by employing the error propagation theory. Then, a linear goal programming model is used to seek the ideal

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weight vector and the evaluation values of all candidate blackstart schemes can be obtained. A risk attitude factor based method is presented to sort the schemes. Finally, case studies are carried out for the Guangdong power system in south China to demonstrate the developed model and method. Simulation results of the proposed method and comparisons with different decisionmaking methods demonstrate that the developed interval-value based black-start decision-making method can deal with the uncertainty of the index values and indexes’ weights well and could better describe the actual decision-making process. Acknowledgments This work is jointly supported by National High Technology Research and Development Program of China (863 Program) (2015AA050202), National Natural Science Foundation of China (51377005), Zhejiang Key Science and Technology Innovation Group Program (2010R50004). References [1] Andersson G, Donalek P, Farmer R, et al. Causes of the 2003 major grid blackouts in North America and Europe, and recommended means to improve system dynamic performance. IEEE Trans Power Syst 2005;20(4):1922–8. [2] Makarov YV, Reshetov VI, Stroev A, et al. Blackout prevention in the United States, Europe, and Russia. Proc IEEE 2005;93(11):1942–55. [3] Lindenmeyer D, Dommel HW, Adibi MM. Power system restoration—a bibliographical survey. Int J Electr Power Energy Syst 2001;23(3):219–27. [4] Adibi MM. Power system restoration: methodologies and implementation strategies. New York: Wiley-IEEE Press; 2000. [5] Joglekar JJ, Nerkar YP. A different approach in system restoration with special consideration of islanding schemes. Int J Electr Power Energy Syst 2008;30 (9):519–24. [6] Carvalho PMS, Ferreira LAFM, Barruncho LMF. Optimization approach to dynamic restoration of distribution systems. Int J Electr Power Energy Syst 2007;29(3):222–9. [7] Liu Y, Gu XP. Skeleton-network reconfiguration based on topological characteristics of scale-free networks and discrete particle swarm optimization. IEEE Trans Power Syst 2007;22(3):1267–74. [8] Kirschen DS, Volkmann TL. Guiding a power system restoration with an expert system. IEEE Trans Power Syst 1991;6(2):558–66. [9] Adibi MM, Kafka LRJ, Milanicz DP. Expert system requirements for power system restoration. IEEE Trans Power Syst 1994;9(3):1592–600. [10] Nourizadeh S, Nezam Sarmadi SA, Karimi MJ, et al. Power system restoration planning based on wide area measurement system. Int J Electr Power Energy Syst 2012;43(1):526–30. [11] Islam S, Chowdhury N. A case-based Windows graphic package for the education and training of power system restoration. IEEE Trans Power Syst 2001;16(2):181–7. [12] Zeng SQ, Lin ZZ, Wen FS, et al. A new approach for power system black-start decision-making with vague set theory. Int J Electr Power Energy Syst 2012;34 (1):114–20. [13] Lee SJ, Lim SI, Ahn BS. Service restoration of primary distribution systems based on fuzzy evaluation of multi-criteria. IEEE Trans Power Syst 1998;13 (3):1156–63. [14] Kostic T, Cherkaoui R, Germond A, et al. Decision aid function for restoration of transmission power systems: conceptual design and real time considerations. IEEE Trans Power Syst 1998;13(3):923–9. [15] Liu WJ, Lin ZZ, Wen FS, et al. Intuitionistic fuzzy Choquet integral operatorbased approach for black-start decision-making. IET Gener Transm Distrib 2012;6(5):378–86. [16] Lin ZZ, Wen FS, Huang JS, et al. Evaluation of black-start schemes employing entropy weight based decision-making theory. J Energy Eng 2010;136 (2):42–9. [17] Weber EU, Blais AR, Betz NE. A domain-specific risk-attitude scale: measuring risk perceptions and risk behaviors. J Behav Decis Making 2002;15(4):263–90. [18] Verma SP, Andaverde J, Santoyo E. Application of the error propagation theory in estimates of static formation temperatures in geothermal and petroleum boreholes. Energy Convers Manage 2006;47(20):3659–71. [19] Feltes JW, Grande-Moran C. Black start studies for system restoration. In: Proceedings of IEEE power and energy society general meeting, Pittsburgh, Pennsylvania, USA; 2008. p. 1–8. [20] Goh CH, Tung YC, Cheng CH. A revised weighted sum decision model for robot selection. Comput Ind Eng 1996;30(2):193–9. [21] Moradijoz M, Parsa Moghaddam M, Haghifam MR, et al. A multi-objective optimization problem for allocating parking lots in a distribution network. Int J Electr Power Energy Syst 2013;46:115–22. [22] Pal BB, Moitra BN, Maulik U. A goal programming procedure for fuzzy multiobjective linear fractional programming problem. Fuzzy Sets Syst 2003;139 (2):395–405.