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CHAPTER 2 TELLEGEN‟S – THEOREM BASED POWER FLOW...
Transcript of CHAPTER 2 TELLEGEN‟S – THEOREM BASED POWER FLOW...
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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