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I. Overview of methodologyTo realise this project, an interconnection of the Southern Interconnected Grid (SIG) and
Northern Interconnected Grid (NIG) of Cameroon is realised with a HVDC connection with the
aim of minimising the active power transmission losses in the AC networks, while studying the
voltage stability of the HVDC link in terms of voltage values between nodes.
I.1 Active Power minimizationIn this section, the basic concept applied for the minimization of active power transmission
losses is explained. The following steps are then followed in order to attain the objective of (see
flow chart below);
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Apply the particle swarm Optimization method to the OPFControl variables
Reactive power generated bycapacitor banks
Position of OLTC trans formers
Voltages at the generator buses
Output variables
Optimal position of control variables forminimum trans mission loss es
Formulation of the Optimal Power Flow Problem (OPF)
Objective function Network constraints
Solution of the Power Flow Problem by the Newton Raphson methodInput variables
Bus admittance matrix
Active and reactive power generatedand consumed at the different nodes
Output variables
Voltage and phas e at the differentnodes
Active and reactive power generated
Formulate the Power Flow Problem
Formulate the bus admittance matrix Power Flow Equation
Impedance model of the different network elements
Impedance model of transmissionlines
Impedance model of transformers
Adaptation of the SIG and NIG mode
Hypothesis for themodeling of the SIG
Determine the numberof nodes
One line diagram of theSIG model
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I.2 Model of the SIGGeneral Hypothesis for SIG and NIG
The following assumptions were made in modelling the SIG:
The base apparent power is
and the base
corresponds to the nominal
voltage at a particular node;
For two transmission lines in parallel, the equivalent resistance and reactance are
calculated. This implies for two nodes connected by two lines in parallel, only one
equivalent transmission line is considered;
In all power plants (except Song Loulou and Lagdo) with two or more generators, the
latter are considered as one and the power generated is the sum of the power
generated by these different generators. This signifies the power generated by
individual generators is not specified;
The power consumed at a node varies as a function of the voltage across the node. In
this model, the loads were considered constant irrespective of the voltage across them
Generally, the reactive power injected by a capacitor bank varies as a function of its
capacitance and the voltage across it. This was not taken into account in this model. The
reactive power injected by a capacitor bank was considered constant regardless of the
voltage across it.
All the HV substations with capacitor banks have two capacitor banks each. Only one
capacitor bank is indicated with a capacity equal to the sum of the reactive power
injected by the two capacitor banks.
Diagram of the SIG model
The figure below is a diagram showing the different bus bars, generators and transformers of
the SIG model considered. A total of 46 buses and 50 branches (branches are transmission lines
or transformers). Table below indicates the generator nodes together with their corresponding
node indices and Table shows the different transformers as well as their departure and arrival
nodes;
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Ta b le 1: Generators in the m odel and their corresponding node indicesLocation of Power
plant
Number of generators
considered
Name used in model Node index as
represented in the model
Song Loulou 4 SLL1 1
SLL2 2SLL3 3
SLL4 4
Oyomabang 2 Oyo HFO 11
Oyo LFO 12
Edea 1 Edea PP 23
Dibamba 1 Dibamba PP 27
Logbaba 2 Logbaba HFO 29
Logbaba LFO 30
Bassa 2 Bassa2 34
Bassa3 35Limbe 1 Limbe HFO 41
Bafoussam 1 Bafoussam PP 44
Ta b le 2: Transformers and the ir corresponding arrival and departure node indices
Location of transformerDeparture node Arrival node
Song Loulou 1 5
2 5
3 5
4 5
Oyomabang Substation 9 10
9 18
Oyomabang HFO 10 11
Oyomabang LFO 10 12
Mangombe Substation 8 25
Logbaba Substation 7 28
Logbaba HFO 28 29
Logbaba LFO 28 30
Bekoko Substation 6 39
Bassa2 33 34
Bassa3 33 35
Edea Power plant 22 23
Dibamba 26 27
Limbe 41 42
Bafoussam 44 45
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Some of the transmission lines are of particular interest because the amount of power that
flows through them is very high, and others by virtue of the fact that they are very long and
thus have very high resistances. These two factor affect considerably the active power
transmission losses in these lines. The length of the 225kV and some 90kV transmission lines
are indicated in the figure below.
10
LOGBABABEKOKO MANGOMBE
OYOMABANG
B
NSI
AHALANGOUSSO
KONDENGUI
NJOCK-
KONG
EDEA PP
ALUCAM
NGODI BAKOKO
KOUMASSI
DIBAMBA
MAKEPE
BASSA
DEIDO
BONABERI
LIMBE
NKONGSAMBA
BAFOUSSAM
BAMENDA
LIMBE HFO
1
SONGLOULOU
7
1
96 8
18
19
14
30
27
21
22
20
35
32
3334
38
40
37
41
46
45
2 3
5
11 12
24
23
26
25
28
31
29
36
39
43
42
44
4
15
DR NSIMALEN
Key
Hydo electric power gen
HFO thermal generato
LFO thermal generato
225kV____________
90kV_____________
HV capacitor banks
EDEA
(41.5km) (58km)
(65km)
(58km)(168.38km)
(94km)
(94k
m)
(70km)
Formulating the power flow problem
Formulation of the bus admittance matrix ()The first step in developing the mathematical model describing the power flow in the network is the
formulation of the bus admittance matrix. The bus admittance matrix is an matrix (where is thenumber of buses in the system) constructed from the admittances of the equivalent circuit elements of
the segments making up the power system. Most system segments are represented by a combination of
shunt elements (connected between a bus and the reference node) and series elements (connected
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between two system buses). Formulation of the bus admittance matrix follows two simple rules Error!
Reference source not found.:
Consider 2 nodes and , and be the corresponding admittance between and .If and k are linked by a transmission line or transformer
Else
For a node
Whereis the set of nodes connected to bus by a line or transformer and the reactance connectedto the node (shunt of reactance).In polar form, is expressed in the form || || Where are the real and imaginary parts of the admittance respectively.
Formulation of the power flow problem
The power flow problem is used to determine the values for all state variables (voltage magnitude and
angle) by solving an equal number of power flow equations based on the input data specifications. In
this section, the different input data known at each of the nodes are specified.The voltage at a bus of the system in polar coordinates is as follows: The voltage at another bus is simi larly written by changing the subscript from to .The net current injected into the network at bus in terms of the elements of the admittance matrixis given by
Let
and
denote the net real and reactive power injected at the bus
. The complex conjugate of
the power injected at bus is By substituting equations and in equation we obtain:
Expanding this equation into real and imaginary parts:
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Equations and constitute the polar form of the power-flow equations; they provide calculatedvalues for the net real power and the net reactive power injected at bus .Let denote the scheduled power generated at bus and denote the scheduled power demand.Then, is the net scheduled power being injected at bus . If the calculated value of isdenoted by , we can define the mismatch, as the scheduled value minus the calculatedvalue .i.e. Likewise, the reactive power mismatch at bus is:
Mismatches occur in the process of solving power flow problems when calculated values ofand do not coincide with the scheduled values and . If the calculated values and match the scheduled values and perfectly, then the mismatches and are zeroat the bus , i.e. If there is no scheduled value for bus , then the active power mismatch cannot be defined andthere is no requirement to satisfy equation in the course of solving the power flow problem.Similarly, if
is not specified at bus
, then equation
does not have to be satisfied.
Four unknown quantities associated with each bus are , voltage angle and the voltagemagnitude. For a system with N nodes, there are:N voltage amplitudes N voltage angles N active power at the node N reactive power at the node
4N unknowns can be identified. There are at most two equations available for each node, and so the
number of unknown quantities must be reduced to agree with the number of available equations before
beginning to solve the power flow problem. The general practice in power flow is to identify three types
of buses in the network. At each bus, two of the variables , andare specified and theremaining two are calculated.
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Ta b le 3: Summary of power-flow problemBus type Number of
buses
Quantities
specified
Number of available
equations
Number of state
variables( ,)Slack (1)
Voltage controlled (2) Load (3) 2( ) 2( )Totals
The power flow formulation was applied to the SIG. As seen in Error! Reference source not
found., buses. The node types were defined as follows:All thermal generators, interconnection nodes and load nodes are PQ buses giving a
total of
buses. Nodes with thermal generators are PQ nodes because the amount of
power generated by these plants are considered fixed, in order to respect the cost
effective schedule made by QSOM ;
The generators in Edea and six generators of Song Loulou are PV buses giving a total of 4
PV buses;
Two generators in Song Loulou are connected to the slack bus.
Table gives a summary of the number of buses of each type and the number of corresponding
equations.
Ta b le 4: Node Configuration for the SIGBus type Number of buses Number of available
equations
Slack (1) Voltage controlled (2) Load (3) Totals
There are different computational techniques used by different programs to determine the state
variablesError! Reference source not found.. These are: the Gauss-Seidel Method, the Newton Raphson
Method and the Fast Decoupled method. In this work the Newton Raphson method will be applied
because the number of i terations required by the Newton Raphson method is independent of the
number of buses whi le the computation time using Gauss-Seidel increases almost directly with the
number of buses.
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Solution of Load flow using Newton Raphsons method
Taylor's series expansion for a function of two or more variables is the basis for the Newton-Raphson
method of solving the power flow problem Error! Reference source not found. . To apply the Newton-
Raphson method to the solution of the power-flow equations, we express bus voltages and line
admittances in polar form i.e.
These equations are differentiated with respect to the voltage angles and the magnitudes (state
variables). For active power we have
A similar mismatch equation can be written can written for the reactive power ,
Each non slack bus has two equations for . Collecting all the mismatch equations intovector-matrix form yields
(
[
] [
]
[
] [ ])
The partitioned form of the equation above emphasizes the four different types of partial
derivatives which enter into the Jacobian . The solution to this equation is found by iteration as follows:1. Estimate values for the state variables
and
(generally,
=0 and
;
2. Use the estimate to calculate from equations and and the mismatchesfrom equations and and the partial derivatives elements of the Jacobian;3. Solve equation below for initial corrections ;4. Add the solved corrections to the initial estimates to obtain
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5. Use the new values
as starting values for iteration 2.
Formulas for the update of state variables are as follows:
This process is repeated until the mismatch is less than a tolerance or when the number of iterations is greater than the maximum number of iterations (generally, the
tolerance is 0.005 and the maximum number of iterations is 12).
The NIG model
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Ta b le 5: Generators in the m odel and their corresponding node indicesLocation of Power
plant
Number of generators
considered
Name used in model Node index as
represented in the model
Lagdo 2 G1 1
G2 2Garoua 1 Gen 3 11
Ta b le 6: Transformers and the ir corresponding arrival and departure node indices
Location of transformerDeparture node Arrival node
Lagdo 1 Lagdo
2 Lagdo
Ngaoundere Substation Ngaoundere 6
6 7
Garoua substation Garoua1 8
Garoua substation 9 8
Garoua substation 8 10
Garoua substation 8 11
Guider 12 Guider
Maroua substation 12 Maroua
Maroua substation Maroua 15
Maroua substation 15 16
DC link modification on the N-R algorithm
The work to be realise is a study on the interconnected link using the HVDC, hence the AC-DC newton
Raphson load flow algorithm is given below
Where
Yac: the admittance matrix for the AC network
Ydc: the admittance matrix for the DC line
Jac: Jacobian matrix for AC network
Jac: Jacobian matrix for DC line
X: values obtained from iterations
: tolerance
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