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.Aspect 5 (The conventional electric power transmission network.
Analytical methods used.
What is an active electric power transmission network?
Use of numerical simulations.
Why is a new analytical method required?
"PP0Why a new method?(!The odes along with the nonlinear transformation can solved by packages such as EMTP or EMTDC.
However, one cannot rule out the possibility of numerical instability providing erroneous results.
Numerical instability arises from the existence of zero eigenvalues in a nonlinear system [5].
" 1Why a new method?(The system of ode s have zero eigenvalues.
The inclusion of nonlinear transformations and the presence of zero eigenvalues will give rise to bifurcation leading to several periodic and aperiodic (numerical) oscillations.8Why a new method?( Problem: The numerical solution of a stable (physical) system may be unstable. The conditions that may initiate any of these numerical instability depend on the initial conditions, which are never completely known. & 2
Aspect Six
(The conventional electric power transmission network.
Analytical methods used.
What is an active electric power transmission network?
Use of numerical simulations.
Why is a new analytical method required?
Some advantages of the new approach.P3Advantages of the new approach. (Provide tractability to a nonlinear system with large dimension such as a power system.
The large dimensional nonlinear system can be modeled as an equivalent reduced order system. 4ConclusionsIn summary, the challenges involved in modelling active EPTN s involve:
Determining the number of eigenvalues with zero real parts for a large scale system;
Eliminating the fast transient by determining the invariant manifold while still retaining their influence on the nonlinear behaviour of the system;
Eliminating time dependence by the method of averaging.
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7Challenges in Modelling Active Electric Power NetworkshDr. S.K. Chakravarthyg RC:\Program Files\Microsoft Office\Templates\Presentation Designs\high voltage.potrDr. S.K. Chakravarthyso24 Microsoft PowerPointso@@@s@@ͦGPoM P('& &&#TNPP0v
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Department of Elect. Engg., KFUPM98(, Aspects to be addressed(The conventional electric power transmission network.
Analytical methods used.
What is an active electric power transmission network?
Use of numerical simulations.
Why is a new analytical method required?
Some advantages of the new approach.P
Aspect One
(An electric power transmission network is designed to transfer bulk power between two points.
Performance of an electric power transmission network is obtained from the nominal Pi-equivalent circuit.
What is an active electric power network?
#
Aspect Two
(P The conventional electric power transmission network.
Analytical methods used.
"PPPQAnalytical Methods Used(
Analytical Methods Used(For initi
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Department of Elect. Engg., KFUPM98(,
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8DocumentSummaryInformation8 Aspects to be addressed(The conventional electric power transmission network.
Analytical methods used.
What is an active electric power transmission network?
Use of numerical simulations.
Why is a new analytical method required?
Some advantages of the new approach.P
Aspect One
(An electric power transmission network is designed to transfer bulk power between two points.
Performance of an electric power transmission network is obtained from the nominal Pi-equivalent circuit.
What is an active electric power network?
#
Aspect Two
(P The conventional electric power transmission network.
Analytical methods used.
"PPPQAnalytical Methods Used(
Analytical Methods Used(For initial values of control vector u and load vector p one can find the state variable x.
State variable x represents the bus voltages in the network. "Analytical Methods Used(Knowing the initial values of ( x, u,p) one obtains the solution at a future instant t.
This solution represents the slowly evolving dynamics of the system (due to the moment of inertia of rotating machines). %Aspect Three
(The conventional electric power transmission network.
Analytical methods used.
What is an active electric power transmission network?
"PP
#An active electric power network? $!(The extent of power that can be transferred between two points can be controlled by FACTS or Static VAR Systems.
By the use of these local controllers, the surge impedance and propagation constant change rapidly. #An active electric power network? $!(The changes in these parameters can be enforced at least once every cycle.
Consequently, the network parameters become time dependent (that is, they need to be represented by differential equations). The Problem(In modeling:
The resonant frequencies of the network will (dynamically) change.
The switching operation of the controllers makes the transmission network nonlinear.
There is a distinct possibility of the occurrence of switching bifurcations. *
)Aspect Four(The conventional electric power transmission network.
Analytical methods used.
What is an active electric power transmission network?
Use of numerical simulations.
,PP*=Analytical Solution (system involving fast and slow dynamics)>=(
-=Analytical Solution (system involving fast and slow dynamics)>>$
.Aspect 5 (The conventional electric power transmission network.
Analytical methods used.
What is an active electric power transmission network?
Use of numerical simulations.
Why is a new analytical method required?
"PP0Why a new method?(!The odes along with the nonlinear transformation can solved by packages such as EMTP or EMTDC.
However, one cannot rule out the possibility of numerical instability providing erroneous results.
Numerical instability arises from the existence of zero eigenvalues in a nonlinear system [5].
" 1Why a new method?(The system of ode s have zero eigenvalues.
The inclusion of nonlinear transformations and the presence of zero eigenvalues will give rise to bifurcation leading to several periodic and aperiodic (numerical) oscillations.8Why a new method?( Problem: The numerical solution of a stable (physical) system may be unstable. The conditions that may initiate any of these numerical instability depend on the initial conditions, which are never completely known. & 2
Aspect Six
(The conventional electric power transmission network.
Analytical methods used.
What is an active electric power transmission network?
Use of numerical simulations.
Why is a new analytical method required?
Some advantages of the new approach.P3Advantages of the new approach. (Provide tractability to a nonlinear system with large dimension such as a power system.
The large dimensional nonlinear system can be modeled as an equivalent reduced order system. 4ConclusionsIn summary, the challenges involved in modelling active EPTN s involve:
Determining the number of eigenvalues with zero real parts for a large scale system;
Eliminating the fast transient by determining the invariant manifold while still retaining their influence on the nonlinear behaviour of the system;
Eliminating time dependence by the method of averaging.
LHUH"7-: ConclusionsSince the solutions are dependent on the choice of initial conditions, numerical methods must be integrated with symbolic processing software;
Methods are required for dimensionally reducing the problem. /$&+ /
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Aspect ThreeNo Slide Title$An active electric power network? $An active electric power network? The ProblemAspect Four>Analytical Solution (system involving fast and slow dynamics)>Analytical Solution (system involving fast and slow dynamics) Aspect 5Why a new method?Why a new method?Why a new method?Aspect Six Advantages of the new approach.ConclusionsConclusionsConclusionsFonts UsedDesign TemplateEmbedded OLE Servers
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_PID_GUIDAN{8C703200-0861-11D4-B974-0080C867al values of control vector u and load vector p one can find the state variable x.
State variable x represents the bus voltages in the network. "Analytical Methods Used(Knowing the initial values of ( x, u,p) one obtains the solution at a future instant t.
This solution represents the slowly evolving dynamics of the system (due to the moment of inertia of rotating machines. %Aspect Three
(The conventional electric power transmission network.
Analytical methods used.
What is an active electric power transmission network?
"PP
#An active electric power network? $!(The extent of power that can be transferred between two points can be controlled by FACTS or Static VAR Systems.
By the use of these local controllers, the surge impedance and propagation constant change rapidly. #An active electric power network? $!(The changes in these parameters can be enforced at least once every cycle.
Consequently, the network parameters become time dependent (that is, they need to be represented by differential equations). The Problem(In modeling:
The resonant frequencies of the network will (dynamically) change.
The switching operation of the controllers makes the transmission network nonlinear.
There is a distinct possibility of the occurrence of switching bifurcations. *
)Aspect Four(The conventional electric power transmission network.
Analytical methods used.
What is an active electric power transmission network?
Use of numerical simulations.
,PP*=Analytical Solution (system involving fast and slow dynamics)>=(
-=Analytical Solution (system involving fast and slow dynamics)>>$
.Aspect 5 (The conventional electric power transmission network.
Analytical methods used.
What is an active electric power transmission network?
Use of numerical simulations.
Why is a new analytical method required?
"PP0Why a new method?(!The odes along with the nonlinear transformation can solved by packages such as EMTP or EMTDC.
However, one cannot rule out the possibility of numerical instability providing erroneous results.
Numerical instability arises from the existence of zero eigenvalues in a nonlinear system [5].
" 1Why a new method?(The system of ode s have zero eigenvalues.
The inclusion of nonlinear transformations and the presence of zero eigenvalues will give rise to bifurcation leading to several periodic and aperiodic (numerical) oscillations.8Why a new method?( Problem: The numerical solution of a stable (physical) system may be unstable. The conditions that may initiate any of these numerical instability depend on the initial conditions, which are never completely known. & 2
Aspect Six
(The conventional electric power transmission network.
Analytical methods used.
What is an active electric power transmission network?
Use of numerical simulations.
Why is a new analytical method required?
Some advantages of the new approach.P3Advantages of the new approach. (Provide tractability to a nonlinear system with large dimension such as a power system.
The large dimensional nonlinear system can be modeled as an equivalent reduced order system. 4ConclusionsIn summary, the challenges involved in modelling active EPTN s involve:
Determining the number of eigenvalues with zero real parts for a large scale system;
Eliminating the fast transient by determining the invariant manifold while still retaining their influence on the nonlinear behaviour of the system;
Eliminating time dependence by the method of averaging.
LHUH"7-: ConclusionsSince the solutions are dependent on the choice of initial conditions, numerical methods must be integrated with symbolic processing software;
Methods are required for dimensionally reducing the problem. /$&+ /
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Department of Elect. Engg., KFUPM98(, Aspects to be addressed(The conventional electric power transmission network.
Analytical methods used.
What is an active electric power transmission network?
Use of numerical simulations.
Why is a new analytical method required?
Some advantages of the new approach.P
Aspect One
(An electric power transmission network is designed to transfer bulk power between two points.
Performance of an electric power transmission network is obtained from the nominal Pi-equivalent circuit.
What is an active electric power network?
#
Aspect Two
(Analytical methods used.
"PPAnalytical Methods Used(
Analytical Methods Used(For initial values of control vector u and load vector p one can find the state variable x.
State variable x represents the bus voltages in the network. "Analytical Methods Used(Knowing the initial values of ( x, u,p) one obtains the solution at a future instant t.
This solution represents the slowly evolving dynamics of the system (due to the moment of inertia of rotating machines. %Aspect Three
(7What is an active electric power transmission network?
"7PP8
#An active electric power network? $!(The extent of power that can be transferred between two points can be controlled by FACTS or Static VAR Systems.
By the use of these local controllers, the surge impedance and propagation constant change rapidly. #An active electric power network? $!(The changes in these parameters can be enforced at least once every cycle.
Consequently, the network parameters become time dependent (that is, they need to be represented by differential equations). The Problem(In modeling:
The resonant frequencies of the network will (dynamically) change.
The switching operation of the controllers makes the transmission network nonlinear.
There is a distinct possibility of the occurrence of switching bifurcations. *
)Aspect Four(Use of numerical simulations.
,PP*=Analytical Solution (system involving fast and slow dynamics)>=(
-=Analytical Solution (system involving fast and slow dynamics)>>$
.Aspect 5 ()Why is a new analytical method required?
")PP*0Why a new method?(!The odes along with the nonlinear transformation can solved by packages such as EMTP or EMTDC.
However, one cannot rule out the possibility of numerical instability providing erroneous results.
Numerical instability arises from the existence of zero eigenvalues in a nonlinear system [5].
" 1Why a new method?(The system of ode s have zero eigenvalues.
The inclusion of nonlinear transformations and the presence of zero eigenvalues will give rise to bifurcation leading to several periodic and aperiodic (numerical) oscillations.8Why a new method?( Problem: The numerical solution of a stable (physical) system may be unstable. The conditions that may initiate any of these numerical instability depend on the initial conditions, which are never completely known. & 2
Aspect Six
($Some advantages of the new approach.%P%3Advantages of the new approach. (Provide tractability to a nonlinear system with large dimension such as a power system.
The large dimensional nonlinear system can be modeled as an equivalent reduced order system. 4ConclusionsIn summary, the challenges involved in modelling active EPTN s involve:
Determining the number of eigenvalues with zero real parts for a large scale system;
Eliminating the fast transient by determining the invariant manifold while still retaining their influence on the nonlinear behaviour of the system;
Eliminating time dependence by the method of averaging.
LHUH"7-: ConclusionsSince the solutions are dependent on the choice of initial conditions, numerical methods must be integrated with symbolic processing software;
Methods are required for dimensionally reducing the problem. /$&+ /
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(bEquation Equation.30,Microsoft Equation 3.00,bEquation Equation.30,Microsoft Equation 3.0 /0|DTimes New RomanbbDb0hbhb0DImpactew RomanbbDb0hbhb0" DMonotype SortsbbDb0hbhb00DArial NarrowsbbDb0hbhb0"@DSymbolarrowsbbDb0hbhb0
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!!#$%42$ygz2$t0[kW[suz2$V}6:SY8qEz2$H)ْpX"z2$KT\De34{z2$qMsSY<zb$>i2tddw zc$@867g4%d%dtb0lbppp@<4!d!dpvb<4BdBdpvbg4ddtb0lbp@pp?%O=6Challenges in Modelling Active Electric Power Networks8Dr. S. K. Chakravarthy
Department of Elect. Engg., KFUPM98(, Aspects to be addressed(The conventional electric power transmission network.
Analytical methods used.
What is an active electric power transmission network?
Use of numerical simulations.
Why is a new analytical method required?
Some advantages of the new approach.P;!
Aspect One
(5The conventional electric power transmission network.6P6
Aspect One
(An electric power transmission network is designed to transfer bulk power between two points.
Performance of an electric power transmission network is obtained from the nominal Pi-equivalent circuit.
What is an active electric power network?
#
Aspect Two
(Analytical methods used.
"PPAnalytical Methods Used(
Analytical Methods Used(For initial values of control vector u and load vector p one can find the state variable x.
State variable x represents the bus voltages in the network. "Analytical Methods Used(Knowing the initial values of ( x, u,p) one obtains the solution at a future instant t.
This solution represents the slowly evolving dynamics of the system (due to the moment of inertia of rotating machines. %Aspect Three
(7What is an active electric power transmission network?
"7PP8
#An active electric power network? $!(The extent of power that can be transferred between two points can be controlled by FACTS or Static VAR Systems.
By the use of these local controllers, the surge impedance and propagation constant change rapidly. #An active electric power network? $!(The changes in these parameters can be enforced at least once every cycle.
Consequently, the network parameters become time dependent (that is, they need to be represented by differential equations). The Problem(In modeling:
The resonant frequencies of the network will (dynamically) change.
The switching operation of the controllers makes the transmission network nonlinear.
There is a distinct possibility of the occurrence of switching bifurcations. *
)Aspect Four(Use of numerical simulations.
,PP*=Analytical Solution (system involving fast and slow dynamics)>=(
-=Analytical Solution (system involving fast and slow dynamics)>>$
.Aspect 5 ()Why is a new analytical method required?
")PP*0Why a new method?(!The odes along with the nonlinear transformation can solved by packages such as EMTP or EMTDC.
However, one cannot rule out the possibility of numerical instability providing erroneous results.
Numerical instability arises from the existence of zero eigenvalues in a nonlinear system [5].
" 1Why a new method?(The system of ode s have zero eigenvalues.
The inclusion of nonlinear transformations and the presence of zero eigenvalues will give rise to bifurcation leading to several periodic and aperiodic (numerical) oscillations.8Why a new method?( Problem: The numerical solution of a stable (physical) system may be unstable. The conditions that may initiate any of these numerical instability depend on the initial conditions, which are never completely known. & 2
Aspect Six
($Some advantages of the new approach.%P%3Advantages of the new approach. (Provide tractability to a nonlinear system with large dimension such as a power system.
The large dimensional nonlinear system can be modeled as an equivalent reduced order system. 4ConclusionsIn summary, the challenges involved in modelling active EPTN s involve:
Determining the number of eigenvalues with zero real parts for a large scale system;
Eliminating the fast transient by determining the invariant manifold while still retaining their influence on the nonlinear behaviour of the system;
Eliminating time dependence by the method of averaging.
LHUH"7-: ConclusionsSince the solutions are dependent on the choice of initial conditions, numerical methods must be integrated with symbolic processing software;
Methods are required for dimensionally reducing the problem. /$&+ /
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!!#$%42$ygz2$t0[kW[suz2$V}6:SY8qEz2$H)ْpX"z2$KT\De34{z2$qMsSY<zb$>i2tddw zc$@867g4%d%dtb0lbppp@<4!d!dpvb<4BdBdpvbVg4ddtb0lbp@pp?%O=6Challenges in Modelling Active Electric Power Networks8Dr. S. K. Chakravarthy
Department of Elect. Engg., KFUPM98(, Aspects to be addressed(The conventional electric power transmission network.
Analytical methods used.
What is an active electric power transmission network?
Use of numerical simulations.
Why is a new analytical method required?
Some advantages of the new approach.P;!
Aspect One
(5The conventional electric power transmission network.6P6
-Purpose of conventional transmission networks.-(An electric power transmission network is designed to transfer bulk power between two points.
Performance of an electric power transmission network is obtained from the nominal Pi-equivalent circuit.#
Aspect Two
(Analytical methods used.
"PPAnalytical Methods Used(
Analytical Methods Used(For initial values of control vector u and load vector p one can find the state variable x.
State variable x represents the bus voltages in the network. "Analytical Methods Used(Knowing the initial values of ( x, u,p) one obtains the solution at a future instant t.
This solution represents the slowly evolving dynamics of the system (due to the moment of inertia of rotating machines. %Aspect Three
(7What is an active electric power transmission network?
"7PP8
#An active electric power network? $!(The extent of power that can be transferred between two points can be controlled by FACTS or Static VAR Systems.
By the use of these local controllers, the surge impedance and propagation constant change rapidly. #An active electric power network? $!(The changes in these parameters can be enforced at least once every cycle.
Consequently, the network parameters become time dependent (that is, they need to be represented by differential equations). The Problem(In modeling:
The resonant frequencies of the network will (dynamically) change.
The switching operation of the controllers makes the transmission network nonlinear.
There is a distinct possibility of the occurrence of switching bifurcations. *
)Aspect Four(Use of numerical simulations.
,PP*=Analytical Solution (system involving fast and slow dynamics)>=(
-=Analytical Solution (system involving fast and slow dynamics)>>$
.Aspect 5 ()Why is a new analytical method required?
")PP*0Why a new method?(The odes along with the nonlinear transformation can solved by packages such as EMTP or EMTDC.
However, one cannot rule out the possibility of numerical instability providing erroneous results.
Numerical instability arises from the existence of zero eigenvalues in a nonlinear system.
1Why a new method?(The system of ode s have zero eigenvalues.
The inclusion of nonlinear transformations and the presence of zero eigenvalues will give rise to bifurcation leading to several periodic and aperiodic (numerical) oscillations.8Why a new method?( Problem: The numerical solution of a stable (physical) system may be unstable. The conditions that may initiate any of these numerical instability depend on the initial conditions, which are never completely known. & 2
Aspect Six
($Some advantages of the new approach.%P%3Advantages of the new approach. (Provide tractability to a nonlinear system with large dimension such as a power system.
The large dimensional nonlinear system can be modeled as an equivalent reduced order system. 4ConclusionsIn summary, the challenges involved in modelling active EPTN s involve:
Determining the number of eigenvalues with zero real parts for a large scale system;
Eliminating the fast transient by determining the invariant manifold while still retaining their influence on the nonlinear behaviour of the system;
Eliminating time dependence by the method of averaging.
LHUH"7-: ConclusionsSince the solutions are dependent on the choice of initial conditions, numerical methods must be integrated with symbolic processing software;
Methods are required for dimensionally reducing the problem. /$&+ /
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(bEquation Equation.30,Microsoft Equation 3.00,bEquation Equation.30,Microsoft Equation 3.0 /0|DTimes New RomanbbDb0hbhb0DImpactew RomanbbDb0hbhb0" DMonotype SortsbbDb0hbhb00DArial NarrowsbbDb0hbhb0"@DSymbolarrowsbbDb0hbhb0
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!!#$%42$ygz2$t0[kW[suz2$V}6:SY8qEz2$H)ْpX"z2$KT\De34{z2$qMsSY<zb$>i2tddw zc$@867g4%d%dtb0lbppp@<4!d!dpvb<4BdBdpvbVg4ddtb0lbp@pp?%O=6Challenges in Modelling Active Electric Power Networks8Dr. S. K. Chakravarthy
Department of Elect. Engg., KFUPM98(, Aspects to be addressed(The conventional electric power transmission network.
Analytical methods used.
What is an active electric power transmission network?
Use of numerical simulations.
Why is a new analytical method required?
Some advantages of the new approach.P;!
Aspect One
(5The conventional electric power transmission network.6P6
-Purpose of conventional transmission networks.-(An electric power transmission network is designed to transfer bulk power between two points.
Performance of an electric power transmission network is obtained from the nominal Pi-equivalent circuit.#
Aspect Two
(Analytical methods used.
"PPAnalytical Methods Used(
Analytical Methods Used(For initial values of control vector u and load vector p one can find the state variable x.
State variable x represents the bus voltages in the network. "Analytical Methods Used(Knowing the initial values of ( x, u,p) one obtains the solution at a future instant t.
This solution represents the slowly evolving dynamics of the system (due to the moment of inertia of rotating machines. %Aspect Three
(7What is an active electric power transmission network?
"7PP8
#An active electric power network? $!(The extent of power that can be transferred between two points can be controlled by FACTS or Static VAR Systems.
By the use of these local controllers, the surge impedance and propagation constant change rapidly. #An active electric power network? $!(The changes in these parameters can be enforced at least once every cycle.
Consequently, the network parameters become time dependent (that is, they need to be represented by differential equations). The Problem(In modeling:
The resonant frequencies of the network will (dynamically) change.
The switching operation of the controllers makes the transmission network nonlinear.
There is a distinct possibility of the occurrence of switching bifurcations. *
)Aspect Four(Use of numerical simulations.
,PP*=Analytical Solution (system involving fast and slow dynamics)>=(
-=Analytical Solution (system involving fast and slow dynamics)>>$
.Aspect 5 ()Why is a new analytical method required?
")PP*0Why a new method?(The odes along with the nonlinear transformation can solved by packages such as EMTP or EMTDC.
However, one cannot rule out the possibility of numerical instability providing erroneous results.
Numerical instability arises from the existence of zero eigenvalues in a nonlinear system.
1Why a new method?(The system of ode s have zero eigenvalues.
The inclusion of nonlinear transformations and the presence of zero eigenvalues will give rise to bifurcation leading to several periodic and aperiodic (numerical) oscillations.8Why a new method?( Problem: The numerical solution of a stable (physical) system may be unstable. The conditions that may initiate any of these numerical instability depend on the initial conditions, which are never completely known. & 2
Aspect Six
($Some advantages of the new approach.%P%3Advantages of the new approach. (Provide tractability to a nonlinear system with large dimension such as a power system.
The large dimensional nonlinear system can be modeled as an equivalent reduced order system. 4ConclusionsIn summary, the challenges involved in modelling active EPTN s involve:
Determining the number of eigenvalues with zero real parts for a large scale system;
Eliminating the fast transient by determining the invariant manifold while still retaining their influence on the nonlinear behaviour of the system;
Eliminating time dependence by the method of averaging.
LHUH"7-: ConclusionsSince the solutions are dependent on the choice of initial conditions, numerical methods must be integrated with symbolic processing software;
Methods are required for dimensionally reducing the problem. /$&+ /
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Department of Elect. Engg., KFUPM98(, Aspects to be addressed(The conventional electric power transmission network.
Analytical methods used.
What is an active electric power transmission network?
Use of numerical simulations.
Why is a new analytical method required?
Some advantages of the new approach.P;!
Aspect One
(5The conventional electric power transmission network.6P6
-Purpose of conventional transmission networks.-(An electric power transmission network is designed to transfer bulk power between two points.
Performance of an electric power transmission network is obtained from the nominal Pi-equivalent circuit.#
Aspect Two
(Analytical methods used.
"PPAnalytical Methods Used(
Analytical Methods Used(For initial values of control vector u and load vector p one can find the state variable x.
State variable x represents the bus voltages in the network. "Analytical Methods Used(Knowing the initial values of ( x, u,p) one obtains the solution at a future instant t.
This solution represents the slowly evolving dynamics of the system (due to the moment of inertia of rotating machines. %Aspect Three
(7What is an active electric power transmission network?
"7PP8
#An active electric power network? $!(The extent of power that can be transferred between two points can be controlled by FACTS or Static VAR Systems.
By the use of these local controllers, the surge impedance and propagation constant change rapidly. #An active electric power network? $!(The changes in these parameters can be enforced at least once every cycle.
Consequently, the network parameters become time dependent (that is, they need to be represented by differential equations). The Problem(In modeling:
The resonant frequencies of the network will (dynamically) change.
The switching operation of the controllers makes the transmission network nonlinear.
There is a distinct possibility of the occurrence of switching bifurcations. *
)Aspect Four(Use of numerical simulations.
,PP*=Analytical Solution (system involving fast and slow dynamics)>=(
-=Analytical Solution (system involving fast and slow dynamics)>>$
.Aspect 5 ()Why is a new analytical method required?
")PP*0Why a new method?(The odes along with the nonlinear transformation can solved by packages such as EMTP or EMTDC.
However, one cannot rule out the possibility of numerical instability providing erroneous results.
Numerical instability arises from the existence of zero eigenvalues in a nonlinear system.
1Why a new method?(The system of ode s have zero eigenvalues.
The inclusion of nonlinear transformations and the presence of zero eigenvalues will give rise to bifurcation leading to several periodic and aperiodic (numerical) oscillations.8Why a new method?( Problem: The numerical solution of a stable (physical) system may be unstable. The conditions that may initiate any of these numerical instability depend on the initial conditions, which are never completely known. & 2
Aspect Six
($Some advantages of the new approach.%P%3Advantages of the new approach. (Provide tractability to a nonlinear system with large dimension such as a power system.
The large dimensional nonlinear system can be modeled as an equivalent reduced order system. 4ConclusionsIn summary, the challenges involved in modelling active EPTN s involve:
Determining the number of eigenvalues with zero real parts for a large scale system;
Eliminating the fast transient by determining the invariant manifold while still retaining their influence on the nonlinear behaviour of the system;
Eliminating time dependence by the method of averaging.
LHUH"7-: ConclusionsSince the solutions are dependent on the choice of initial conditions, numerical methods must be integrated with symbolic processing software;
Methods are required for dimensionally reducing the problem. ="Conclusions
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