Capacitors dispatch for quasi minimum energy loss in distribution systems using a loop-analysis based method

Electrical Power and Energy Systems 32 (2010) 543–550

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

Capacitors dispatch for quasi minimum energy loss in distribution systems using a loop-analysis based method W.C. Wu *, B.M. Zhang, K.L. Lo Department of Electrical Engineering, Tsinghua University, Room III-120, West Main Building, Beijing 100084, PR China

a r t i c l e

i n f o

Article history: Received 4 August 2008 Received in revised form 29 September 2009 Accepted 6 November 2009

Keywords: Capacitor optimization Energy loss minimization Distribution systems Loop-analysis

a b s t r a c t The dispatch of capacitors in distribution systems for daily operation is investigated in this paper. The objective is to determine the next day operating schedule of the capacitors so as to minimize the total daily energy loss. A loop-analysis based analytic algorithm is developed in this paper to efficiently calculate the optimal settings of capacitor with time-varying load. Firstly, capacitors switching times are determined by a heuristic method. Then, the optimal settings of capacitors for all operating times are calculated by an iterative algorithm. Numerical simulations were done and the results show that the proposed algorithm has an approximately linear convergence and is efficient. Ó 2009 Elsevier Ltd. All rights reserved.

1. Introduction Capacitors have been widely employed in distribution systems for reactive power compensation to reduce power loss and to improve voltage profile. Optimal capacitor sizing is an integer-programming problem, and many algorithms have been proposed to solve this problem. However, the applicability of these algorithms to a practical system is formerly limited either due to suffering time-varying load or due to exhausting calculation burden. In a practical power system, the load demands vary continuously and hence a practical and efficient method for capacitor optimization should consider time variation of load demand. What time and how much to switch off/on capacitors so as to minimize whole day energy loss should be researched. This is a complex non-linear integer-programming problem. Two aspects are concerned. First is to consider what time to switch so that daily capacitors switching times satisfy the maximum allowable daily switching frequency (MADSF). Second is to consider how much to switch so that a temporal–spatial optimization result is obtained. This is a temporal–spatial integer non-linear programming problem and no efficient solution method is available. In order to reduce calculation burden, simplified models and heuristic methods were proposed to achieve a compromised optimum result. A dynamic programming based approach was proposed to produce an optimal capacitor dispatching schedule in which the operating * Corresponding author. E-mail address: [email protected] (W.C. Wu). 0142-0615/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2009.11.010

times of every capacitor is defined as state variables [1]. Lu and Hsu [2] applied heuristic rules to choose only limited preferable state variables for each hour to reduce the whole solution space. However, the computation time increases exponentially with the increasing of the number of switched capacitors. It is difficult for these methods to apply to a practical large-scale system due to exhausting computing burden. In 2001, another dynamic programming based approach was developed to dispatch capacitors and TCUL of main transformer [3]. The generic algorithm is used to optimize capacitors with fuzzy multi-objective is proposed in Ref. [4]. An improved tabu search approach with heuristic rules was developed for optimal capacitor placement in Ref. [5]. However, these methods are also computationally time consuming. Ref. [6] hypothesized that the more intensely the load demand varies, the greater the demand for the dispatching of capacitors would be. Based on this assumption, a heuristic rule was proposed to optimize capacitor switching time. In actual practice, this is not always true. Recently, a multi-objective model and a TS-based approach were proposed to provide decision support in the capacitor location problem [7]. And in Ref. [8,9], an ant direction hybrid differential evolution combining with integer-programming method was proposed to solve large capacitor placement problems in distribution systems. But, these two approaches focus on off-line planning and are not suitable for real-time application. In Ref. [13], a decomposition method was developed to optimize capacitor operation considering substation voltage harmonics in radial distribution system. In our previous study, a loop-analysis method was proposed for the solution of optimal distribution network capacitors operation

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[8]. It was a static optimization method with a constant load model and is not suitable for real-time application. In this paper, this method is further extended to include a time-varying load model by solving a dynamic optimization problem. This improved algorithm is particularly suitable for use in the analysis of radial or weakly meshed distribution systems. The numerical tests show the convergence character of this algorithm is nearly linear and efficient.

g1 g2 g3

g5

2. A loop-analysis based method for minimal power loss

g4

To make this paper more readable, some basic definitions of loop-analysis method in Ref. [8] are briefly reviewed in this section.

Fig. 1. A network with a tree and a link.

2.1. Loop-analysis based method For a distribution network with N nodes, b branches and m loops, a source node numbered as N is specified as the root node, we have n = N  1 and b = n + m. If the network is radial (that is, m = 0), then, in this special case, b = n. A ‘path’ is an important concept in loop-analysis. Any node in the network can always be traced by a single path to a root node. The ‘path’ for a node can be defined as the set of branches along this route. In order to understand the loop-analysis method, some incident matrices describing the topology of the distribution network are defined as follows: T is the node-path incident matrix of n  b, which is known as the path matrix. T(i, j) = ±1 if branch j is on the path of node i, otherwise T(i, j) = 0. The sign ‘+’ indicates that the reference direction of branch j are the same as that of path i, otherwise the sign ‘’ is used. B is the loop-branch incident matrix of m  b, which is known as the loop matrix. B(i, j) = ±1 if branch j is on the ith loop, otherwise B(i, j) = 0. The sign ‘+’ indicates that the reference direction of branch j are the same as that of loop i, otherwise ‘’ is used. Ig is the node injection current vector of n  1. Ib is the branch current flow vector of b  1. For a connected graph, starting from a leaf node with degree 1, we number the nodes in the network by using Tinney 2 ordering method. All radial tree branches can be ordered at this step. Then at the second step, we use Width First Search (WFS) method to identify tree and link branches in the remaining network graph. The direction of a branch is defined as from a small number terminal node to a large number terminal node. The branch number is taken as the starting node number of the branch. Along this tree, a path starting from any node to the root node can be determined and a loop can also be determined by closing a link on the tree. Thanks to this kind of node ordering rule and definition of branch direction above, the non-zeros in matrix T are all 1 and centralize to upper triangular part in T. Direction of link branch is defined as the direction of loop. An example to explain this principle is given in Fig. 1 and corresponding matrices are given in formula (1). In Fig. 1, the path of node 1 includes branches 1, 3 and 5 and the path of node 2 includes branches 2, 4 and 5. Branch 6 is a link branch which is closing the single loop consisting of branches 3, 4 and 6 in the network. Eqs. (1) and (2) are the corresponding matrices of the network in Fig. 1, Tt, Bt and Tl, Bl are sub-matrices of T and B respectively corresponding to tree and link branches.

Because a path is defined as a set of tree branches, the link branch sub-matrix Tl in Eq. (1) is always zero. Based on the definition of branch direction given above, it can be seen that Tt in Eq. (1) is a lower triangular matrix with the value of 1 in the appropriate positions.

r s T¼ t u v

B¼r

.. 21 2 3 4 5 . 63 . 1 1 .. 7 61 6 . 7 6 1 1 1 .. 7 7 6 6 .. .. 7 7 6 6 1 1 . 7 ¼ ½T t .T l  7 6 .. 7 6 7 6 1 1 . 5 4 .. 1 .

h1 2 3 4 5 6i . ...1 ¼ ½Bt ..Bl  1 1

ð1Þ

ð2Þ

Node current injections in Fig. 1 can be denoted by the n  1 vector Ig. From Kirchhoff’s Current Law, branch current vector Ib of b  1 can be expressed by,

I b ¼ T T I g þ BT I L

ð3Þ

where IL is the loop current vector of m  1 (loop current is equivalent to link branch current in a weakly meshed distribution network). It can be seen from Eq. (3) that the branch current Ib is the joint contribution of the node current injection Ig and the loop current IL. From Ohm’s law,

V b ¼ Zb I b

ð4Þ

where Vb is the branch voltage vector of b  1, Zb is a diagonal matrix of branch impedances. And from Kirchhoff’s Voltage Law,

BV b ¼ 0

ð5Þ

Combining Eqs. (3)–(5) yields:

BZ b BT I L þ BZ b T T I g ¼ 0 Defining a loop impedance matrix of m  m as

Z L ¼ BZ b BT

ð6Þ

Loop current IL can be calculated by T I L ¼ Z 1 L BZ b T I g

ð7Þ

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Substituting Eq. (7) into Eq. (3) for branch current Ib, then T I b ¼ ðT T  BT Z 1 L BZ b T ÞI g

3. Formulation of optimal capacitor daily schedule

ð8Þ

2.2. An algorithm to optimize capacitor sizing for minimum power loss According to Eq. (8), active power loss Ploss of a distribution system with m switchable capacitors, b branches and n nodes can be formulated as:

Ploss ¼

b X

Rbi jI bi j2 ¼

i¼1

b X

Rbi jI 0bi þ

i¼1

m X

Eij DI gj j2

3.2. Formulation of optimization model

j¼1

where Ibi is the current of branch i, Rbi the resistance of branch i, the original current of branch i, Eij the element of matrix E, where T in accordance to Eq. (8), DI gj is the current E ¼ T T  BT Z 1 L BZ b T variation node j. If the injected reactive power at node i is changed by an increment DQi to minimize the real power losses through switching on a capacitor connecting to node i, then the capacitor will produce an increment in node current injection DIgi. If voltage at node i is V i ¼ V i \hi , then the DIgi caused by DQi can be expressed as:

DI gi ¼

jDQ i Vi



  DQ  ¼  i \hi  90 ¼ jDI gi j\hi  90 Vi

Thus, the phase angle of node current injection DIgi sented by the voltage phase angle hi. The increment tive power injection would slightly change the phase angle, and an approximate formula is used to power loss Ploss defined in Eq. (9):

Mb ¼ J

ð10Þ can be reprein node-reacnode-voltage minimize the

ð11Þ

Here, b is a p  1 vector of the magnitude of matching injection flow jDI gi j for the m nodes connected with capacitors.

2

M 11

6 M ¼ 4 ... M p1

3 J1 6 . 7 7 7 . . . 5 and J ¼ 6 4 .. 5 Mpp Jp

. . . M 1p ... ...

2

3

b X i¼1

Jj ¼

b X

hjk 0

Rbi Eij Eik cos hjk 

b X

Rbi Eij Eik

ð12Þ

i¼1

  0i Rbi Eij I 0r bi sin hj  I bi cos hj

Due to the limitation of a capacitor’s life span, the number of daily switching times should not exceed its maximum allowable daily switching frequency (MADSF). The variable NS is used here to represent MADSF. Considering a distribution system with m switchable capacitors, the capacitor dispatch problem can be mathematically formulated as the following non-linear optimization problem:

8 N P > > min Ploss ðQ jc1 ; Q jc2 ; . . . ; Q jcm ÞDT > > > j¼1 > > > < s:t: V min 6 V 6 V max > I 6 I b max > > > > > N 6 N > i S > : j Q ci ¼ k  U si ; j ¼ 1; 2 . . . ; N; i ¼ 1; 2; . . . ; m

ð14Þ

where N is the number of load segments for 1 day, DT the time interval of each segment, Vmin 6 V 6 Vmax the voltage magnitude the branch current limit constraints, Ni the total constraint, I 6 Imax b switching number for 1 day of capacitor i, Q jci the capacity of capacitor i at load level j, Q jci ¼ 0 means that capacitor i is switched off at load level j, Usi the size (kVAR) of the ith capacitor bank and K is an integer number. This is a mixed integer-programming problem, which can be decomposed into a master problem and a slave sub-problem. The master problem is to schedule capacitors’ switching times, while the slave sub-problem is to determine capacitors setting values at each switching time. 4. A heuristic method to schedule switching times

where

M jk ¼

Proper dispatch of capacitors considering a time-varying load demand can significantly reduce total daily energy loss. Since the continuously changing load demand curve cannot be directly used in the dynamic optimization, the daily load demand curve is approximated into a number of load segments, and within each segment, the demand level is kept constant [6].

ð9Þ I 0bi



3.1. Time-varying load model

ð13Þ

i¼1 0i I 0r bi and I bi are respectively the real and imaginary parts of the original current I 0bi of branch i. The magnitude of matching injection flow b is obtained by solving Eq. (11), which is used to adjust the control variables to improve the initial solution. Based on Eq. (11), an iterative algorithm is developed to determine the optimal capacitor sizing for a specified operating time [10]. Please see the details in paper [10]. This algorithm is a static optimization technique with a constant load model and is not suitable for real-time application. In the following paragraphs an improved method is developed to reduce total daily energy loss, which uses a time-varying load model.

In this section, a heuristic method is proposed to solve the master problem. According to the algorithm presented in Ref. [10], the optimal matching injected reactive power for minimal power loss can be efficiently calculated for a specific load level. Thus, it is convenient to form the optimal matching injected reactive power curve of each node that a capacitor is connected, as shown in Fig. 2A. Q ji represents the optimal matching injected reactive power of the ith capacitor at time segment j, and then the variation of optimal matching reactive power between the two Q ji  Q j1 i

Q 4 3

Q

8

5 6 7

1 2 3

2 1

8 4 5 6 7

t

t

A.Original curve of optimal

B.The merge result curve of

matching reactive power

optimal matching reactive power

Fig. 2. An example of piecewise curve merging.

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adjacent time segments j and j  1 is the ith capacitor’s switching capacity. Due to the limitation of maximum allowable daily switching frequency (MADSF), it is important to optimize their switching times. A heuristic algorithm is presented below: (1) Set n = 24. (2) One day is divided into n segments, and the optimal matching reactive power curve for every capacitor is calculated according to the algorithm presented in Ref. [10]. (3) For i = 1, 2, . . . , n. of optimal matching Find the smallest variation Q ji  Q j1 i reactive power of capacitor i between adjacent time segments j and j  1, and then merge the time segment j and j  1 into one. The following equation is used to calculate of the new time the optimal matching reactive power Q j1 i segment.

Q j1 ¼ i

5. Optimize capacitors’ optimal setting for minimum energy loss Once the daily switching times of capacitors are determined, an iterative algorithm is to be developed to optimize capacitors’ setting values at all time segments. 5.1. General model To optimal matching injected flow for minimum power loss is not practical because load demands in distribution systems vary continuously with time. A more practical approach is needed to match injected flow for minimum energy loss. To achieve this goal, a novel algorithm is developed in this section. According to Eq. (9), the system power loss at time t is:

i¼1

Rbi jI t0 bi þ

m X

Eij DI tgj j2

24 P

¼

DT t Ptloss

DT t

t¼1

b P i¼1

Rbi jI t0 bi

þ

m P j¼1

!

ð16Þ

Eij DDI tgj j2

From Eq. (10), if the voltage angle of node j, to which a capacitor is connected, is htj , then t

DI tgj ¼ ij ðsin htj  j cos htj Þ

ð17Þ

t

where ij is the magnitude of DI tgj . Substitute Eq. (17) into Eq. (16), and after rearrangement, it becomes

W loss ¼

24 X

2

DT t 4

t¼1

b X i¼1

0 Rbi @ I 0rt bi 

m X

!2 t Eij ij

cos htj

j¼1

!2 1 3 m X t 0it t Eij ij sin hj A5 þ I bi þ

ð18Þ

j¼1

Fig. 2A is the original optimal matching injected reactive power at each time segment and Fig. 2B is the result of segments merging using the described method. If the capacitor’s MADSF is two per day, the start of time segments 4 and 8 are selected to switch the capacitors. The method proposed in Ref. [6] hypothesized that the more acutely the load demand varies, the greater would be the demand for dispatching of capacitors, in which case time segments 7 and 8 will be selected. But from the curves shown in Fig. 2, the former solution is obviously more reasonable. However, it is not possible to guarantee that the results of the optimal solution obtained by our proposed approach are always reasonable.

b X

24 P t¼1

j Q j1 i DT j1 þ Q i DT j DT j1 þ DT j

And DTj1 = DTj1 + DTj, DTk = DTk+1, k = j, j + 1, . . . ,n  1, where DTj1 and DTj are the time intervals of segment j  1 and j respectively. (4) n = n  1; if n > MADSF, then go to (3), otherwise stop.

Ptloss ¼

W loss ¼

ð15Þ

j¼1

where DI tgj is the current variation caused by the switching on/off capacitor connecting to node j at time t. It is assumed that 1 day can be divided equally into 24 time segments (DTi represents the time interval of segment i, i = 1, 2, . . . ,24), and the load at each time segment is kept constant. Thus, the total daily energy loss Wloss is:

t0i where I t0r bi ; I bi are the real and imaginary parts of the original curt0 rent I bi of branch i at time t.

5.2. Matching injected flow for minimum energy loss If capacitor j is switched on/off N j times in 1 day, then T(j, 1), T(j, 2), . . . ,T(j, Nj) represent the time at which a capacitor is respectively switched on/off. Hence, T(j, i) represents the time of capacitor j at the ith switching. It is assumed that the current from each capacitor is constant at N 1 2 each time segment, and ij ; ij ; . . . ; ij j represent the current magnitudes of capacitor j at Nj time segments respectively. It is worth noting that the necessary condition for minimum Wloss is given below,

@W loss @W loss þ ¼0 @I @h h

ð19Þ iT

where I ¼ i11 . . . iN1 1 ; . . . i1m . . . iNmm is the magnitude of the corresponding current vector of capacitors at all time segments. And h is the voltage angle vector of nodes to which capacitors are connected (ai = hi  90°, and ai is the phase angle of the corresponding current of capacitor). In Eq. (19), h are state variables which vary with the variation of the value of I, and therefore it cannot be controlled directly. The only control variable in this equation is I which determines the setting values of capacitors in this problem. So the second term can be ignored, and the necessary condition to minimize Wloss is simplified as:

@W loss @W loss ðIÞ @W loss ðhÞ @h þ  ¼0 ¼ @I @h @I @I

ð20Þ

Because the variation of the injected reactive power has only a small effect on the phase angle h of node voltage, thus

@h 0 @I @W loss ðIÞ ¼0 @I

ð21Þ

Substituting Eq. (18) into Eq. (21) and rearranging yields:

Wg ¼ C

ð22Þ Pm

where g ¼ ½ g11 . . . gN1 1 ; . . . g1m . . . gNmm T is a i¼1 N i  1 vector of the magnitude of matching injected flow at all load levels for minimum daily energy loss, and

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W.C. Wu et al. / Electrical Power and Energy Systems 32 (2010) 543–550

 C ¼ C11

. . . CN1 1

. . . C1m

. . . CNmm

T

where

8   Tði;kþ1Þ1 b > 0it t t < C k ¼ P DT P E R I 0rt t ji bj i bj sin hi  I bj cos hi j¼1 t¼Tði;kÞ > : i ¼ 1; 2; . . . ; m; k ¼ 1; 2; ::; N i

ð23Þ

and

2

W11 11

6 6 0 6 6 6 0 6 6 W ¼ 6 ... 6 6 11 6 Wm1 6 6 4 ...

0

0

...

W11 1m

...

m W1N 1m

...

0

...

...:

...

... N 1 Nm 1m

N1 N1 11

... W

...

...

...

...

...

...

...

...

1 W1N m1

...

W11 mm

0

0

...

...

...

0

...

0

...

0

0

m Nm WNmm

0

W

N1 1 1m

WNm1m 1 . . . WNm1m N1

W

(3) From notes (1) and (2), we know that elements of W and C can be calculated from M and J at different load levels respectively. Thus, the main calculation process is similar to the algorithm for minimum power loss presented in Ref. [10]. (4) As a result of neglecting the second term h in the partial derivative condition of Eq. (19), the solution obtained from Eq. (22) is an approximate one. The algorithm developed in the next section is able to yield a more precise solution.

3 7 7 7 7 7 7 7 7 7 7 7 7 7 5

iteration counter k=1 Perform a full distribution-load flow to obtain the branch currents and node voltages at

η (k )

Calculate the

η (k ) = −

(k )

the injected reactive power at

where

8 Tði;lþ1Þ1 b P P > > > Wllii ¼ DT t E2ki Rbk ; > > > t¼Tði;lÞ k¼1 > < l ¼ 1; 2; . . . ; N i ; i ¼ 1; 2; . . . ; m > > > l1 l2 > Wii ¼ 0; l1 ! ¼ l2 ; > > > : l1 ; l2 ¼ 1; 2; . . . ; Ni ; i ¼ 1; 2; . . . ; m 8 li lj > TP > b u P > li lj > > Wij ¼ DT t Ekj Eki Rbk cos htij > > > li lj k¼1 > t¼T > > l > > > > < ht 0 T li lj b u P P ij ll ll  DT t Ekj Eki Rbk T ui j > T li j > > li lj k¼1 > > t¼T l > > > > > li lj ll ll > > W ¼ 0; T ui j 6 T li j > > > ij > : i; j ¼ 1; 2; . . . ; m

(k ) Wloss

(k ) ( k −1) fabs (Wloss − Wloss )<ξ

Y

ð24Þ

N

ΔY ( k )

η (k ) Y (k )

Y ð25Þ

(k )

Y ( k ) = Y ( k −1) + ΔY

(k )

Y Fig. 3. Flow diagram of the algorithm to calculate optimal capacitor’s initial setting values for minimum energy loss.

In Eq. (25), Use the method proposed in sectionIV to calculate the switching times for every capacitor

ll

T ui j ¼ minðTði; li þ 1Þ  1; Tðj; lj þ 1Þ  1Þ ll T li j

¼ maxðTði; li Þ; Tðj; lj ÞÞ

li ¼ 1; 2; . . . ; Ni ;

iteration counterk=1

lj ¼ 1; 2; . . . ; Nj

For the above derivation, the following four points are worth noting, Pm Pm (1) W is approximately a i¼1 N i  i¼1 N i constant matrix because htij is generally small. Its relation with M in Eq. (11) can be expressed by:

8 Tði;lþ1Þ1 P > ll > > DT t Mii > Wii ¼ > > t¼Tði;lÞ <

Cli ¼

ð26Þ ll

Yi j

(k )

ll

>> Yinstalli

i¼1 N i 1

| Yi j ( k ) |< ε , ∀ε > 0

Y

N

Yi

j (k)

<< − Yinstalli N

Pm

Tði;lþ1Þ1 X

|Yi j ( k ) |< Ys

T ui j > T li j

t¼T l

(2) C is a

Y

k=k+1

N

li lj

TP > u ll > > > Wiji j ¼ DT t M ij ; > > li lj :

Use the algorithmdescribed in section5.3 to calculate the optimal compensating admittance Y (k)

vector. Its relation with J in Eq. (11) is:

Y

switch on all the capacitor banks connecting to node i at time segment j

switch off all the capacitor banks connecting to node i at time segment j

switch on/off the capacitor banks whose capacity is nearest toY j ( k ) i

DT t Jti

ð27Þ

END

t¼Tði;lÞ

where Jti is the ith element of J of load level at time segment t.

Fig. 4. Flow diagram of the method for optimal capacitor dynamic sizing with timevarying load model.

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Table 1 Daily load values of the three-feeder system. Time

Node 4

5

6

7

8

9

10

11

12

13

14

15

16

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

0.75 0.72 0.83 0.83 0.83 0.87 0.88 0.89 0.89 1 1.22 1.3 1.23 1.2 1.1 1.03 1.1 1.14 1.15 1.14 1.1 1.01 0.9 0.8

0.8 0.8 0.8 0.8 0.8 0.9 0.9 0.9 0.9 1 1.2 1.3 1.2 1.2 1.1 1 1.1 1.1 1.2 1.1 1.1 1 0.9 0.8

0.74 0.78 0.87 0.81 0.83 0.87 0.88 0.87 0.91 1 1.24 1.3 1.23 1.2 1.1 1.03 1.1 1.14 1.15 1.14 1.1 1.01 0.9 0.8

0.8 0.8 0.8 0.8 0.9 0.9 0.9 0.9 0.9 1 1.2 1.3 1.2 1.2 1.1 1 1.1 1.1 1.2 1.1 1.1 1 0.9 0.8

0.8 0.7 0.9 0.8 0.8 0.9 0.9 0.9 0.9 1 1.3 1.3 1.2 1.2 1.1 1 1.1 1.1 1.2 1.1 1.1 1 0.9 0.8

0.65 0.75 0.82 0.82 0.83 0.87 0.88 0.91 0.92 1 1.2 1.3 1.23 1.2 1.1 1.03 1.1 1.14 1.15 1.14 1.1 1.01 0.9 0.8

0.75 0.73 0.82 0.81 0.87 0.87 0.88 0.88 0.93 1 1.26 1.3 1.23 1.2 1.1 1.03 1.1 1.14 1.15 1.14 1.1 1.01 0.9 0.8

0.72 0.75 0.83 0.81 0.83 0.87 0.88 0.89 0.94 1 1.2 1.3 1.23 1.21 1.1 1.03 1.1 1.14 1.15 1.14 1.1 1.03 0.9 0.8

0.75 0.75 0.82 0.87 0.88 0.87 0.88 0.88 0.95 1 1.2 1.3 1.23 1.2 1.1 1.03 1.1 1.14 1.15 1.14 1.1 1.03 0.9 0.8

0.76 0.76 0.86 0.81 0.83 0.87 0.88 0.85 0.92 1 1.26 1.3 1.23 1.2 1.1 1.03 1.1 1.14 1.15 1.14 1.1 1.03 0.9 0.8

0.79 0.73 0.82 0.86 0.83 0.87 0.88 0.88 1.2 1.1 1 1.1 1.1 1.1 1.01 1.03 1.1 1.14 1.15 1.14 1.1 1.03 0.9 0.8

0.74 0.79 0.8 0.81 0.88 0.83 0.88 0.88 1.3 1.2 1 1.1 1.1 1.1 1.08 1.03 1.1 1.14 1.15 1.14 1.1 1.03 0.9 0.8

0.5 0.8 0.8 0.8 0.8 0.9 0.9 0.9 1.2 1.1 1.1 1.1 1.1 1.1 1.1 1 1.1 1.1 1.2 1.1 1.1 1 0.9 0.8

5.3. Algorithm steps to obtain capacitors’ initial setting values When the daily switching times of capacitors are determined, the algorithm to obtain capacitors’ initial setting values at all time segments can be described as follows (see the flow diagram of the algorithm in Fig. 3). 5.4. The iterative steps to determine capacitors’ optimal setting for minimum energy loss Based on the algorithm presented in Section 5.3, the following algorithm steps are developed to dispatch capacitors for all time segments (see the flow diagram of the algorithm in Fig. 4): (1) Use the method proposed in Section 4 to calculate the switching times for every capacitor. (2) Set the iteration counter k = 1. (3) Use the algorithm described in Section 5.3 to calculate the optimal compensating admittance Y(k). jðkÞ jðkÞ (4) If jY i j < Y s orjY i j < e; 8e > 0 (‘‘i = 1, 2, . . . ,m” represent capacitors; ‘‘j = 1, 2, . . . ,Ci ” represent time segments), then go to END, otherwise go to the next step. Here, Ys indicates the equivalent admittance of one capacitor bank; m is the total number of switchable capacitors; and Ci is the total number of times that capacitor i is switched on/off in 1 day. (5) Three cases should be considered for i = 1, 2, . . . ,m; and j = 1, 2, . . . ,Ci. jðkÞ

Case 1: If Y i Y installi , then switch on all the capacitor banks connecting to node i at time segment j and exit the optimization procedure. Yinstalli is the total equivalent admittance of all banks installed at node i. jðkÞ jðkÞ Case 2: If jY i j Y installi and Y i < 0, then switch off all the capacitor banks connecting to node i at time segment j and exit the optimization procedure; Case 3: Otherwise, switch on/off the capacitor banks whose jðkÞ capacity is nearest to Y i ; k = k + 1, go to step (3). This method can calculate the optimal setting values of all capacitors at all time segments in a single calculation process.

Computational experience has indicated that the calculation process produces no oscillation and the convergence is usually smooth. 6. Numerical results Based on the proposed approach, a C++ program with threephase models is developed and tested. A three-feeder system [11] and a 69-node system [12] are used to validate the approach. Table 2 Some parameters of capacitors of the three-feeder system with small bank size. Capacitor no.

Node no.

Initial size (kVAR)

Max size (kVAR)

Bank’s size (kVAR)

5 6 9 11 12 14 16

5 6 9 11 12 14 16

1100 1200 1200 600 3700 1800 1800

3000 3000 5000 2400 4000 2500 3600

50 50 30 40 50 10 40

Table 3 Optimal results of capacitor dispatching under the constraint of three allowable switching (three-feeder system with small bank size). Capacitor no.

Switching time (h)

Banks switch on (number)

Capacitor

Switching time (h)

Banks switch on (number)

5

1st 10th 22nd

+13 +12 12

12

1st 10th 22nd

38 +12 13

6

1st 10th 22nd

+18 +15 14

14

1st 9th 22nd

81 +33 31

9

1st 10th 22nd

+102 +24 24

16

1st 9th 22nd

4 +14 13

11

1st 10th 22nd

13 +1 1

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Table 4 Some iterative results of optimal matching injected current g (Phase A) of capacitors at its first switching time.

g

Capacitor

5 6 9 11 12 14 16

Iter. 1 (PU)

Iter. 2 (PU)

Iter. 3 (PU)

0.21302 0.30540 1.02828 0.17310 0.64320 0.27235 0.05812

0.00234 0.00480 0.01662 0.00254 0.01043 0.00177 0.00117

0.00003 0.00007 0.00027 0.00004 0.00017 0.00001 0.00002

Table 5 Some parameters of capacitors of the three-feeder with a large bank size. Capacitor no.

Node no.

Initial size (kVAR)

Max size (kVAR)

Bank’s size (kVAR)

5 6 9 11 12 14

5 6 9 11 12 14

1500 1500 1500 800 4000 2000

3000 3000 5000 2400 4000 2500

500 500 500 400 500 200

the size of each bank of capacitor 11 is 40 kVAR. Thus, ‘‘13, +1, 1” for capacitor 11 in Table 3 means that capacitor 11 will switch off 13  40 kVAR at the 1st h, and then switch on 40 kVAR at the 10th h, and finally switch off 40 kVAR at the 22nd h. For the three-feeder system with a small bank size, part of the optimal matching injected current g (Phase A) of each capacitor at its first switching time is summarized in Table 4. The table also shows that the algorithm has an approximately linear convergence, since the absolute values of optimal matching injected current g (Phase A) of each capacitor show a significantly descending trend. Fig. 5 shows the relationship of MADSF to the total daily energy loss reduction of a three-feeder system with a small bank size. Fig. 6 shows the relationship of a three-feeder system with a larger bank size. Fig. 7 shows the results of the 69-node system illustrating the relationship of MADSF to the total daily energy loss reduction. These results indicate that the reduction in daily energy loss usually increases with the increase of MADSF. One salient point in the results, which should be noted, is that the relationships shown in Figs. 5–7 are almost monotonic increasing. This indicates that the algorithm has a good ability to search for the global optimum. Figs. 5–7 also shows that fewer MADSF value could yield the most benefit of daily energy loss reduction, which is evident in the first four switching of figures. For MADSF value greater than 5, the curve becomes relatively flat. This conforms to practical experience. To further illustrate the optimization results, the root mean square (RMS) of bus voltages in PU values before and after the

600

Reduced Energy Loss (kWH)

The parameters of the three-feeder system of Ref. [9] are not repeated here but those that have been changed are listed in Tables 1, 2 and 5 in this section. The daily load values of the three-feeder system are shown in Table 1. In this table, the values represent the ratio between the corresponding load values and their normal values respectively. The load normal values are cited from Ref. [9]. Some parameters of capacitors of the three-feeder system are listed in Tables 2 and 5 respectively. It is assumed that a capacitor is composed of several banks, and each bank has a fixed size and can be operated separately. The MADSF value is taken as 3, and the optimized results of capacitors daily operating times and their optimal setting values are listed in Tables 3 and 6 respectively. In these tables the values in columns 2 and 5 give the optimal switching times for capacitors in 1 day. For example, ‘‘1st, 10th, and 22nd” in Table 3 for capacitor 5 means that capacitor 5 will be operated at the 1st, 10th, and 22nd h of the 24 h clock. Columns 3 and 6 give the number of capacitor banks switched on/off at the corresponding switching times. The ‘negative sign’ means ‘switch off’ and ‘positive sign’ means ‘switch on’. For example, it can be seen from Table 2 that

580 560 540 520 500 480

1

3

5

7

9

11

13

15

17

19

21

23

Maximum allowable daily switching frequency Fig. 5. The relationship of energy loss reduction with maximal allowable switching frequency (three-feeder system with small bank size).

Table 6 Optimal results of capacitor dispatching under the constraint of three allowable switching (three-feeder system with a large bank size).

5

6

9

11

Switching time (h)

Banks switch on (number)

Capacitor no.

Switching time (h)

Banks switch on (number)

1st 10th 23rd

+1 +1 1

12

1st 11th 23rd

4 +1 1

1st 10th 23rd

+2 +1 1

14

1st 9th 23rd

5 +2 1

1st 11th 23rd

+7 0 0

16

2nd 9th 23rd

+1 +1 1

1st 11th 23rd

2 0 0

380

Reduced Energy Loss (kWH)

Capacitor no.

360 340 320 300 280 260 240

0

3

6

9

12

15

18

21

24

Maximum allowable switching frequency Fig. 6. The relationship of energy loss reduction with maximal allowable switching frequency (three-feeder system with a large capacitor bank size).

550

W.C. Wu et al. / Electrical Power and Energy Systems 32 (2010) 543–550

Reduced Energy Loss (kWH)

PU is shown in Fig. 9. From Fig. 9, we can see that the current on branches are reduced after capacitor dispatch. This means that capacitor dispatch can to some extent relieve the branch current violation. 7. Conclusion

RMS of bus voltages(P.U)

Fig. 7. The relationship of energy loss reduction with maximal allowable switching frequency (69-node system).

before dispatch

after dispatch

0.016 0.014 0.012 0.01 0.008 0.006 0.004 0.002 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

time Fig. 8. RMS of bus voltages in PU (three-feeder system).

before dispatch

after dispatch

sum of all branches’ current (PhaseA,P.U)

30 25 20 15 10 5 0

1 2 3 4 5 6 7 8 9 10 11 12 1314 15 16 17 18 19 20 21 22 23 24

time Fig. 9. Sum of all branches’ Phase A current magnitude in PU (three-feeder system).

capacitor dispatch for every operation are illustrated in Fig. 8. Fig. 8 shows that the voltages become more flat after capacitors dispatch. The sum of all branches’ Phase A current magnitude in

Capacitor dispatch optimization in daily operation is a very complex problem. Many methods have been developed, and most of them are very time consuming such as dynamic programming or soft computing algorithms. Based on a loop-analysis method, an efficient daily optimization approach for capacitor dispatch is proposed in this paper. This algorithm can calculate the optimal setting values of all capacitors at all time segments in one calculation process. Meanwhile, a practical heuristic method is also proposed to optimize capacitors’ daily operating time. All these algorithms are implemented in C++ program with three-phase models. The numerical tests show that the algorithm has an approximately linear convergence and has good ability to search for optimal solution. Acknowledgment This work was supported in part by National Science Foundation of China (50823001). References [1] Hus HY, Kou HC. Dispatch of capacitors on distribution system using dynamic programming. IEEE Proc – C 1993;140(6):433–8. [2] Lu Fengchang, Hsu Yuanyih. Fuzzy dynamic programming approach to reactive power/voltage control in a distribution substation. IEEE Trans Power Syst 1997;12(2):681–8. [3] Liang Ruey-Hsun, Cheng Chen-Kuo. Dispatch of main transformer ULTC and capacitors in a distribution system. IEEE Trans Power Syst 2001;16(4):625–30. [4] Das D. Optimal placement of capacitors in radial distribution system using a Fuzzy-GA method. Int Electric Power Energy Syst 2008;30(6):361–7. [5] Gallego Ramon A, Monticelli Alcir José, Romero Rubén. Optimal capacitor placement in radial distribution networks. IEEE Trans Power Syst 2001;16(4):630–7. [6] Deng Youman, Ren Xiaojuan, Zhao Changcheng. A heuristic and algorithmic combined approach for reactive power optimization with time-varying load demand in distribution systems. IEEE Trans Power Syst 2002;17(4): 1068–72. [7] Dulce Fernão Pires, António Gomes Martins, Carlos Henggeler Antunes. A multiobjective model for VAR planning in radial distribution networks based on tabu search. IEEE Trans Power Syst 2005;20(2):1089–94. [8] Chiou Ji-Pyng, Chang Chung-Fu, Su Ching-Tzong. Ant direction hybrid differential evolution for solving large capacitor placement problems. IEEE Trans Power Syst 2004;19(4):1794–800. [9] Chiou Ji-Pyng, Chang Chung-Fu, Su Ching-Tzong. Capacitor placement in largescale distribution systems using variable scaling hybrid differential evolution. Int Electric Power Energy Syst 2006;28(10):739–45. [10] Zhang BM, Wu WC. A loop-flow-based method for capacitor optimization in distribution systems. Control Eng Pract 2005;13(12):1545–52. [11] Civanlar S, Grainger JJ, Yin H, et al. Distribution feeder reconfiguration for loss reduction. IEEE Trans Power Deliv 1988;3(3):1217–23. [12] Baran ME, Wu FF. Optimal capacitor placement on radial distribution systems. IEEE Trans Power Deliv 1989;4(1):725–34. [13] Wu X, Lo KL. Optimal choice of fixed and switched capacitors in radial distributor with distorted substation voltage. IEE Proc, Genera Transm Distribut 1995;142:24–8.