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CHAPTER 2

TELLEGEN‟S – THEOREM BASED POWER FLOW

METHOD FOR 1-PHASE RADIAL DELIVERY NETWORK

2.1 Introduction

This chapter formulates Backward-Sweep (BS) equations based on

Tellegen‘s-Theorem (TT) to resolve power flow solution during Forward-

Sweep (FS) for 1-phase Radial Delivery Networks (RDN). The Network

Topology (NT) based algorithm reported by J.H.Teng [17] encounters

problems, due to development of two-complex matrices like Bus

Injection-Branch Current (BIBC) and Branch Current-Bus

Voltage‘(BCBV) for large RDN with system components. It also takes

more Net Execution Time (NET) to converge the solution, when compared

with the (FB) mode [25] and Ladder Network style (LN)[19] as expressed

in A.G.Bhutad et. al [10] for the RDN data [16]. A string of

interconnected Ladder-Network(LN) [18-19] was found less efficient to

update or modify in sorted form due to the complexity of numbering

scheme and [20] reports that LN Technique is found to be fastest, but did

not got converged-solution in 5 out-of 12 tests performed.

A new power flow algorithm is proposed for 1- RDN based on

Tellegen‘s Theorem (TT). A set of iterative power flow equations are

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developed to find power and current during FB sweeps, respectively.

The accurate value of injected current computation from up stream to

down stream RDN using TT and KVL leads to faster convergence, when

compared to [17, 24].

TT ‗states that the algebraic sum of complex powers‘ meeting at a

node is zero. Using TT, the Backward-Sweep of nodal power and element

loss are computed from downstream to up-stream of a network. This

competent method is formulated during the FB sweeps, which is

helpful to accurate downstream element-currents. Finally, directly using

KVL in the Forward-Sweep to obtain distribution power flow solution.

The validation of proposed algorithm is carried out using [MATLAB] for

the RDN data given in [8].

INDEX

k = Node or Element Number

i = Injection

Si(k) = Power Injected at node-k

Sd(k) = Power Demand at node-k

Sl(k) = Power loss in element-k

V(1) = Voltage at Node-1

V(k) = Unknown voltage Node-k

I1 = Current injected at Node-1

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Id(k) = Current drawn at node-k

Z(k) = Element-Impedance referred to node-k of Receiving End

n = Number of nodes

b = Number of elements (= n-1)

p = Number of iteration

x = Network Termination Level (NTL)

2.2 Tellegen‟s Theorem based–Methodology

The node oriented numbering for a typical ‗radial distribution

network‘ shown in Figs. (2.1- 2.2), having n nodes and b (= n-1)

elements.

Fig. 2.1 Typical RDN structure with levels.

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2.2.1 Node numbering technique

The network nodes are numbered level-by-level from left to right side

of the RDN till the end of NTL as shown in Fig. 2.1. In case of network

branch, which lies between kth top-node and (k+1)th bottom-node, it is

suggested that the element-number is same as downstream node-

number itself, because network is of radial delivery in nature.

2.2.2 Backward-Sweep to compute nodal power injection

Use TT during Backward-Sweep to compute power injection from leaf

node to node-1. The RDN shown in Figs. (2.2-2.3) is so configured with

laterals or sub-laterals.

(a) Power Injection at nth node: In Fig. 2.2 at leaf-nodes the power

injection iS is equal to load dS i.e.

i dS (n) =S (n) (2.1)

Fig. 2.2 Main feeder with load and loss in the RDN.

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Further, iS calculations are carried out by Backward-Sweep from the

sub-lateral to lateral till the node-1 is reached.

i i l dS(k) =S(k+1)+S(k+1)+S (k) (2.2)

where,

upstream injected power

downstream injected power

line loss along the branch

upstream power demand

i

i

l

d

S (k)

S (k+1)

S (k +1)

S (k)

l

i2

S (k +1)S (k+1) =abs Z(k +1)

V(k +1) (2.3)

Fig. 2.3 RDN with laterals-sub-laterals.

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(b) Simplifying equation (2.2): Fig. 2.4 shows that Power Injection in the

up-stream is equal to the summation of powers and losses in the

downstream. Hence, three terms i.e. belongs to (2.2) can be reduced to

two terms as

i

n-upstream n-upstream

nodes branches

k=downstream k=downstream

nodes branches

S (k) = Loads + Losses (2.4)

i d l

n-upstream

nodes

k=downstream k=downstream

nodes nodes+1

n-upstream

nodes

S (k) = S (k)+ S (k) (2.5)

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Fig. 2.4 RDN with load at node and loss along the element.

It is observed that first-term (loads) of (2.5) is independent of assumed

voltage, whereas second-term (losses) is ‗dependent‘ on square of

absolute value of the assumed voltage. In equation 2.5, line-

losses(second-term) contributes less (i.e. 8%), when compared to (loads)

first-term. Because of error due the assumed voltage with second term,

the iterative convergence results are accurate.

2.2.3 Forward-Sweep to compute nodal current injection

An improved FS-technique computes the currents injected at

downstream node-k using the accurate up-stream currents and

downstream originating currents.

(a) The injection-current at kth node is

i iI k S (k)/V(k)*

( ) = (2.6)

(b) Calculation of element-currents: The downstream element-current

Ii(k+1) is equal to recent up-stream value of current incoming at node-k

minus the currents originating from downstream at the same node.

(p)

(p-1)emanatingbranches

k=k+1,k=k+2

I (k+1) = I (k) - (Lateral+Node) Currentsi i (2.7)

where, p = Iteration No. i.e. p = 1, 2.3..

Then node-currents of the RDN of equation (2.7) can be written as

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emanating branches

juncupdow dow

k=k+1i i di

I (k+1) = I (k) - I (k+2) - I (k) (2.8)

The current-injection Ii at node-k shown in Fig.2.5 is

i i d

emanating node

k=k+1

* *I (k+1) = S (k) /V(k+1) - Si(k+2) /V(k) - S (k) /V(k)[ ] [ ] (2.9)

Fig. 2.5 Current entering and leaving at node-k of a RDN.

‚Applying KVL to update node-voltage for the RDN shown‘ Fig.2.6 at

node-(k+1) as

V(k+1) =V(k) - Z(k+1)I(k+1) (2.10)

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Fig. 2.6. Nodal voltage levels in RDN.

Eqs. (2.6), (2.9), and (2.10) are to be executed repeatedly until the

convergence is reached. The voltage mismatch at node ‗k‘ can be

expressed as:

(p+1) (p+1) (p)ΔV(k) =V(k) -V(k) (2.11)

2.3 Comparison of Proposed-Method

The three-step power flow methods explained in [17] and [24] are

compared with each other in Table 2.1.

Table 2.1 Comparison of NT and LN based power flow methods [17, 24]

Sl.

No.

[17] Procedure [24] Procedure

A Nodal current injection ‗Ii (k)‘ at

iteration-p

*

)(

)(

kV

kSiIi

Nodal currents: Current

injection

‗Ii (k)‘ is

*

)(

)(

kV

kSiIi

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B Bus-Injection to Branch-Current

(BIBC): Branch-Currents‘

=[BIBC] [ iI ] Here [BIBC] with 0‘s and 1‘s information helps to find

Element-Currents in terms of Node-Currents‘

Backward-Sweep: ‗Expression

for branch currents‘

emanatingnodes(p)

(P)

i ik=1I (k +1) = In (k)

where In= nodal currents

C ‗Branch-Current to bus-Voltage‘ (BCNV): ‗Bus-voltage is computed using‘

[BCBV]& [BIBC] and [Ii] is

i 1 i[V (k)] = [V ]- [BCBV][BIBC][I (k)]

Forward-Sweep: Nodal voltages are computed in the forward sweep as

i i

i

V (k +1) = V (k)

- Z(k +1)I (k +1)

Then A to C points discussed in Table 2.1 are compared with the

proposed-method tabulated in Table 2.2. It is observed that minimum

three steps are required to find the final load flow solution.

Table.2.2 Tellegen‘s-Theorem based power flow procedure

Sl. No. Proposed-Method

1 Backward-Sweep: Compute ‗power Si(k)‘ using (2.5) by separating flat

start-voltage dependent-term and ‗independent-term as load and losses,

respectively. It is known that the ‗loss‘ in a network is about 8% of the loads, which tends to minimize error in the solution.

2 Forward-Sweep: The ‗recent values‘ of up-stream node-currents from (2.6)

are used to compute up-stream currents from (2.9), which causes improved

rate of-convergence.

3 Forward-Sweep: Node voltage is directly computed using KVL

i i iV (k+1) =V (k) - Z(k+1)I (k+1)

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Fig. 2.7 Flow chart of Tellegen‘s Theorem based method

The detailed flow-chart of the TT based method is presented in Fig.2.7.

Table 2.3 Performance test of TT based method for15-node RDN

Power Flow methods No. of Iterations Flexibility

Method [17] 3 No

Method [24] 3 Yes

Proposed Method 2 Yes

Table 2.4 Performance test of TT based method for 28-node RDN

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Methods No. of Iterations Set-Accuracy

Method [17] 4 0.001

Method [24] 3 0.001

Proposed Method 2 0.001

2.4 Results and Discussions

TT based method improves the algorithm efficiency during Forward-

Sweep, when compared to Forward-Backward sweep methods of [25].

With this the results obtained are better than that of Network-Topology

[17] and Ladder-Network [24] for the standard-data [8, 23].The proposed

algorithm presents in simplification of (2.2) and tried to formulate

current injection in (2.9) The converged solution mentioned in Tables of

(2.3, 2.4) were simulated in [MATLAB] Ver. 7.01 with system

configuration of 512MB-RAM, Intel Pentium IV-Processor, 1.73 GHz-

Speed. The advantage of the TT based method has been confirmed by

considering element losses into account given in equation (2.2) and up-

dating latest node-current in (2.9). Then node-voltage is found using

(2.11). Thus the novel-method has been verified to be of better-quality in

accuracy, number of iterations and robustness as per the performance

seen in Tables 2.3 to 2.6.

Table 2.5 Converged Node-Voltage performance for 15- Node RDN

Node- Voltage

Proposed- Procedure

(P)

Procedure [24] Procedure [17]

% Accuracy % Accuracy

(S) (T) (P-S)/P (P-T)/P

V1 1.0000 1.0000 1.0000 1.0000 1.0000 V2 0.9706 0.9697 0.9734 0.0029 0.0009

V3 0.9697 0.9688 0.9717 0.0021 0.0009

V4 0.9554 0.9541 0.9591 0.0039 0.0014

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V5 0.9551 0.9537 0.9587 0.0038 0.0015

V6 0.9552 0.9535 0.9585 0.0035 0.0018

V7 0.9405 0.9389 0.9461 0.0060 0.0017

V8 0.9250 0.9231 0.9325 0.0081 0.0021

V9 0.9137 0.9115 0.9223 0.0094 0.0024

V10 0.9087 0.9064 0.9172 0.0094 0.0025

V11 0.8968 0.8942 0.9049 0.0090 0.0029

V12 0.9685 0.9668 0.9697 0.0012 0.0018

V13 0.9676 0.9655 0.9683 0.0007 0.0022

V14 0.9666 0.9639 0.9668 0.0002 0.0028

V15 0.9664 0.9637 0.9666 0.0002 0.0028

Table 2.6 Power-flow result for IEE 28-Node RDN

Voltage Procedure [24] Proposed-Method Procedure [17]

V1 1.0000 1.0000 1.0000

V2 0.9544 0.9568 0.9569

V3 0.9100 0.9120 0.9126

V4 0.8820 0.8882 0.8891

V5 0.8706 0.8731 0.8744

V6 0.8101 0.8160 0.8187

V7 0.7702 0.7795 0.7838

V8 0.7606 0.7615 0.7668

V9 0.7200 0.7303 0.7380

V10 0.6900 0.6921 0.7038

V11 0.6607 0.6677 0.6628

V12 0.6507 0.6571 0.6539

V13 0.6304 0.6305 0.6319

V14 0.6093 0.6099 0.6060

V15 0.5913 0.5975 0.5974

V16 0.5817 0.5887 0.5823

V17 0.5804 0.5812 0.5800

V18 0.5709 0.5787 0.5710

V19 0.9470 0.9495 0.9516

V20 0.9408 0.9476 0.9505

V21 0.9418 0.9450 0.9496

V22 0.9426 0.9431 0.9506

V23 0.9010 0.9060 0.9072

V24 0.9033 0.9026 0.9044

V25 0.8922 0.8991 0.9019

V26 0.8123 0.8122 0.8161

V27 0.8110 0.8109 0.8154

V28 0.8103 0.8103 0.8154

2.5 Conclusions

With use of Tellegen‘s theorem, backward sweep of node injection and

power loss are computed for the radial delivery network. This competent

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method is used to formulate element-currents during the Forward-Sweep

and the inclusion of ‗power variable‘ in the algorithm during Backward

Sweep leads to more accurate results. The proposed TT based power-flow

solution is verified to be fast converging, with reduced number of

iterations for different 1-phase Radial Distribution Network.

CHAPTER 3

DIRECTED–GRAPH BASED POWER-FLOW

ALGORITHM FOR 1-PHASE RDN