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technical report
SR 95/######
Technical report of FLOWC program
p.
1
technical report
SR 95/######
p.
INDEX
1.
GENERALS ......................................................................................................................4
1.1 Network models .......................................................................................................5
1.2 Algorithm of solution................................................................................................6
1.3 Functional possibilities ..............................................................................................6
1.4 Limits of the program ...............................................................................................7
2.
LOAD-FLOW solution ......................................................................................................8
2.1 Generals ...................................................................................................................8
2.2 Scope and validity of the model.................................................................................8
2.3 Description of the model ...........................................................................................8
2.4 Other algorithms of solution in the program ............................................................11
2.5 References ..............................................................................................................12
3.
LOAD-FLOW SOLUTION THROUGH THE LOAD ANALYSIS METHOD ................14
3.1 Generals .................................................................................................................14
3.2 Scope and validity of the model...............................................................................14
3.3 Description of the model .........................................................................................14
3.4 References ..............................................................................................................16
4.
MODEL OF THE NETWORK COMPONENTS.............................................................17
4.1 Model of the lines ...................................................................................................17
4.2 Model of the transformers and LTC (on load tap-changer) ......................................18
4.2.1 Model of the transformers with the ratio out of the rated value in p.u. ............18
4.2.2 LTC model - Derivatives of the parameters with respect to the
regulation variables ........................................................................................20
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4.6.1 Generals ........................................................................................................30
4.6.2 Scope and validity of the model .....................................................................30
4.6.3 Description of the model................................................................................31
5.
MODELS FOR THE REACTIVE POWER CONTROL..................................................33
5.1 Generals .................................................................................................................33
5.2 Scope and limits of the model .................................................................................33
5.3 Description of the models .......................................................................................34
5.4 References ..............................................................................................................37
6.
MODEL FOR THE HVDC SYSTEM .............................................................................38
6.1 Description of the model .........................................................................................38
6.2 References ..............................................................................................................44
7.
CONTINGENCY CALCULATION IN THE FLOWC PROGRAM ................................45
7.1 Generals .................................................................................................................45
7.2 Scope and validity of the model...............................................................................45
7.3 Description of the model .........................................................................................45
7.4 References ..............................................................................................................48
8.
CALCULATION OF THE DISTANCE FROM THE VOLTAGE COLLAPSE...............49
8.1 Generals .................................................................................................................49
8.2 Definition of the sensitivities and of the distance from the collapse ..........................49
8.3 Implementation of the method in the FLOWC program ...........................................52
8.4 References ..............................................................................................................53
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1.
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p.
GENERALSErrore. Il segnalibro non è definito.
The scope of FLOWC program is the solution of the steady state power flow distribution and of
the voltage profiles in electrical power systems; these systems can be AC transmission network
or integrated AC-DC networks.
Exploiting the main characteristics of the solution algorithm it's possible, without any loss of
precision, to analyze the steady state of electrical distribution systems and, more in general, of
industrial plants.
In the following some of the most interesting aspects of the load-flow analysis are synthetized:
1) The generation must satisfy the load plus the transmission losses, so this generation must be
optimally distributed among the generation groups from an economic point of view.
2) The connections can transmit a certain power and then it's necessary that the working point
isn't near to the stability limit or to the thermal limit.
3) It is necessary to verify that the voltage levels of some busbars are within the limits. This
control can be performed by an adequate reactive power dispatching.
4) If the power system is part of an interconnected system it is necessary that the scheduled
power exchanges among the areas are respected.
5) The network disturbances due to faults may cause the out of service of some components;
the effects of these emergency situations may be reduced through proper load-flow
simulations in n-1 conditions (contingency analysis).
As said before the scope of the load-flow is the calculation of the power flows and voltage
profiles of the network.
A single-line representation of the network can be considered advantageous since electrical
systems are generally symmetric. Four quantities are associated to each busbar: the active and
reactive power and the voltage module and phase. In the load-flow calculation three different
busbar types are represented and in each busbar two of the four quantities are fixed. One of
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p.
1.1 Network modelsErrore. Il segnalibro non è definito.
The models available for transmission networks components are:
###
AC lines, represented by lumped or distributed parameters models or by
generalized ### model;
###
two or three winding transformers with or without tap-changer;
###
quadrature booster (phase-shifter) with the on-line control of the angle or power;
###
DC multiterminal connections with different transmission schemes (mono-phase,
bi-phase, with grounded return, etc.); converter representation through a base module
foreseeing a completely controlled bridge connected to the AC system through a conversion
transformer. The transformer is represented with a tertiary winding to which a synchronous
compensator, AC filters or other devices for reactive power control may be connected.
###
FACTS (Flexible AC Transmission System) devices as: Phase-Shifter,
Tap-Changer (LTC or TCUL), series controlled reactance (RANI) and controlled
compensator (SVC).
The models available to represent loads are:
###
constant power load;
###
constant impedance load;
###
voltage dependent load through exponential law;
###
voltage dependent load through tabular data defined by the user or through the
point defined by POWASY (this last type is useful for asynchronous machine simulation).
The models available for generators are:
###
generation busbar with controlled voltage and active power (P,V type). The user
has available an analytical equation that allows the simulation of the capability curve of the
machine to define the relevant limits;
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1.2 Algorithm of solutionErrore. Il segnalibro non è definito.
The FLOWC program adopts an algorithm of solution based on the NEWTON-RAPSHON
method with the complete Jacobian matrix, using during the iterative cycle of solution the
technique of symbolic bifactorization of the system to be solved.
In the matricial system the equations relevant to the nodes, the tap-changers, the phase-shifters
and series controlled reactance are present; each equation has as unknowns respectively the
module and phase of the node voltage, the tap ratio, the complex ratio of the phase-shifter and
the series impedance.
The method proved very effective in situations of weak and badly conditioned networks and
with radial configurations, above all in networks with industrial characteristics, where the
hypothesis X>>R (the series reactance largely greater than the resistance) on the connections is
no longer valid.
1.3 Functional possibilitiesErrore. Il segnalibro non è definito.
Besides the classical load-flow solution, the FLOWC program has available the following further
functionalities:
###
load-flow solution of an AC network with DC multilink;
###
load-flow calculation in contingency conditions (n-1) of lines, transformers and
generators. The contingencies may be requested in a generalized way, that is on all the
components of a certain type, on one or more components indicated by the user, or
automatically on all the components resulting overcharged from the load-flow calculation;
###
solution of a bounded load-flow with predefined limits on voltage and reactive
powers of the generation busbars;
###
calculation of the distance from the voltage collapse for all the load busbars of the
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1.4 Limits of the programErrore. Il segnalibro non è definito.
The present limits of the program are:
-
2000 busbars;
2400 AC lines;
1000 transformers (including also the LTCs e Phase-Shifters, maximum number of
LTC = 500 of PHS = 50);
1000 generators;
200 non-linear loads;
200 asynchronous machines;
200 restrictions in the solution of the bounded load-flow;
30 HVDC converters;
75 lines in the DC transmission networks;
75 neutral lines in the DC networks.
10 FACTS devices
The program is completely parametrized, then the above limits can be modified while creating
the executable module. It is to be noted that increasing these limits also the memory request for
the program execution is increased.
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2.
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p.
LOAD-FLOW solutionErrore. Il segnalibro non è definito.
2.1 GeneralsErrore. Il segnalibro non è definito.
The calculation of the active and reactive power flow in the network connections is performed
solving an algebraic system with the complete Jacobian matrix through the Newton-Raphson
iterative method. In alternative the decoupled method is available [1, 4], eventually assisted by a
preliminary solution performed through the Gauss-Sidel iterative method [4].
2.2 Scope and validity of the modelErrore. Il segnalibro non è definito.
-
The scope of the model is the calculation of all the voltage modules and phases and of the
power flows, starting from the voltage solution.
The model representing the network is of the single-line type and, then, doesn't consider the
possible dissymmetries between the phases.
The model proved very robust both in networks with X>>R (transmission networks) and in
networks with X###R (industrial plants).
The solution is performed assuming the network in steady-state.
Methods to threat sparse matrixes and also a symbolic solution performed only once at the
beginning of the execution have been used to fasten the solution.
2.3 Description of the modelErrore. Il segnalibro non è definito.
In the following the models developed to solve the system of equations for the load-flow
calculation is demonstrated in general terms.
In a load-flow calculation the variables in the electrical network can be classified as follows:
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technical report
N
Ai = V i ⋅ ∑ Yij* ⋅V j* =Vi 2 ⋅ ( Gii − jBii ) +
j =1
N
∑
j =1
j≠ i
SR 95/######
V i ⋅V j ⋅ (Gij − jBij )⋅ (cos(θ i − θ j ) + jsin (θ i − θ j ))
p.
(2.1)
hence:
Pi = + Vi 2 ⋅Gii −
N
∑
j =1
j≠ i
Qi = − Vi 2 ⋅ Bii −
where:
Ai
:
Pi
:
Qi
:
Vi
:
Vj
:
Gii :
Bii
:
Gij :
Bij
:
###i :
###j :
N
∑
j =1
j≠ i
Vi ⋅V j ⋅ ( + Gij ⋅cos(θ i − θ j ) + Bij sen (θ i − θ j ))
V i ⋅V j ⋅ (− Bij ⋅cos(θ i − θ j )+ Gij sen (θ i − θ j ))
(2.2)
Apparent power injection in the node i of the network
Active power injection in the node i of the network
Reactive power injection in the node i of the network
Voltage module in the node i of the network
Voltage module in the node j of the network
Shunt conductance in the node i of the network
Shunt susceptance in the node i of the network
Conductance of the connection between the node i and the node j
Susceptance of the connection between the node i and the node j
Phase of the voltage of the node i with respect to the reference node
Phase of the voltage of the node j with respect to the reference node
The above equations give rise, in a network with n nodes, to a non-linear system with 2n-2
equations and 2n-2 unknowns, being known the voltage and phase values in the slack busbar.
The solution of such a system is performed using a numerical iterative Newton-Raphson method
[1, 4].
Applying this method to the system representing the network in matricial form gives:
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where:
JP,### : Submatrix with the derivatives of the active powers with respect to the voltage phases.
JP,V : Submatrix with the derivatives of the active powers with respect to the voltage modules.
JQ,### : Submatrix with the derivatives of the reactive powers with respect to the voltage phases.
JQ,V : Submatrix with the derivatives of the reactive powers with respect to the voltage modules.
###V : Unknown vector of the voltage modules corrections.
###### : Unknown vector of the voltage phases corrections.
###P : Vector of the known terms, active power residuals
###Q : Vector of the known terms, reactive power residuals
Each term of the Jacobian submatrixes is found differentiating the equation 2.2 with respect to
the state variables V and ### and has the following expressions:
∂
∂
Pi
=−
θi
∂
∂
Pi
= + 2V i ⋅ Gii −
Vi
∂
∂
Qi
=−
θi
∂
∂
Qi
= − 2V i ⋅ Bii −
Vi
∂
∂
Pi
= − V i ⋅V j ⋅( − Bij ⋅cos θ ij + G ij ⋅sen θ ij )
θj
∂
∂
Pi
= − V i ⋅(+ G ij ⋅cos θ ij + Bij ⋅sen θ ij )
Vj
∂
∂
Qi
= V i ⋅V j ⋅( + G ij ⋅cos θ ij + Bij ⋅sen θ ij )
θj
N
∑
j =1
j≠ i
V i ⋅V j ⋅ ( + Bij ⋅ cos θ ij − G ij ⋅ sen θ ij )
N
∑
j =1
j≠ i
N
∑
j =1
j≠i
V j ⋅ ( + G ij ⋅ cos θ ij + Bij ⋅ sen θ ij )
V i ⋅V j ⋅ ( + Gij ⋅ cos θ ij + Bij ⋅ sen θ ij )
N
∑
j =1
j≠i
V j ⋅ ( − Bij ⋅ cos θ ij + Gij ⋅ sen θ ij )
(2.4)
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p. 11
The terms ###V and ###### represent the unknowns and are the corrections to be applied to the
vectors of the phases and modules of the nodal voltages.
The known term results from the difference between the values of the parameters P (load in
MW and MVAr) and the values of the active and reactive powers calculated through the (2.2)
with the current values of V e ### .
The method solves the system (2.3) applying the complete Jacobian matrix with the four
submatrixes [JP,###], [JP,V ], [J,Q###], [JQ,###], calculated at each iteration.
At the i-th iteration the solution of the (2.3) gives the ###V and ###### corrections to be applied
to the solution of the previous iteration and the state variables result::
V (i ) = V (i − 1) + ∆ V
θ (i ) = θ (i − 1) + ∆ θ
(2.5)
The characteristics of sparsity of the admittance matrix representing the system are also typical
of the Jacobian matrix. Then all the techniques to solve systems characterized by sparse matrixes
are also applied to the system (2.3) [2].
This, together with the use of a symbolic solution of the system (the network topology remains
unchanged during the solution), makes the model particularly fast also in the solution of large
networks.
Besides, the model is assisted, in case of convergence difficulties or of oscillations around the
solution, by a correction method minimizing the mistakes contained in the solution vectors ###V
e ###### [3].
When elements such as TCUL, Phase-shifters or controlled series impedances are present in the
network, the system (2.3) is modified to include the new state variables of these devices
(transformation ratio, PHS angle, etc.).
For further details see the chapter relevant to the models of each component.
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p. 12
as regards the Jacobian, the terms JP,V e JQ,### are equal to zero. So two linear systems, that can
be solved separately, are obtained.
J P ,θ ⋅ ∆ θ = ∆ P
(2.7)
J Q,V ⋅ ∆ V = ∆ Q
The Jacobian terms and the known terms of the system are calculated as in the (2.2) and (2.4)
and are not calculated at each iteration.
To improve the convergence of the method the value of the ###P and ###Q residuals has been
maintained with the following equation, computed at the k-th Newton iteration:
N K − th =
∑ [( ∆ P ) + ( ∆ Q ) ]
2
K − th
2
K − th
(2.8)
The (2.8), called residual function, is used as an index of convergence; if NK >NK-1 the values of
###V and ###### obtained in the k-th iteration are reduced of an amount and, with these new
values, the k-th iteration is repeated.
After three attempts, if the method continues to diverge, the program calculate again the
Jacobian starting from the solution with the minimum residual function.
The decoupled method can be assisted by a Gauss-Sidel solution performed at the beginning or
among the Newton iterations. This solution performed at the beginning allows to find a starting
point for V and ### for a better convergence of the decoupled method.
The Gauss method can be used among the Newton-Raphson iterations with the decoupled
matrix in case of convergence difficulties.
The Gauss method can also be used before the solution with the Newton complete method. In
this particular case the solution with Gauss doesn't give particular advantages, while it increase
the calculation time.
The Gauss method can be applied only if negative series reactances (series compensation) are
technical report
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p. 13
[3] T. Hartkopf:
"A MODIFIED NEWTON METHOD FOR POWER FLOW CALCULATION
USING
A
MINIMISATION
TECHNIQUE
WITH
UNIVERSAL
CONVERGENCE"
[4] R. Marconato:
"SISTEMI ELETTRICI DI POTENZA" VOL.1-VOL.2 CLUP - MILANO.
technical report
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p. 14
3. LOAD-FLOW SOLUTION THROUGH THE LOAD ANALYSIS METHOD
4.
3.1 Generals
For some problems it could be sufficient to know, also not precisely, only the active power
flows in the network connections. It could be the case of:
1. a first phase of a transmission network planning, in which, with fixed generations and loads,
different network arrangements must be verified in order to guarantee the active power
transmission within the security limits;
2. simplified studies of static security, to calculate line or transformer overloads due to
network disturbances as the opening of a connection or the loose of generation groups.
The method of solution used in these cases is the so called Load Analysis Method.
3.2 Scope and validity of the model
The scope of the model is the calculation of the active power flows in the network connections.
This method takes origin from the simplification of the network admittance matrix, possible
when the longitudinal reactance of the lines is greater than the related resistance (typical of
transmission networks). Besides the longitudinal branches, being essentially reactive, doesn't
influence the active power flows. From this follows the hypothesis that, to calculate these flows,
the elements of the system can be represented using only the longitudinal reactances.
The hypothesis and limits of the model are the following:
- The method can only be applied in the electrical systems with the hypothesis X>>R verified.
- The differences of the voltage phases at the ends of a connection are small.
- The voltage modules are near 1 p.u. (as the model uses voltage modules of 1 p.u.)
- As a consequence of the hypothesis adopted for the impedance of the connections the
transmission losses can't be calculated.
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p. 15
Xij : reactance of the connection between the node i and j
###ij : voltage phase difference between the node i and j
In the hypothesis of small differences in the voltage phase and unitary modules the (3.1)
becomes:
Pij =
1
⋅ϕ ij
X ij
(3.2)
The previous equation is analogous to the DC Ohm law for a resistance Xij with a current Pij
flowing in it, between the nodes i and j with an applied voltage ###ij .
From the (3.2) the active power injection in the i-th node of a network with N nodes in p.u. is
equal to:
Pi =
∑
Pij =
j
∑
j
1
⋅ϕ ij
X ij
(3.3)
If the N-th node is assumed as the active power slack node, being the network without losses,
we have:
PN = −
N− 1
∑
Ph
(3.4)
h= 1
Now, assuming:
 [P ]T = [P1 , P1 ,..., PN − 1 ]
 T
[ϕ ] = [ϕ 1 ,ϕ 1 ,...,ϕ N − 1 ]
(3.5)
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p. 16
reduced matrix of the network longitudinal susceptances, including lines and interconnection
transformers
this allows, being the active power injections known, the calculation of all the voltage phases
and then, through the (3.2), of the active power flows in all the connections.
The load-flow solution is then reduced to the solution of a linear system of N-1 equations and
need no iteration.
3.4 References
[1] R. Marconato:
"SISTEMI ELETTRICI DI POTENZA" VOL.1-VOL.2 CLUP - MILANO.
technical report
4.
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p. 17
MODEL OF THE NETWORK COMPONENTS
4.1 Model of the lines
The representation of the lines in FLOWC program is of the single-line type and is obtained with
a ### model.
The double bipole representing the connection between two nodes of the network can be
symmetric or dissymmetric.
Three different models can be used:
1. symmetric bipole with lumped parameters
2. symmetric bipole with distributed parameters
3. dissymmetric bipole or with generalized parameters.
The following figure represents the model of the lines.
Z12
1
Y10
2
Y20
For the model 1 the parameters of the bipole are:
z12 = ( r12 + jx12 )⋅l
1
y10 = y 20 = ( g + j ω c )⋅l
2
For the model 2 the parameters of the bipole are:
(4.1)
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p. 18
: line capacitance in nF/km
For the model 3 the parameters of the bipole are:
Z 12 = Z 21 = ( R12 + jX 12 )
Y10 = ( G10 + j ω C10 )
Y 20 = ( G 20 + j ω C 20 )
where:
R12
X12
G10,G20
C10,C20
:
:
:
:
(4.3)
longitudinal resistance in ###
longitudinal reactance in ###
transverse conductances relevant to the ends 1 and 2 in S
transverse capacitances relevant to the ends 1 and 2 in S
4.2 Model of the transformers and LTC (on load tap-changer)
4.2.1
Model of the transformers with the ratio out of the rated value in p.u.
In general the equations of the double bipole are:
v1 = (1 + z ⋅ y 02 )⋅v 2 + z ⋅i2
i1 = ( y 01 + y 01 ⋅ z ⋅ y 02 + y 02 )⋅ v 2 + (1 + z ⋅ y 01 ) ⋅i 2
(4.4)
Taking a transformer with the following rated parameters and with a tap changer on the primary
( mT1 ) or secondary ( mT 2 ) winding(the two possibilities are reciprocally exclusive):
AN
V ,V
:machine rated apparent power
:primary and secondary rated voltages
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AB
V1B ,V2 B
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p. 19
:base power of the system
:system rated voltages at the machine terminals
Besides we indicate:
miN =
ViN
ViB
( i = 1, 2 )
(4.7)
Defining again the circuital parameters in the p.u. of the system, we have:
zB =
AB
zN
AN
yB01 =
AN
y N 01
AB
yB02 =
AN
y N 02
AB
(4.8)
And also defining:
m1 = mT 1m1N ;
m2 = m2 N
(4.9)
with tap on the side 1 (analogous expressions for tap on the side 2). It can be noted that the new
variable m1 includes both the effect of the rated voltage different from the system base voltage
( m1N ) and the effect of the taps ( mT1 ).
Using the equation of the generic double bipole and assuming the machine parameters with the
taps in the "0" position and in p.u. of the system, the parameters of the ### circuit result:
z = m1m2 z B
y01 =
m2 − m1 m2 z B
1
m − m1 1
+
⋅ ⋅ y B 01 = 2 ⋅ y B01 + 2
⋅
m ⋅z
m z
m
m
z
(4.10)
technical report
z 12
1
y01
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p. 20
2
y02
With the taps on the secondary winding, the same result would have reached.
4.2.2
LTC model - Derivatives of the parameters with respect to the regulation variablesErrore.
Il segnalibro non è definito.
The equations (4.10) will be used to express the derivatives of the active and reactive power
with respect to the tap position. This derivatives will be inserted, in the proper positions, in the
Jacobian matrix to be used in the load-flow.
The derivatives of the ### circuit parameters of the tap transformer are:
∂y
=− y
∂m1
∂y
m1 01 = − 2 y01 − y
∂m1
∂ y02
m1
= y
∂m1
m1
∂y
= − y
∂m2
∂y
m2 01 = y
∂m2
∂y02
m2
= − 2 y02 − y
∂m2
m2
(4.11)
If only the four elements with indexes i, j with i,j=1,2 of the admittance matrix of a network
with the LTC transformer between the nodes 1 and 2 are considered:
Y11 = y 01 + y +
∑ (y
01k
+ y1k ) Y12 = − y
technical report
∂Y11
∂Y
= + y − y = 0 m2 12 = + y
∂m2
∂m2
∂Y21
∂Y
m2
m2 22 = − 2 y02 − 2 y
=+ y
∂m2
∂m2
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p. 21
m2
(4.14)
The derivatives of the active power injected in the nodes are:
a) primary taps:
m1
∂
∂
∂
m1
∂
(
)
A1
*
= − 2 y 01
+ y * ⋅V12 − y * ⋅V1 ⋅V 2 ⋅ exp ( j (θ 1 − θ 2 ))
m1
A2
= − y * ⋅V 2 ⋅V1 ⋅ exp ( j (θ 2 − θ 1 ))
m1
(4.15)
b) secondary taps
m2
∂
∂
∂
m2
∂
A1
= − y * ⋅V1 ⋅V 2 ⋅ exp( j (θ 1 − θ 2 ))
m2
(
)
A2
*
= − 2 y 02
+ y * ⋅V 22 − y * ⋅V 2 ⋅V1 ⋅ exp( j (θ 2 − θ 1 ))
m2
(4.16)
Finally it must be remembered that the variables m1 e m2 doesn't directly give the tap value,
because:
mi = mTi miN ( i = 1,2)where :
miN = ViN ViB
mTi = 1 ± ε i = tapvalue
(4.17)
technical report
m1
∂
∂
P2
= − V1 ⋅V 2 ⋅ ( + g ⋅ cos(θ 1 − θ 2 ) − b ⋅ sen (θ 1 − θ 2 ))
m1
m1
∂
∂
Q2
= − V1 ⋅V 2 ⋅ ( − b ⋅ cos(θ 1 − θ 2 ) − g ⋅ sen (θ 1 − θ 2 ))
m1
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p. 22
2) with taps on the secondary winding:
m2
∂
∂
∂
m2
∂
∂
m2
∂
m2
∂
∂
P1
= − V1 ⋅V 2 ⋅ ( + g ⋅ cos(θ 1 − θ 2 ) + b ⋅ sen (θ 1 − θ 2 ))
m2
Q1
= − V1 ⋅V 2 ⋅ ( − b ⋅ cos(θ 1 − θ 2 ) + g ⋅ sen (θ 1 − θ 2 ))
m2
P2
= − 2( g 02 + g ) ⋅V 22 − V1 ⋅V 2 ⋅ ( + g ⋅ cos(θ 1 − θ 2 ) − b ⋅ sen (θ 1 − θ 2 ))
m2
(4.19)
Q2
= + 2( b02 + b ) ⋅V 22 − V1 ⋅V 2 ⋅ ( − b ⋅ cos(θ 1 − θ 2 ) − g ⋅ sen (θ 1 − θ 2 ))
m2
These expressions, both if derivatives with respect to m1 and to m2 , must be placed in the cells
relevant to the derivatives with respect to V1 or V2 , depending on the controlled node.
4.2.3
Three windings transformers
The presence of three windings transformers of LTC type makes the calculation of the
admittance matrix and of the Jacobian even more complex, above all considering that in the
FLOWC program the triangular model is used.
The equations have been consequently modified.
technical report
SR 95/######
p. 23
4.3 Model of Phase-Shifters (PHS)
4.3.1
Model of the double bipole in p.u.Errore. Il segnalibro non è definito.
Assuming the following parameters for a PHS, inserted in an electric system:
ViN
rated voltage of the transformer(the effects of the LTC taps included) at the i ,
ViB
system rated voltage at the i ,
mi = ViN ViB
i = 1, 2 terminals
i = 1, 2 terminals
p.u. rate between these voltages;
noticing that:
mi = miT ViN 0 ViB
miT
is the effect of LTC
ViN 0 rated voltage with the tap in the middle position
zN
short circuit impedance in p.u. of the machine
AN
rated power
AB
system rated power
With these parameters the following equations can be written, in p.u. of the system, for the ###
equivalent circuit, as known, is dissymmetric, in the sense that the longitudinal impedance
assumes different values if seen from the first or the second end:
y01 =
1 − n(1 + j tan α )
zecc + n 2 1 + tan 2 α z ser
(
n(1− j tan α
)
)
y12 =
n(1 + j tan α )
zecc + n2 1 + tan 2 α z ser
(
(
)
)
n 1 + tan α − n(1 − j tan α
2
2
)
(4.20)
technical report
m1
m2
n=
z ser = m22 z N
AN
AB
SR 95/######
p. 24
(4.21)
The expressions have been obtained with the first side regulating (node 1); with the regulation
on the secondary winding (node 2) analogous expressions can be written, assuming:
m2
m1
n=
4.3.2
z ser = m12 z N
AN
AB
(4.22)
Expressions of the active and reactive power
In general we can write:
A12 = V1 ⋅( y 01 ⋅V1 + y12 ⋅(V1 − V 2 )) =
*
(
)
*
= y 01
+ y12* ⋅V12 − y12* ⋅V1V 2 ⋅( cos(θ 1 − θ 2 ) + j sen (θ 1 − θ 2 ))
A21 = V 2 ⋅( y 02 ⋅V2 + y 21 ⋅(V2 − V1 )) =
(4.23)
*
(
)
*
*
*
= y 02
+ y 21
⋅V 22 − y 21
⋅V1V 2 ⋅( cos(θ 2 − θ 1 ) + j sen (θ 2 − θ 1 ))
Then it can be useful to define the terms of the admittance matrix, relevant to the only PHS
component:
Y11 Y12   y 01 + y12

= 
Y21 Y22   − y 21
− y12 
y 02 + y 21 
with these elements we can, more simply, write:
(4.24)
technical report
SR 95/######
p. 25
then:
Y11 =
Y21 =
z ecc
1
+ n 1 + tan 2 α z ser
2
(
)
− n(1 − j tan α )
zecc + n 2 1 + tan 2 α z ser
(
)
Y12 =
Y 22 =
z ecc
− n(1 + j tan α )
+ n 2 1 + tan 2 α z ser
z ecc
n 1 + tan α
+ n 2 1 + tan 2 α z ser
(
2
(
)
2
(
)
(4.26)
)
Then the powers are:
A12 =
*
z ecc
V12
n(1 − j tan α )⋅V1V 2 ⋅( cosθ 12 + j sen θ 12 )
−
2
2
*
*
*
z ecc
+ n 2 1 + tan 2 α z ecc
+ n 1 + tan α z ecc
(
(
)
)
(
)
(
)
n 1 + tan α ⋅V
n(1 + j tan α )⋅V 2V1 ⋅( cosθ 21 + j sen θ 21 )
A21 = *
−
2
2
*
*
*
z ecc + n 1 + tan α z ecc
z ecc
+ n 2 1 + tan 2 α z ecc
2
2
(
2
2
)
(4.27)
Analogously with the regulation on the secondary winding. Now it isn't opportune to divide the
denominator in its real and imaginary part. Instead the derivatives of the powers with respect to
the state variables can be expressed in the complex form.
4.3.3
Expressions of the derivatives
For shortness we indicate:
t = tan α ;
ze = zecc ;
zs = zser
(4.28)
The following quantities are used as state variables:
θ 1 , θ 2 , V1 , V2 , t , n
(4.29)
technical report
(
∂ Y12 − jn z e + n 2 (1 + jt )
=
∂t
ze + n 2 1 + t 2
(
2
(
(
)
zs
∂ Y21 jn z e + n (1 − j )
=
∂t
ze + n2 1 + t 2
)
zs
2t ⋅n z e
∂ Y22
=
2
∂t
ze + n 1 + t 2
)
(
2
2
(
zs
)
2
2
(
(
)
)
zs
2
zs
2
)
)
SR 95/######
(
(
p. 26
) )
)
∂ Y12 (1 + jt ) − ze + n2 1 + t 2 z s
=
2
∂n
z e + n2 1 + t 2
zs
(
(
(
)
(
) )
)
∂ Y21 (1 − jt ) − ze + n 1 + t z s
=
2
∂n
ze + n 2 1 + t 2
zs
(
2
(
)
2n z e
∂ Y22
=
2
∂n
ze + n 1 + t 2
(
(
)
2
zs
(4.30)
)
2
Analogous formula can be written in the case of regulation on the secondary winding.
These are the expressions of the derivatives of the admittances. The expressions of the
derivatives of the powers can be immediately derived from (4.25). Differentiating with respect to
θ i or Vi , we have the usual expressions, in which the admittances are multiplying factors;
differentiating with respect to t and n , it is sufficient to substitute in the (4.25) the expressions
of the derivatives of the admittances as expressed in (4.30). In the program the derivatives are
calculated in complex form, to take the real part for the active and the imaginary part for the
reactive.
4.4 Model of the loads
The loads can be modelled as:
1. loads with constant active and reactive power
2. loads with constant impedance
3. non-linear loads with power exponentially dependent from voltage
4. non-linear loads with power dependent from voltage with tabular law.
The non-linear loads of type 3 are characterized by the following relations:
technical report
SR 95/######
p. 27
As regards the non-linear loads of type 4 the values of the active and reactive power are taken
from the tables given by the user or read from the file created by the POWASY program, in the
case of loads simulating asynchronous motors. For voltage values between two tabular points,
the corresponding power values are estimated through linear fitting.
The active and reactive power absorbed by non-linear loads is calculated at each NewtonRaphson iteration using the voltage resulting from the previous solution.
4.5 Model of the generators
The generators are modelled as active and reactive power injections.
In the generation nodes of P,Q type this injection is assumed constant during the NewtonRaphson iterative cycle, as in the case of P,V nodes the active power P is a control variable
(scheduled value) as the reactive power Q is a dependent variable.
It is possible to introduce the limits of the capability curve with the maximum and minimum
value of the active and reactive power. Reached the solution of the network, the programs
verifies if the generators are violating the reactive limits and if the option of the generated
reactive power control is active. The values of the capability curve can be automatically
modified depending on the state of each generator with the following relations:
- Overexcited generator:
for
for
0 ≤ PG ≤ P =
2
2
PAPP
− QMAX
2
2
PAPP
− Q MAX
≤ PG ≤ PMAX
results
QG ≤ QMAX
results
2
QG ≤ PAPP
− PG2
results
QG ≥ Q MIN
- Underexcited generator:
for
0 ≤ PG ≤ PMAX
where:
P
: generated active power
technical report
SR 95/######
p. 28
The active power produced by the thermal groups can be optimized through the criterion of load
distribution with equal incremental costs [1].
Before the execution of the load-flow solution, the generated active power is re dispatched
among the groups considered controllable. The criterion is based on the minimisation of the total
generation cost of the generators, given by:
C=
N
∑
(4.32)
ci
i= 1
where:
C
: Total generation cost of the controllable active power
ci
: generation cost of each group in $/h
N
: Number of the controllable generation groups
Making the hypothesis that the cost ci only dependent on the active power generated by the
group, the (4.32) can be expressed as follows:
C=
N
∑ c ( P ) = c ( P )+ c ( P )+
i= 1
i
Gi
1
G1
2
G2
.... + c N ( PGN )
(4.33)
where:
PGi : Active power injected by the i-th controllable generation group
A set of variables PGi must be found to minimize the (4.33) and contemporary observing the
following restrictions:
N
∑
i= 1
PGi − PD ≡ f ( PG1, PG 2 ,..., PGN ) = 0
PGi ,min ≤ PGi ≤ PGi ,max
(4.34)
(4.35)
technical report
SR 95/######
C* = C − λ ⋅ f
p. 29
(4.36)
where:
f
: is the restriction (4.34)
### : Lagrange multiplier
and differentiating it we obtain the formulation of the problem of bounded minimum:
∂C *
∂C
∂f
=
− λ⋅
=0
∂PGi ∂PGi
∂PGi
where:
∂C
dci
=
∂PGi dPGi
∂f
=1
∂PGi
The terms
(4.37)
(4.38)
(4.39)
dci
represent the incremental generation costs of the i-th group and are expressed in
dPGi
$/MWh.
The cost curves ci(PGi) are generally empirically determined and are approximable by a
polynomial that, for the FLOWC program, is limited to the 2nd degree:
Ci ( PGi ) = α i + βi ⋅ PGi + γi ⋅ PGi2
(4.40)
Determining all the partial derivatives through the (4.38) the new values of the PGi can be
determined solving the following system in the hypothesis of no group at the maximum or
minimum of the capability:
technical report
SR 95/######
L ( PG1 , PG 2 ,..., PGN , λ,π G1 ,π G 2 ,...,π GN ,π G1 ,π G 2 ,...,π GN ) =
p. 30
(4.42)
The Kuhn-Tucker conditions which gives the optimal point are:
dCi
 ∂L
 ∂P = dP − λ − π Gi + π Gi = 0 i = 1,..., N
 Gi N i
 ∂L =
P − PD = 0
 ∂λ ∑i = 1 i
P ≤ P ≤ P
Gi
Gi
 Gi
π Gi ( PGi − P Gi ) = 0

π Gi ( PGi − PGi ) = 0
π Gi ≥ 0

π Gi ≥ 0
4.5.2
ReferencesErrore. Il segnalibro non è definito.
[1] Olle I. Elgerd:
"ELECTRIC ENERGY SYSTEMS THEORY" MCGRAW-HILL BOOK COMPANY.
4.6 Models of FACTS componentsErrore. Il segnalibro non è definito.
4.6.1
GeneralsErrore. Il segnalibro non è definito.
The FACTS (Flexible AC Transmission Systems) devices are conceived to increase the
"flexibility degrees" [1] of electric power systems both in normal and in emergency conditions.
In the following some FACTS devices are described from a modellistic point of view.
4.6.2
Scope and validity of the modelErrore. Il segnalibro non è definito.
The model of the FACTS device implemented in the FLOWC program is based on the following
technical report
4.6.3
SR 95/######
p. 31
Description of the modelErrore. Il segnalibro non è definito.
The FACTS of the type PHS and/or LTC (diagonal booster)
Some FACTS components optimally implement the function of the Phase-Shifters with control
on active power flow or angle and of the tap LTC transformers (with continuos regulation
instead of discreet).
The component will be treated for all the calculation phase as any other phase-shifter, eventually
of diagonal type (also with voltage control and variable transformation ratio).
Then for the models and the generals on the implementation are valid the descriptions already
reported in the chap. 4.3, relevant to the model of the phase-shifter.
The FACTS of the type Series Impedance (ZSE, RANI)
This FACTS component is used to control the active power flow between its two end nodes.
The impedance of the component is considered only as reactance, positive or negative, without
resistance. The value of the reactance is limited between a maximum (positive) and a minimum
(negative) value. Then the equation of the component are very simple. The π circuit has no
transverse branches but only the longitudinal parameter, and this one has only the imaginary
component. Then:
y01 = 0;
y02 = 0
y12 = y21 = y = jb
where, besides:
(4.44)
technical report
b≤
SR 95/######
−1
−1
; b≥
x max
xmin
p. 32
(4.47)
The apparent power flowing is:
(
Aij = b ⋅Vi
V j ⋅sen (θ i − θ j )+ jb ⋅ Vi
Pij = b ⋅Vi
V j ⋅sen (θ i − θ j )
V j ⋅cos(θ i − θ j )− Vi 2
)
(4.48)
then:
(
V j ⋅ cos(θ i − θ j ) − Vi 2
Qij = b ⋅ Vi
)
(4.49)
Then the susceptance b is used as state variable for this component .
Then the derivatives are:
∂
∂
Pi
∂b
Pj
∂b
∂
∂
V j ⋅sen (θ i − θ j )
∂
∂
= Vi
V j ⋅cos(θ i − θ j ) − Vi 2
Qj
= Vi
V j ⋅cos(θ i − θ j ) − V i 2
∂b
V j ⋅ sen (θ i − θ j )
Pij
= + Vi
V j ⋅ cos(θ i − θ j )
∂
= − Vi
V j ⋅ cos(θ i − θ j )
∂
Pij
Qi
∂b
= Vi
∂θ i
∂θ j
= − Vi
V j ⋅sen (θ i − θ j )
Pij
∂b
∂
= Vi
(4.50)
=
Pij
= V j ⋅sen (θ i − θ j )
∂Vi
∂V j
V j ⋅sen (θ i − θ j )
Pij
technical report
5.
SR 95/######
p. 33
MODELS FOR THE REACTIVE POWER CONTROLErrore. Il segnalibro non è definito.
5.1 GeneralsErrore. Il segnalibro non è definito.
The control of the violations of the reactive power limits for the generation groups (reactive
control) is performed after the calculation of the network state variables (module and phase of
the voltages).
The program can follow three different types of criteria for the reactive control, described in
detail in the following.
5.2 Scope and limits of the modelErrore. Il segnalibro non è definito.
The scope of the model is to keep the values of the reactive power generated by groups within
the capability limits through an iterative process between the Newton-Raphson solution and the
violation control. The possible types of reactive control are:
- Transformation of the P,V nodes in violation in P,Q nodes.
The limitation of this method is that the number of reactive slack nodes (P,V) is reduced
and is consequently reduced the voltage control possibility.
- Modification of the scheduled voltages in the P,V nodes through an esteemed sensitivity
coefficient.
The limitations of this model are the following:
1) It doesn't make any correction for all the generation nodes with voltage values violating
the VMIN o VMAX limit.
2) The sensitivity coefficient is very variable with the network topology and then its
correct valuation is very difficult.
- Search of an admissible functioning region with neither reactive nor voltage violations on
the generation P,V nodes.
technical report
SR 95/######
p. 34
5.3 Description of the modelsErrore. Il segnalibro non è definito.
The reactive control acts through the modification of some control variables of the network (f.i.
the nodal topology, the scheduled voltage for the P,V nodes, etc.). After these modifications the
state of the network is calculated again through the Newton-Raphson solution and the eventual
violations on the reactive power are verified. This process is iterated till all the generator are
functioning within their capability curve (if this point exists).
In the following each of the three methods is described.
-
Method i)
The P,V nodes with generators in violations are converted to P,Q type and the generated
reactive power is fixed at the capability limit in the sense of the violation (f.i. QMAX if the
power of the solution was > QMAX).
The total values of the reactive power exceeding the upper and lower limit respectively are
calculated. If the sum of the values violating the upper limit is greater than that violating the
lower limit, all the P,V generators violating the upper limit are converted to P,Q type with a
value of generated reactive power equal to the generator QMAX; vice versa if the sum of the
values violating the lower limit is greater than that violating the upper limit.
With these modifications the load-flow calculation with Newton-Raphson is executed again
to find a new network state. For the eventual node still in violation the procedure is iterated
until all the nodes are within the limits (supposing that this functioning point exists,
otherwise a maximum number of iterations is fixed).
-
Method ii)
Based on th extent of the reactive power violations the scheduled voltage in the generation
nodes (P,V) is modified using an esteemed sensitivity value ( ∆ V ∆ Q ) equal for all the
nodes given by the user (0.2 kV/MVAr as default).
technical report
SR 95/######
∆ Vi = ∆ Qi ⋅S XVQ
p. 35
(5.1)
where:
###Vi : variation of the scheduled voltage for the i-th node
###Qi : difference between the generated reactive power and the violated limit for the i-th node
SXVQ : esteemed sensitivity (###V/###Q)
- for the i-th generation node belonging to the critical set or to the set OK we have:
∆ V j = M j ⋅S XVQ ⋅
∑
∑
∆ Qj
Mj
(5.2)
where:
Mj : reactive power margin in the i-th node with respect to the limit QMAXi or QMINi violated by
the critical set
###Qj :
difference between the generated reactive power and the limit violated by the critical set
This way the scheduled voltage values for all the generation nodes (P,V and V,###) can
be defined again in the k-th iteration between the Newton-Raphson solution and the
reactive control as follows:
Vi ( K ) = Vi ( K − 1) + ∆ Vi ( K )
-
(5.3)
Method iii)
This criterion, besides controlling the reactive power generated by groups, also controls the
possible voltage violations on the generation nodes.
The method linearly solves a system described by inequalities which extremes are the
maximum and minimum values of voltage and reactive power in the generation nodes.
The solution of the system by the SOLIS1 (core of the constrained system calculation) finds
technical report
SR 95/######
Node type
U
State variables
X
P,Q
V,###
P,V
Q,###
V,###
P,Q
Q,###
P,V
p. 36
Linearizing the (5.4) around the vector of solution (X0,U0) we have:
∂F
∂X i
∆ Xi +
( X 0 ,U0 )
∂F
∂U j
∆Uj = 0
(5.5)
( X 0 ,U 0 )
From the (5.5) the variation of a state variable in the i-th node when the control variable of
the j-th node of the system is varied can be found:
∂F
∆ Xi = −
∂X i
−1
⋅
( X 0 ,U 0 )
∂F
∂U j
∆U j
(5.6)
( X 0 ,U0 )
The (5.6) in matricial form becomes::
∆ X = − JX
−1
⋅ JU ⋅ ∆ U
where:
[JX]-1 : inverse of the Jacobian matrix
[JU] : matrix of the derivatives with respect to the control variables
(5.7)
technical report

 X0 +

X 0 +

∂X
(U − U 0 ) ≥ X MIN
∂U
∂X
(U − U 0 ) ≤ X MAX
∂U
SR 95/######
p. 37
(5.8)
Operating on the system (5.8) we have:
 ∂X
 ∂U U ≥ ( X MIN − X 0 ) +
 ∂X

U ≤ ( X MAX − X 0 ) +
 ∂U
∂X
U0
∂U
∂X
U0
∂U
(5.9)
In this form it is possible to identify a system of the type [A][X]###[B] to be given as input
to the SOLIS module.
Remembering that the solution given by the SOLIS derives from a linearization of the
problem (5.5), some iterations between the SOLIS itself and Newton-Raphson can be
necessary to reach a operating state without violations.
5.4 ReferencesErrore. Il segnalibro non è definito.
[1]
A. Garzillo, M. Innorta, P. Marannino, F. Mognetti:
"HOW TO SUPPLY APPROPRIATE VAR COMPENSATION PROGRAMS TO THE
PLANNING OF AN ELECTRIC NETWORK BY THE SOLUTION OF LINEAR
INEQUALITY SYSTEM" PROC. 9TH PSCC, CASCAIS, SEPTEMBER 1987, PP.
788-792.
technical report
6.
SR 95/######
p. 38
MODEL FOR THE HVDC SYSTEMErrore. Il segnalibro non è definito.
6.1 Description of the modelErrore. Il segnalibro non è definito.
Within the FLOWC program the converters are simulated as equivalent loads connected to the
secondary winding of the conversion transformer (at the converter side).
Three different load models are present to simulate the HVDC; they are adopted depending on
the type of control for the tap-changer of the conversion transformer and the control of the
converter angle. For all the three models P and Q of the load are fixed at a value resulting from a
previous solution of the DC network. The three models are:
a) When the transformer tap-changer is of the continuous type, isn't fixed at one limit and the
converter angle is fixed, the converter node is converted to P,Q,V type. The control of the
voltage at the DC side at the scheduled value is obtained through the tap-changer on the
transformer.
b) When the transformer tap-changer is of the discrete type or is fixed at one limit and
converter angle is variable, the control of the voltage at the DC side is performed regulating
the value of the angle ### of the conversion station.
c)
When the transformer tap-changer is of the discrete type or is fixed at one limit and the
converter angle is fixed, the DC voltage can't be controlled.
In the case c) or when the limit for one control variable is reached, an iterative process between
the solution of the AC network and that of the DC one is necessary. The following steps are
performed:
i) Solution of the AC network
The converters are simulated as loads. The network solution gives the commutating voltage
technical report
SR 95/######
p. 39
It is necessary that each DC network within a mixed AC-DC electric system has only one
conversion node with the control variables fixed (tap of the transformer and angle ### of the
converter).
The fundamental equations for the three models are described in the following. For a more
detailed description see the technical report of the FLOWDC program.
-
Model A
The solution of the DC network equations allows the calculation of the voltage and of the
DC current of the converter (Ud, Id).
The following equations are valid2:
3 2
18 
3
E c cos α = U d +  X c + 2 Rc  I d + ∆ V
π
π
π

3 2
18 
 3
−
Ec cos γ = U d +  −
X c + 2 Rc  I d − ∆ V
π
π
 π

where:
Ec : Commutating voltage
### :
### :
Ud : DC voltage
Id : DC current
Xc : Commutating reactance
Rc : Commutating resistance
###V :
Rectifier
(6.1)
Inverter
(6.2)
Rectifier control angle
Inverter control angle
Voltage drop on the valves
The fundamental component of the current absorbed by the converter is:
technical report
SR 95/######
p. 40
 P2 N = U d ⋅ I d ± ∆ V ⋅ I d

18
2
2
 P1 = P2 N + 3 ⋅ Rc ⋅ I AC = P2 N + 2 ⋅ Rc ⋅ I d = P2 N + ∆ P
π

=
⋅
⋅
A
E
I
3
 1
c
AC


 Q1
= A12 − P12

18
2
 Q2 N
= Q1 − 3 ⋅ X c ⋅ I AC
= Q1 − 2 ⋅ X c ⋅ I d2 = Q1 − ∆ Q

π
2
2
U
= P2 N + Q2 N
3 ⋅ I AC
 2N
(
where:
P2N :
Q2N :
U2N :
P1 :
Q1 :
A1 :
)
Active power scheduled at the converter terminals (valve side)
Reactive power scheduled at the converter terminals (valve side)
Voltage module scheduled at the converter terminals (valve side)
Active power at the commutating busbar
Reactive power at the commutating busbar
Apparent power at the commutating busbar
The busbar 1 in the preceding figure represents the commutating point that is an ideal
point in the network, as the busbar 2 is the converter side of the transformer (valve side).
If the transformer has a continuous tap-changer without limits, the current and voltage
(Ud, Id ) conditions of the DC network can be maintained (compatibly with the existing
limits on the active and reactive power transfer) with the converter operating with a fixed
angle. In fact the tap-changer can vary the way of control of the voltage Ec within the
scheduled values.
3
(6.4)
technical report
###N and ###N
-
SR 95/######
p. 41
: scheduled angles of the rectifier and the inverter respectively.
Model B
When the tap-changer is fixed or discrete it is not possible to obtain a voltage EcN suitable
to keep the DC voltage at its scheduled value. To obtain this it is necessary to keep
constant the quantity:
Ec cos α
Ec cos γ
or
The problem can be subdivided into three parts:
-
-
Determination of the AC quantities connected with the converter operation and
influencing Ud and Id through the angle ### (### or ### for rectifier and inverter
respectively).
Determination of a possible functioning point compatible with the AC network.
Determination of all the AC quantities left.
The converter absorbs a constant current IAC, the active power losses are constant (they
only depends on the current) and then P2 is fixed. Then:
(
Q2 = 3( I AC ⋅U 2 ) − P22
2
Q2
)
12
= Q2 (U 2 )
(6.7)
technical report
SR 95/######
p. 42
The validity of the curve is limited to values of ### satisfying ###<###min<90°.
The relation between ### and Q2 is the following:
Q2 (θ ) = Q1 (θ ) − ∆ Q =
cosθ N 

2
3 I AC EcN
 − P1 − ∆ Q
cos
θ


2
(6.8)
From EcN, ###N and ###min the value of Qmin and the relevant U2min is obtained:
U 2 min =
P22 + Q22min
(6.9)
3I AC
Then the characteristic Q2(U2) is:
Q2
2
3(IcN
U22 )-P22
Q 2min
U2min
U2
Below Q2min the model is no longer valid, since ###<###min.
Besides the contribution ∂Q ∂U to the Jacobian matrix results:
technical report
SR 95/######
p. 43
If the resulting voltage is smaller than U2min the solution is not valid and it is necessary to
modify the tap or use the model C. If, on the contrary, the voltages U2 are within the limits
all the quantities relevant to the converters are calculated:
 P1 = P2 + ∆ P
Q = Q + ∆ Q
2
 1

P12 + Q12
E
=
 c
3I AC


E
 cosθ = cosθ N ⋅ cN
Ec


 2P

1
 u = arccos
− cosθ  − θ
 P2 + Q2


1
 1


( 6.11)
In fact the overlap is varied because of the variation of ###. If u>60### more than two
valves are simultaneously commutating and the method is no longer valid (the program
execution is stopped).
technical report
-
SR 95/######
p. 44
Model C
The AC network is solved with the load values P2 and Q2 as described for the model A.
From the resulting DC voltage U2 it is possible to obtain the commutating voltage Ec in
the following way:
Ec =
(U
2
2
)
+ Rc P2 + X cQ2 + ( P2 X c − Q2 Rc )
U 22
2
2
(6.12)
The control equations for the converter with the tap and the angle fixed are:
3 2
⋅ Ec ⋅cosα N = U d +

 π

 3 2 ⋅ E ⋅cos γ = U +
c
N
d
 π
18 
3
 X c + 2 Rc  ⋅ I d + ∆ V Rectifier
π

π
18 
 3
X c + 2 Rc  ⋅ I d − ∆ V Inverter
−
π

 π
(6.13)
The model of the converter used for the DC network solution is represented by the
following equivalent circuit:
R
+
Ed
-
The solution of the DC network gives the new values of Ud and Id and, consequently, the
technical report
7.
SR 95/######
p. 45
CONTINGENCY CALCULATION IN THE FLOWC PROGRAMErrore. Il segnalibro non
è definito.
7.1 GeneralsErrore. Il segnalibro non è definito.
The term contingency calculation is referred to the solution of the load-flow problem in
situations of the network with an element lacking (n-1 conditions).
The contingency calculation in the FLOWC program can be performed on the branches (lines or
transformers) or on the network generators.
7.2 Scope and validity of the modelErrore. Il segnalibro non è definito.
-
The scope of the calculation is to verify the operation of the network after a variation in the
topology owing to the out of order of an element.
The validity conditions of the model for the contingency calculation in the branches are the
same which characterize the solution of the complete load-flow.
In the calculation of the generation contingencies three distinct load-flow are solved,
approximately describing the instants after the out of order of the generation group; these
load-flows are defined as follows:
### Inertial load-flow: it represents the state of the network in the first instant after the
contingency (before the operation of the regulators);
### Primary power/frequency regulation load-flow: it represents the state of the network
after the end of the first transient period (due to the primary regulation);
### Secondary power/frequency regulation load-flow: it represents the state of the
network after the end of the transient due to the secondary regulation;
These load-flows are intended as "snapshots" of the state of the network during the
transient due to the contingency.
technical report
SR 95/######
p. 46
complete load-flow as starting point. The symbolic solution of the system calculated for the
network in normal conditions is applied in full to the contingency calculation simply putting
to zero the admittance of the branch out of order.
-
Model for the calculation of the generation contingencies
For the calculation of the generation contingencies three different models for the three seen
load-flows are used.
Inertial load-flow
The equation of the synchronous machine motion is given by:
& P − P = P
M &δ=
m
e
a
(7.1)
where:
M
= inertia constant
&δ& = secondary derivative of the load angle
Pm = mechanical power
Pe
= electric power
Pa
= Accelerating power
For each machine the inertia constant is:
M=
Ta ⋅ Sn ⋅cosϕ
2π ⋅ f
where:
Ta
J ⋅ω 2n
= starting time of the generator =
Sn ⋅cos ϕ
Sn
= rated power of the generator
cos### = load angle of the generator
(7.2)
technical report
SR 95/######
p. 47
The inertia constant of the generation park (composed by N groups) results:
MT =
NG
∑
Mi
(7.4)
i= 1
From the (7.1) the mead variation of the system frequency can be calculated in case of out
of order of the i-th generator:
∆Ω = −
∆ Pi
MT
(7.5)
where:
###Pi = active power generated by the generator in contingency
This variation in steady-state causes a global increase in the power generated by each
group, given by the following equation:
Pg,new,i = Pg,old ,i − M i ⋅∆ Ω
(7.6)
This way the lack of power due to the out of order of the i-th generator is distributed
proportionally to the inertia constant of each group still in service.
The load-flow solution with this new dispatching causes a difference in the losses with
respect to the base case (with all the generators in operation). This difference is distributed
again with the same law, till the difference between the two load-flow is below a tolerance
value.
Load-flow with primary regulation
technical report
SR 95/######
p. 48
where:
Di
= droop of the i-th machine
The reduction of the network angular speed due to the out of order of a generator is given
by:
∆Ω =
∆P
Ereg,tot
(7.9)
The new operating point for each group participating to the regulation is calculated
through:
Pg,new,i = Pg ,old ,i − Ereg,i ⋅∆ Ω
(7.10)
The difference in the losses with respect to the base case is solved as in the inertial loadflow.
The user can define which machines participate to the primary regulation; the group not
indicated are considered with permanent droop, that is with zero regulating energy.
Load-flow with secondary regulation
The distribution of the lacking power due to the out of order of the generator for the
groups participating to the secondary regulation is performed proportionally to the active
power margin of each machine.
7.4 ReferencesErrore. Il segnalibro non è definito.
[1] R. Marconato:
technical report
8.
SR 95/######
p. 49
CALCULATION OF THE DISTANCE FROM THE VOLTAGE COLLAPSEErrore. Il
segnalibro non è definito.
8.1 GeneralsErrore. Il segnalibro non è definito.
The implemented method is based on the sensitivities calculation to determine the voltage
stability in a power system.
The distance from the voltage collapse expressed in MVA can be considered an optimal
indicator in the definition and in the geographical distribution of the active power reserve of the
electric network. This method is useful both in the planning phase and in the operation of the
power systems.
The algorithm has been implemented after a load-flow execution which gives the informations
relevant to the topology and operation of the network both in steady-state and in contingency.
8.2 Definition of the sensitivities and of the distance from the collapseErrore. Il segnalibro non
è definito.
In the following the models developed to determine the distance from the voltage collapse for a
load node of the network are showed in general terms.
In a load-flow calculation the external variables in an electric network can be classified in the
following way:
XU-
state variables ( voltages |V| and angles ### in the load busbars, P,Q type)
control variables ( voltages |V| and active power in MW in the generation busbars, P,V
type)
P - parameters ( input values: loads in MW and in MVAr, in the load busbars, P,Q type)
W - dependent variables ( generated reactive powers and angles ###, generation busbars, P,V
technical report
dW =
If the term
SR 95/######
∂W ∂U
∂W ∂X
⋅
⋅dP +
⋅
⋅dP
∂U ∂P
∂X ∂P
p. 50
(8.2)
∂U
= 0 (fixed control variables)
∂P
∂W ∂X
⋅
⋅dP
∂X ∂P
(8.3)
dW = SWX ⋅S XP ⋅dP
(8.4)
dW =
Or
in the equation (8.4) can be defined as:
SWX:
SXP:
sensitivity in the dependent variables W due to a variation in the state variables X
sensitivity in the state variables X due to a variation in the input variables P
We will use, as example, to apply what we have said, a network with N busbars (fig. 1).
N-BUSBARS NETWORK
BUS K
BUS I
BUS L
~
(P,Q)
(P,V)
(P,Q)
Fig. 1
technical report
∂Qk
dQk
∂V L
=
∂ϕ k
dϕ k
∂V L
∂Qk ∂V L
∂ϕ L ∂QI
⋅
∂ϕ k ∂ϕ L
∂ϕ L ∂QI
SR 95/######
∂V L
∂PI dQI
⋅
∂ϕ L dPI
∂PI
p. 51
(8.5)
In the equation (8.5) the sensitivities ### VL/### QI and ### ###L/### QI , calculated inverting the
complete Jacobian matrix, are present together with the variations ### QK /### VL and ### QK /
### ###L , which results differentiating the equation expressing the active power injections in
terms of elements of the nodal admittances matrix and of the voltages (in module and phase) [2].
From the equation (8.5) it is possible to have the sensitivity of the generated reactive power as
function of the apparent load in the node I, which, written in general for the generation node k
having j connections becomes:
dQK N. Branches  dQK dVL dQK dϕ L 

= ∑ 
⋅
+
⋅
dQI
L =1
 dVL dQI dϕ L dQI 
(8.6)
dQK N. Branches  dQK dVL dQK dϕ L 

= ∑ 
⋅
+
⋅
dPI
L =1
 dVL dPI dϕ L dPI 
(8.7)
and if
∆ QK =
The variation of the power of the load is
dQK
dQK
⋅∆ PI +
⋅ ∆ QI
dPI
dQI
(8.8)
technical report
SR 95/######
p. 52
The (8.11) determines the generated active power variation due to a load variation. If the active
load increase (eq. (8.9)) is distributed in the network following a criterion which considers the
active power margins in the generation busbars, then the equation (8.11) will have to consider a
further term representing the variation of the dependent variable QI when the active power of
the i-th generator is varying, then we have:
∂QK N . Gen ∂QK
= ∑
∂Pgen
i=1 ∂Qgeni
(8.12)
∂QK dQK
dQK
∂QK
=
⋅cos ϕ load +
⋅sin ϕ load +
∂SI
dPI
dQI
∂Pgen
(8.13)
Calculating the sensitivities in this node (eq. 8.13) for all the P,V generators present in the
network and determining the reactive power margin of each machine (Qmax-Qgen), it is possible
to find which generator of the network must be first saturated when a load variation is
performed in a node (P,Q type).
At each variation the state of the network is updated. The state variables of the other nodes of
the network are determined, exploiting the sensitivity matrix (linearized method), as follows:
Vi (1) = Vi ( 0 ) +
∂Vi
⋅∆ S I +
∂S I
N .Gen
ϕ i(1) = ϕ i( 0 ) +
∂ϕ i
⋅∆ S I +
∂S I
N .Gen
∑
j =1
∑
j =1
∂Vi
⋅∆ Pgen j
∂Pgen j
(8.14)
∂ϕ i
⋅∆ Pgen j
∂Pgen j
(8.15)
The exponent (1) in the (8.14) and (8.15) indicates the update of the variable in the current cycle.
technical report
SR 95/######
p. 53
The method has been implemented after the load-flow solution and is activated through an
option on the user's request.
In the following a flow-chart with the description of the most important steps of the
implemented method is shown.
BASIC LOAD-FLOW
SOLUTION
Reactive power
margin calculation
I=I+1
YES
I > Node number ?
Stop
NO
NO
P,Q node ?
YES
Derivates calculation
∂Qk/∂VL,∂Qk/∂φ L,∂Qk /∂Pgen
Reset on variables
YES
Voltage instability ?
or
Are all generators saturated ?
Calculation of the load increase in the
load-node, update of node voltages
(V and fi) using sensitivities.
Print of results.
Jacobian calculation and
factorization
NO
Jacobian matrix inversion
and sensitivity ∂Qk /∂φ L
calculation. Derivates
∂Qk /∂QI calculation and
identification of generator
to be saturated
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