21

O p t i m u m d is t r ibut ion feeder ba lancing using the ampere-mile vector theory

Equil ibrage de feeder a d is t r ibu t ion opt imale a I'aide de la theorie vector ie l le par mile-ampere

By Bama SzabadOS, MIEEE, Department of Electrical and Computer Engineering, McMaster University, Hamilton, Ontario.

A computer program implementing the novel optimization technique proposed for distribution feeder balancing is presented. The algorithm uses the concept of an adaptive look-ahead target. The choice of various cost functions allows optimization with seasonal variations. The results have been very successfully tested in the field, and the feeders kept their acceptable balance for years.

Un programme sur ordinateur est propose afin de resoudre les techniques realisant l'equilibrage des reseaux de distribution. L'algorithme est fonde sur un but predictif et adaptif. Le choix de plusieurs fonctions de cout permet d'optimiser les variations de charge saisonnieres. Les resultats ont ete verifies par la pratique et les reseaux ainsi equilibres ont maintenu un niveau satisfaisant d'equilibre durant plusieurs annees.

Introduction

With the ever-growing customer loads, the distribution system has moved into a critical phase when the distribution engineer has to operate an often overloaded system. Telephone interference by power lines is an acute concern, and since the customers of the power company are also customers of the telephone company, deterioration of service from either will generate complaints.

Many power utilities settle for the concept of a losing batt le, stating that * 'nothing can be d o n e ' ' . The following paper will sum­marize the joint efforts of the New Brunswick Electric Power Commission (NBEPC) and the New Brunswick Telephone Com­pany (NBTEL) to adequately resolve the dilemma. This work shows that it is indeed possible to configure a distribution system in such a way that neutral current unbalances resulting from single phase localized loads can be optimally compensated.

The Ampere-Mile-Vector Theory 1 is invoked, and a novel technique to balance feeders is proposed. Since the method calls for cumbersome interractive graphic representation, an optimizing algorithm is proposed. This algorithm is implemented on a small LSI11/03 mini-computer, and frees the engineer from time-con­suming vector computat ions. However, the ' ' o p t i m u m ' ' is user definable, and the program acts in a semi-interractive fashion, which leaves the final connection decision to the judgment of the engineer who can interpret the computer analysis. A compromise is often the best solution when based on the various connections proposed by the program.

Local unbalances

The "Cur ren t Unbalance ' ' at one point in the distribution sytem is best defined as the " Z e r o Sequence" for any given frequency. 2

The vector summation of the instantaneous values of the phase currents defines the "Residual Cur ren t " , giving a measure of the cumulative current unbalance at fundamental and harmonic fre­quencies.

Telephone influence is a direct function of the residual current and of the length of the common exposures between power and communication lines. 3 In North America, the use of multi-grounded distribution feeders leads to a mesh of basically single phase loading units which branch off a main three-phase artery. These points are referred to as "single phase take-offs" and are called nodes. The interval between two nodes is a segment. 2

Can. Elec. Eng. J. Vol 6 No 4, 1981

In order to analyze the circuit unbalance along the three-phase artery, it is convenient to take only the nodes. Figure 1 proposes a very simple feeder with single phase take-offs. Each loading is marked on the diagram. Any alteration of connection along the feeder affects the current flow only towards the substation; there­fore, balancing must start at the end of the feeder and proceed towards the source, or the substation.

Figure 2 illustrates the sequence of connections. The 30A load at node 13 is arbitrarily connected to phase A. Next, the 18A load is

15

SUB

10 3 •

32 5 •

56 7

60 9

18 11

13

12 2

68 4

24 6

36 8

12 10

46 12

30 Figure 1: Single phase loading sequence.

2 2 CAN. ELEC. E N G . J. VOL 6 NO 4, 1981

B

1 2 0 A

1 2 0

Figure 2: Principles of nodal unbalances.

connected to phase B at node 12. It is obvious that the unbalance is shown by the vector U\\, the subscrip and superscript indicating the segment along which this unbalance is carried. The next load of 46A connected at phase C will give a lower unbalance U\h on the next segment. At node 10, we have to make a choice for the con­nection of the 60A load: phase A or C causes the unbalance to in­crease, whereas phase B leads to a minimum local unbalance. One can proceed by chosing the phase connection which causes the last point to be nearest to the origin, and hence, minimizes the local un­balance.

Figure 3 shows the connection pat tern. The resultant at the sub­station is U = 52A, which is indeed not good. The obvious reason for this failure comes from the fact that the criteria of minimizing the local unbalance does not forecast a heavy load at subsequent nodes.

Look-ahead with target point

Figure 3 shows that the main problem came from the 60A load after the unbalance at node 4 has been minimized near the origin. To remedy this situation, it is proposed to choose the phase con­nection in a sequence which tends to bring the end of the vector diagram of node 4 to a " ta rge t po in t ' ' (*T4). This will ensure that subsequent connections will close the vector diagram near the origin. The method proves very successful, as depicted in Figure 4, where a 3.6A unbalance is reached at the substat ion. It must be emphasized that the term *'successful balancing ' ' is valid only for the optimization criteria: best current balance at the substation. Note also that the more nodes a system has, the easier it becomes to part i t ion the system into subsets of manageable blocks where target points will enable a linking of the subsets.

, , 1 0 A M P S O F U N B .

Figure 4: Optimizing local target for minimum current unbalance.

Influence of the distance

Voltage drops , line losses and telephone influence are direct functions of the product of the current unbalance by the length of each segment. 4 One can picture that a local unbalance, even very large, carried over a very short span, has no ill effects, whereas a small unbalance carried over a long segment might be the controll­ing factor in the overall interference. The obvious concept to use is, thence, the cumulative Ampere-Mile-Vector theory, as defined in Reference 1.

Figure 5 proposes a feeder configuration with the indication of the nodal distances. The currents and phase connection sequence are those assumed in Figure 4. Figure 6 depicts the cumulative Ampere-Mile-Vector influence derived for the given feeder. Each influence is computed by multiplying the local unbalance, U?\ by

m ZD CO

• ( N o d e s ) 11 12 13

M i l e 0 0 | CDI 0JI oo | o i H oo CO!

L o a d s O J i m O O O ^ r O J C O C O O J t S l t Q O O a ( a m p s ) ^ ^ ( D r H ( \ j ( T ) ( T ) L n ( D ^ h (T) Tips

P h a s e s B C B C 1 i r

A B C

Figure 3: Minimizing the local unbalance. Figure 5: Feeder loading with nodal distances.

SZABADOS: A M P E R E - M I L E VECTOR T H E O R Y 2 3

2 0 A M I

Figure 6: The cumulative ampere-mile-vector influence.

the corresponding nodal distance. It can be seen that the results are very poor. If a telephone line were to parallel the feeder, more than 120 Volts would be induced, even if a 20 per cent screening factor were to be considered. This level of interference is definitely unac­ceptable and shows that the minimum current unbalance at the substation is not a valid criteria for balancing feeders; the Ampere-Mile-Vector criteria is much better suited.

In order to arrive at a satisfactory balancing, one should draw the cumulative Ampere-Mile-Vector at the same time as the un­balance phasor diagram. Now the phase sequence will be dictated by examining the direction of the local current unbalance and the length of the segment involved. The aim is to place the single phase load in such a way that the resulting Ampere-Mile influence (AML) will be subtracted from the present preceeding cumulative influence. Figure 7 illustrates how the choice of connections of the last six line tap-offs in Figure 5 can bring their cumulative Ampere-Mile diagram back towards the origin. This method, however, is obviously more complicated and requires a perceptive analysis.

i 1 2 0 A M I

I , 1 0 A M P S O F U N B .

CNumb«r» Indioat* Fmmdmr Nod**)

9

Figure 7: Ampere-mile balancing method.

Location of the telephone switching center

If the switching center, or telephone central office, is located at the end of the feeder, and a telephone line runs parallel to the total length of the power line (exposure), the voltage induced in a tele­phone pair is derived from the cumulative Ampere-Mile diagram. Figure 8 shows the opt imum A M L unbalance of the feeder with the phase connections as indicated. With the central office ( C O . ) located at the end of the feeder (point A) , the maximum influence is at the substation (point B), with a magnitude of 143 A M L in­fluence, corresponding to 46 Volts induced (using a coupling fac­tor of 0.40 and a screening factor of 20 per cent) . 1 This level is bet­ter than the operating s tandards . 5

Consider now the C O . at the substation (point B). The tele­phone line acts as a voltage integrator of the successive influences within each segment. Even though the end influence remains at 143 A M L , it can be seen in Figure 8 that the A M L influence will pre­sent a maximum of 250 A M L at node 11, corresponding to 80 Volts induced between nodes 11 and 10, which is unacceptable. We must conclude that the relative position of the central office is crucial in the balancing criteria used.

Optimum balancing criteria

The standards adopted by most utilities on the Nor th American continent correspond to a maximum of 160 A M L influence. 5 The aim of any balancing must therefore be to contain the Ampere-Mile influence locus within a circle, centered at the central office and having a radius of 160 A M L . It is, of course, desirable to ar­rive at a radius as small as possible. The circle is centered at the C O . and the entire A M L locus contained therein is called the " m i ­n imum envelope c i r c l e , \ These are shown in Figure 8, where (Ce) is the radius of the minimum envelope circle corresponding to the C O . at the end of the feeder, and (Cs) is the case where the C O . is at the substat ion. Two methods of optimization are therefore re­quired.

Case 1: C. O. at the end of the feeder. The optimization proceeds from the end of the feeder towards the substation, and the phase connection chosen must try to bring the Ami influence phasor towards the origin, as described earlier.

Figure 8: Ampere-mile influence with the CO. at the end.

r—I

24 CAN. ELEC. E N G . J. VOL 6 N O 4, 1981

Case 2: CO. at the substation. Unfortunately, since balancing must start at the end of the feeder, one does not know the center of the envelope circle until the last node is reached. The aim now is to proceed in such a way that all points of the A M L locus be as close as possible to the last point entered. In other words, the last point entered, or the target point, should be the center of gravity of all already existing points, inclusive of the origin. We have to even­tually relocate the target point at completion of each node . Figure 9 illustrates the procedure. The final results of such an optimiza­tion method are given in Figures 10 a, b and c, where the envelope circle has been minimized at 143 A M L and the voltage profile is ex­cellent, since all points are well below the 60 Volts s tandards.

A numerical optimizing algorithm

The optimizing algorithm uses an adaptive target. At each node, the target must be at the center of gravity of all existing points , in­cluding the origin. The total number of combinations of N nodes

2

Figure 9: Optimizing with a relocatable target.

, 1 0 A M P S O F U N B .

5

3 2 Figure 10a: Current unbalance for AML optimization with the CO. at the substation.

• —i 2 0 A M L .

4

Figure 10b: Optimum AML influence.

I N D U C E D

c I M V O L _ T S : >

1 O MILES F R O M C - O .

Figure 10c: Induced voltage.

being the factorial function, it is unthinkable to investigate all possible cases for feeders which have a large number of nodes. However, for a subset of a few nodes, it is no problem to check all possible combinat ions and eventually choose the best connection. Since there is no mathematical algorithm available to minimize the balancing function, we have decided on this last approach . The method is not mathematically elegant, but the end result is ex­cellent. Instead of wasting computer time is sophisticated minimization search techniques, we can use the same C P U time and iron out a solution by "shear brutal fo rce" . In this case, the end justifies the means.

The search is based on target-look-ahead techniques of a given depth level (L). This consists of finding the best combination of the next L nodes which leads to the absolute minimum of the balancing function. When the best combinat ion of L nodes, start­ing at node N is found, the program keeps the best connection at node N and moves to node N + 1, with the next set of L nodes.

SZABADOS: A M P E R E - M I L E VECTOR T H E O R Y 2 5

ENTER: S U M M E R / W I N T E R L O A D S

S E G M E N T L E N G T H S

3 I A S - O F F S E T S

PRINTS: U N B A L A N C E

A M L I N F L U E N C E

COMPUTES

A N D

PRINTS:

ENTER:

Y E S

U N B A L A N C E

A M L I N F L U E N C E

V O L T A G E P R O F I L E

S E A S O N A L P R O J E C T I O N

S E L E C T P L O T O P T I O N S ?

NEW O P T I M

N (

1 Z A T 1 O N ?

PLOTS

END

Figure 11: Program flow chart.

CD D in

- ( N o d e s ) 1 2 3 4 5 6 7 10 11 12 13

\ U U ! | J ! I I 1 M i l e s v - l j o n ! d | ^ \ cxj! 1̂ ooj d | ^ | 1̂ oo | ^ | oo!

WINTER L o a d s ( a m p s )

SUMMER

( D ( \ j m i ^ s m N i D S S i s i s 00 CD CXJ

(\] (D (D (\J S CD CO

P O C \ J i \ O O C \ ] L n C D ^ I I I I I I I I

i n od • ^

i r CD T

CM 00 00 ID CD

I S

a 00

i i r P h a s e s

Figure 12: Seasonal load variations.

This is done until node K is reached, where (K + L) is the last node in the feeder. At this point , the program simply keeps the last L best connections. Practically, six to eight levels are sufficient to ar­rive at the best connection. Figure 11 illustrates the flow chart of the program.

The balancing cost function may have two forms. When the CO. is located at the end of the feeder, the cost function at node N is simply the magnitude of the vector representing the cumulative AML at node N; that is, the distance from the origin to the last node locus. The aim is to determine the envelope circle giving the smallest possible radius. When the central office is located at the substation, the cost function must be changed. It is recommended to use the root square summation of the distances from the node to be connected to each and every one of the existing points, origin in­cluded. This will ascertain that the target point remains wihin the objectives of optimization. We have tried to use other functions, such as the peak minimum cost function, which led to a good end influence but large envelope circles, but the root square summa­tion function always provided the best compromise between mini­mum envelope circle and minimum end influence.

It is also possible to introduce a weighting factor at each node. For instance, given nodes may be located in areas where there can be no customers, and one should not worry about the local peak value of influence. In that case, a weighting coefficient will allow

the envelope circle to disregard these " d e a d s p o t s " . However, it is impor tant to have a long-range load growth forecast in order to apply the weighting factor opt ions.

Seasonal variations

One of the main problems encountered in practical cases, is the effect of load variations. The larger variations usually occur be­tween seasons. When a load factor can be considered constant for all single phase feeders (e.g. residential area), there is no problem since the AML- locus simply varies proportionally to the load fac­tor . However, when the local areas have a very uneven load factor (e.g. summer/winter resorts), the Ampere-Mile-Vector influence varies considerably between seasons. Figure 12 proposes a feeder loading variation which has a very uneven load factor distr ibution. Winter op t imum connection is proposed in Figure 13, but such a connection leads to very bad summer balance. In the example, the winter optimization gives an end influence of 52 A M L with an envelope circle of 121 A M L , which is excellent, but the summer end influence will be 408 A M L with a 520 A M L envelope circle, which is a catastrophy. Hence, when large uneven seasonal varia­tions are present, the best cost function to be considered is the cost function resulting from the summation of each seasonal cost func­tion:

F = Fsu

With such a compounding cost function, a very elegant balanc­ing is reached. In order to analyze the sensitivity of the connection to load factors, one can assume that the load factors are linear with time, and loading can be represented by:

Y = Ys + AY

where

Y = Ys =

AY = a =

present loading. Summer load, variation from Winter to Summer, load factor variation (1 > a > 0).

Note that a — 0 is the Summer loading, and a = 1 is the Winter loading.

6 0 0 A m i

Swi SUMMER LOAD, WINTER OPTIMUM

S s . SUMMER LOAD, SUMMER OPTIMUM

Wsi WINTER LOAD, SUMMER OPT.

Wwi WINTER LOAD, WINTER OPT.

C O 2 0 m l

Figure 13: AML profile with summer/winter loads and summer/winter optimum.

2 6 CAN. ELEC. E N G . J. VOL 6 N O 4, 1981

Flexibility of the program

ENVELOPE C I R C L E

SUMMER

0 . 5

Three phase loads are automatically imbedded in the program by connecting " ze ro segment loadings" ; that is to say, that we have three superposed nodes at the three-phase take-off.

It is sometimes desirable to bias a target at a given point in the system. This is often the case when several distribution substations overlap a communicat ion line. Biasing the target is done by adding " d u m m y l o a d s " with zero length segments if a current unbalance is desired, or with appropria te " d u m m y segments" , where A M L biasing is expected. A n example of such a case is illustrated by Figure 15. Suppose that feeder 1 is to be balanced. The A M L in­fluence at A l is computed, using the exposure of feeder 2, and the sections A4 to A5 of feeder 2 are included in the bias. For instance, if feeder 2 gives an unbalance of 20 Amps along phase A , we can " r e s e t " the unbalance by connecting 20 Amps dummy loads on phases B and C at A l , with zero segment lengths. This way, the algorithm sees the unbalance at A l , but " r e m e m b e r s " the A M L influence created along the exposure of feeder 2.

Figure 14: Probable seasonal projection.

A "probab le seasonal project ion" can be plotted as shown in Figure 14. The evolutions of the end influence and the envelope circle are shown from Winter to Summer. The optimization is suc­cessful, since any time of the year will see a satisfactory balance (al­though in this case, the summer will see a slightly larger influence than the minimum criteria, which is acceptable with good shielded communication circuits.)

C\J m Z D

S U B I

A 2

en u

• u

LU Li_

A s A i

A s

A 4

en LU • LoJ LU I a _

LU

i °

CL LU

" _J L U

DUMMY 2 0 A

2 0 A

' DUMMY 2 0 A

A -

Figure 15: Multi-exposure circuits.

Implementation

The program was written in For t ran for all I / O features, and in machine language for all calculations in order to minimize the run t ime. The program asks for interractive input of feeder data , in­cluding seasonal load factors and possible biased targets for subset linking.

The output provides a printout of unbalances, A M L influences and a probable seasonal projection table. A D / A channel interface is available and the charts can be directed on to a plotter where phasor diagrams and voltage profiles can be hard copied. The operator can analyze the results and make a choice, or a t tempt to use other cost functions or introduce weighting coefficients. Typically, an 8 level predictor will run for 10 minutes for a 15 node subset. The operator interface is designed for maximum flexibility, enabling the program to become a powerful tool for the field engineer.

Conclusions

A novel technique implementing a dynamic balancing of distri­but ion feeders has been described. The algorithm uses the Am­pere-Mile-Vector influence, together with an adaptive look-ahead-target predictor. The minimization of various cost functions allows to optimize for seasonally varying loads. The biasing of the initial conditions of the optimization permits to link subsets, so that multiexposure telephone lines or odd central office location can be dealt with.

The program has been thoroughly tested and some feeders have actually been balanced on the New Brunswick Power Commission Grid, using the above techniques. The results were very positive. Measured values were well within the targets shown by the pro­gram and stability of the balancing remained for years thereafter. Moreover, the optimization provides with a bonus feature of easy updat ing and recordkeeping of feeder loading.

References

1. Szabados, B. , "The Ampere-Mile-Vector Theory Applied to Inductive Coordina­t ion ," IEEE PES, A79095-1 , January 1979.

2. Szabados, B. , and Noble , W . A . , " A New Look at Inductive Coordination," CEA, March 1974.

3. CSA Special Publication C22.3, N o . 3 .1 , 1974. 4. Szabados, B. , "The Neutral Current Phase Shift on Distribution Feeders: Theory

and Field Test ," NTC '76, Dallas, TX, N b . 1 2 - 3 - 1 , proceedings, Vol. 1, 1976. 5. Standard Engineering Practices, N B E P C Sep I I I -2 .00-2 .0 - . 6. Szabados, B. , and Burgess, E.J. , "Optimizing Shunt Capacitor Installations, Us­

ing Inductive Coordination Principles," IEEE trans. PAS, Vol. P A S - 9 6 , Nb. 1, pp. 222-226.