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![Energy Conversion Lab
GAUSE-SEIDAL METHOD
Nature of Gause-Seidal method
solution: if initial estimate x is within convergent
region, solution will converge in zigzag fashion to one
of the roots
no solution: if initial estimate x is outside convergent
region, process will diverge, no solution found
in some case, an acceleration factor α is added to
improve the rate of convergence:
x(k+1) = x(k) +α[g(x(k))-x(k)], where α>1
acceleration factor should not too large to produce
overshoot
see Ex.(6.3) for the acceleration factor used](https://image.slidesharecdn.com/apschap9powerflow-150705002455-lva1-app6891/75/APSA-LEC-9-9-2048.jpg)
![Energy Conversion Lab
NEWTON RAPHSON METHOD FOR n VARIABLES
The Newton-Raphson algorithm for n-
dimensional case is
X(k+1) = X(k) +ΔX(k) = X(k) + [J(k)]-1ΔC(k)
where
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![Energy Conversion Lab
FAST DECOUPLED POWER FLOW
Consider the Newton-Raphson power flow equation
the power flow equation reduces to
ΔP = J1Δδ = [∂P/∂δ]Δδ, ΔQ = J4Δ|V| = [∂Q/∂|V|]Δ|V|
∂Pi/∂δi = -Qi - |Vi|2Bii, Bii = |Yii|sinθii is the imaginary part of the
diagonal elements
since Bii >> Qi, ∂Pi/∂δi (diagonal elements of J1) can be further
reduced to ∂Pi/∂δi = - |Vi|Bii (|Vi|2 ≈|Vi| )
off diagonal element of J1: ∂Pi/∂δi = - |Vi||Vj|Yijsin(θij-δi+δj), since δj-δi
is quite small, θij-δi+δj = θij, J1 = ∂Pi/∂δj = - |Vi||Vj|Bij
since |Vj|≈1, off diagonal elements of J1 = ∂Pi/∂δj = - |Vi|Bij
∆
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VJ
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Q
P δ
4
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![Energy Conversion Lab
FAST DECOUPLED POWER FLOW
Consider the Newton-Raphson power flow
equation
similarly, diagonal elements of J4: ∂Qi/∂|Vi| = - |Vi|Bii
off diagonal elements of J4: ∂Qi/∂|Vj| = - |Vi|Bij
therefore, ΔP and ΔQ has the following forms
B’ and B” are the imaginary part of Ybus
the updated Δδ and Δ|V| can be obtained from
to calculate PQ bus, use simplified J1 and J4 to obtain
solution
to calculate PV bus, J4 can be further eliminated, only
J1 is used to obtain solution
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More Related Content
APSA LEC 9
- 1. Energy Conversion Lab POWERFLOW ANALYSIS Power flow analysis assumption steady-state balanced single-phase network network may contain hundreds of nodes and branches with impedance X specified in per unit on MVA base Power flow equations bus admittance matrix of node-voltage equation is formulated currents can be expressed in terms of voltages resulting equation can be in terms of power in MW
- 2. Energy Conversion Lab BUSADMITTANCE MATRIX Nodal solution nodal solution is based on the Kirchhoff’s current law impedance is converted to admittance Bus admittance equations the impedance diagram: see Fig.6.1 ijijij ij jxrZ y + == 11
- 3. Energy Conversion Lab BUSADMITTANCE MATRIX Bus admittance equations the admittance is based on bus-to- bus: see Fig.6.2 if no connection between bus-to- bus, leave as zero node voltage equation is in the form busbusbus VYI =
- 4. Energy Conversion Lab BUSADMITTANCE MATRIX Node-voltage matrix Ibus=YbusVbus Ibus is the vector of injected currents Vbus is the vector of the bus voltage from reference node Ybus is the bus admittance matrix = n i n i V V V V I I I I 2 1 2 1 nnnin2n1 iniii2i1 2n2i2221 1n1i1211 YYYY YYYY YYYY YYYY
- 5. Energy Conversion Lab BUSADMITTANCE MATRIX Node-voltage matrix diagonal element Yii: sum of admittance connected to bus i off-diagonal matrix Yij: negative of admittance between nodes I and j when the bus currents are known, bus voltages are unknown, bus voltage can be solved as inverse of bus admittance matrix is known as impedance matrix Zbus if matrix of Ybus is invertible, Ybus should be non-singular ij 0 ≠= ∑= n j ijii yY ijjiij yYY −== busbusbus IYV 1− = 1− = busbus YZ
- 6. Energy Conversion Lab BUSADMITTANCE MATRIX Node-voltage matrix admittance matrix is symmetric along the leading diagonal, which result in an upper diagonal nodal admittance matrix a typical power system network, each bus is connected by a few nearby bus, which cause many off-diagonal elements are zero many zero off-diagonal matrix is called sparse matrix the bus admittance matrix in Fig.(6.2) by inspection is − − − = 5.125.1200 5.125.220.50.5 00.575.85.2 00.55.25.8 jj j-jjj jjj jjj Ybus
- 7. Energy Conversion Lab SOLUTIONOF NONLINEAR ALGEBRA EQUATIONS Techniques for iterative solution of non-linear equations Gauss-Seidal Newton-Raphson Quasi-Newton Gause-Seidal method consider a nonlinear equation f(x)=0 rearrange f(x) so that x=g(x), f(x)=x-g(x) or f(x)=g(x)-x guess an initial estimate of x = x(k) use iteration, obtain next x value as x(k+1) = g(x(k)) criteria for stop iteration: |x(k+1)-x(k)| ≤ ε ε is the desired accuracy
- 8. Energy Conversion Lab GAUSE-SEIDALMETHOD Nature of Gause-Seidal method see Ex.(6.2) and Fig.(6.3) Gause-Seidal method needs many iterations to achieve desired accuracy no guarantee for the convergence, depend on the location of initial x estimate
- 9. Energy Conversion Lab GAUSE-SEIDALMETHOD Nature of Gause-Seidal method solution: if initial estimate x is within convergent region, solution will converge in zigzag fashion to one of the roots no solution: if initial estimate x is outside convergent region, process will diverge, no solution found in some case, an acceleration factor α is added to improve the rate of convergence: x(k+1) = x(k) +α[g(x(k))-x(k)], where α>1 acceleration factor should not too large to produce overshoot see Ex.(6.3) for the acceleration factor used
- 10. Energy Conversion Lab GAUSE-SEIDALMETHOD Extend one variable to n variable equations using Gause- Seidal method consider the system of n equations in n variables and solving for one variable from each equation in one time of iteration the updated variable x1 (k+1) calculated in first equation in Eq.(6.12) is used in the calculation of x2 (k+1) in the second equation Ex: in the 2nd iteration x2 (k+1) = c2+g2(x1 (k+1)+x2 (k)+x3 (k)+…+xn (k)) at n iteration to complete n variables, the x1 (k+1),…,xn (k+1) is tested against x1 (k),…,xn (k) for accuracy criterion nnn n n cxxxf cxxxf cxxxf = = = ),,,( ),,,( ),,,( 21 2212 1211 ),,,( ),,,( ),,,( 21 21222 21111 nnnn n n xxxgcx xxxgcx xxxgcx += += +=
- 11. Energy Conversion Lab POWERFLOW SOLUTION Power Flow (Load Flow) operating condition: balanced, single phase model quantities used in power flow equation are: voltage magnitude |V|, phase angle δ, real power P, and reactive power Q system bus classification: slack bus (swing bus): taken as reference where |V| and ∠V are specified. It makes up the loss between generated power and scheduled loads load bus (PQ bus): P and Q are specified, |V| and ∠V are unknown regulated bus (PV bus): P and |V| are specified, ∠V and Q are unknown
- 12. Energy Conversion Lab POWERFLOW EQUATION Power flow formulation consider bus case in Fig.(6.7) current flow into bus i: express Ii in terms of P,Q: the power flow equation becomes the power flow problem results in algebraic nonlinear equations which must be solved by iteration methods ij 10 ≠−= ∑∑ == j n j ij n j ijii VyyVI * i ii i V jQP I − = ij 10 * ≠−= − ∑∑ == j n j ij n j iji i ii VyyV V jQP
- 13. Energy Conversion Lab GAUSS-SEIDELPOWER FLOW EQUATION Gauss-Seidel power flow solution solving Vi: for PQ bus, assume P,Q are known solving Pi: for slack bus, assume V is known solving Qi: for PV bus, assume |V| is known ij )( )(* )1( ≠ + − = ∑ ∑+ ij k kijk i sch i sch i k i y Vy V jQP V ijRe )( 10 )()(*)1( ≠ −= ∑∑ ≠ == + k j n ij j ij n j ij k i k i k i VyyVVP ijIm )( 10 )()(*)1( ≠ −−= ∑∑ ≠ == + k j n ij j ij n j ij k i k i k i VyyVVQ
- 14. Energy Conversion Lab GAUSS-SEIDELPOWER FLOW EQUATION Instructions for Gauss-Seidel solution there are 2(n-1) equations to be solved for n bus voltage magnitude of the buses are close to 1pu or close to the magnitude of the slack bus voltage magnitude at load buses is lower than the slack bus value voltage magnitude at generator buses is higher than the slack bus value phase angle of load buses are below the reference angle phase angle of generator buses are above the reference angle
- 15. Energy Conversion Lab INSTRUCTIONSFOR G-S SOLUTION Instructions for PQ bus solution real and reactive power Pi sch, Qi sch are known starting with an initial estimate of voltage using Vi equation Instructions for PV bus solution Pi sch, |Vi| are specified assume Vi = |Vi|∠0o, solve the Qi equation as below ij )( )(* )1( ≠ + − = ∑ ∑+ ij k jijk i sch i sch i k i y Vy V jQP V ijIm )( 10 )()(*)1( ≠ −−= ∑∑ ≠ == + k j n ij j ij n j ij k i k i k i VyyVVQ
- 16. Energy Conversion Lab INSTRUCTIONSFOR G-S SOLUTION Instructions for PV bus solution when Qi (k+1) is available, solve Vi using equation below since |Vi| is specified, keep imaginary part of Vi, calculate real part of Vi solve Vi stopping criteria ij )( )(* )( )1( ≠ + − = ∑ ∑+ ij k kijk i k i sch i k i y Vy V jQP V { } { }( )2)1(2)1( Re ++ −= k ii k i VimagVV { } { })1()1()1( ImRe +++ += k i k i k i VjVV { } { } { } { } εε ≤−≤− ++ )()1()()1( ImIm,ReRe k i k i k i k i VVVV
- 17. Energy Conversion Lab INSTRUCTIONSFOR G-S SOLUTION Instructions for PV bus solution to accelerate the convergence, using the following approximation after new Vi is obtained α is in the range between 1.3 to 1.7 voltage accuracy in |Vi| and ∠δ is in the range between 0.00001 to 0.00005 ( ))()()()1( k i k cali k i k i VVVV −+=+ α
- 18. Energy Conversion Lab INSTRUCTIONSFOR G-S SOLUTION Instructions for V, ∠δ slack bus solution solve Pi solve Qi accuracy: the largest ΔPΔQ is less than the specified value, typically is about 0.001pu ijRe )( 10 )()(*)1( ≠ −= ∑∑ ≠ == + k j n ij j ij n j ij k i k i k i VyyVVP ijIm )( 10 )()(*)1( ≠ −−= ∑∑ ≠ == + k j n ij j ij n j ij k i k i k i VyyVVQ
- 19. G-S Power flowHomework For the one-line diagram shown below, using the G-S method to determine all bus voltages (magnitude and phase) and show the power flow solution between the buses assume the regulated bus (#2) reactive power limits are between 0 and 600Mvar.
- 20. Energy Conversion Lab NEWTONRAPHSON METHOD Newton Raphson method for solving one variable consider the solution of one-dimensional equation f(x)=c assume x = x(0)+∆x(0) f(x)=f(x(0)+∆x(0))=c use Taylor’s series expansion assume ∆x(0) is very small, higher order terms of expansion can be neglected, Taylor series becomes assume f(x(0))=c-∆c(0), the equation becomes ∆c(0)≅(df/dx)(0)∆x(0) the new approximation of x ( ) cx dx fd x dx df xfxxf =+∆ +∆ +=∆+ 2)0( )0( 2 2 )0( )0( )0()0()0( !2 1 )()( cx dx df xfxxf =∆ +=∆+ )0( )0( )0()0()0( )()( )0( )0( )0()1( ∆ += dx df c xx
- 21. Energy Conversion Lab NEWTONRAPHSON METHOD Newton Raphson method for solving one variable the new approximation of x Newton Raphson algorithm for more information, see Ex.(6.4) Newton’s method converges faster than Gauss-Seidal, the root may converge to a root different from the expected one or diverge if the starting value is not close enough to the root )0( )0( )0()1( ∆ += dx df c xx )()()1( )( )( )( )()( )( kkk k k k kk xxx dx df c x xfcc ∆+= ∆ =∆ −=∆ +
- 22. Energy Conversion Lab NEWTONRAPHSON METHOD FOR n VARIABLES Newton Raphson method for solving n variables nn n nnn nn n n n n cx x f x x f x x f xfxxf cx x f x x f x x f xfxxf cx x f x x f x x f xfxxf =∆ ∂ ∂ ++∆ ∂ ∂ +∆ ∂ ∂ +=∆+ =∆ ∂ ∂ ++∆ ∂ ∂ +∆ ∂ ∂ +=∆+ =∆ ∂ ∂ ++∆ ∂ ∂ +∆ ∂ ∂ +=∆+ )0( )0( )0( 2 )0( 2 )0( 1 )0( 1 )0()0()0( 2 )0( )0( 2)0( 2 )0( 2 2)0( 1 )0( 1 2)0( 2 )0()0( 2 1 )0( )0( 1)0( 2 )0( 2 1)0( 1 )0( 1 1)0( 1 )0()0( 1 )()( )()( )()(
- 23. Energy Conversion Lab NEWTONRAPHSON METHOD FOR n VARIABLES Rearrange in matrix form The matrix can be written as ΔC(k) = J(k) ΔX(k) ∆ ∆ ∆ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = − − − )0( )0( 2 )0( 1 )0()0( 2 )0( 1 )0( 2 )0( 2 2 )0( 1 2 )0( 1 )0( 2 1 )0( 1 1 )0( )0( 22 )0( 11 n n nnn n n nn x x x x f x f x f x f x f x f x f x f x f fc fc fc
- 24. Energy Conversion Lab NEWTONRAPHSON METHOD FOR n VARIABLES The Newton-Raphson algorithm for n- dimensional case is X(k+1) = X(k) +ΔX(k) = X(k) + [J(k)]-1ΔC(k) where − − − =∆ )( )( 22 )( 11 )( k nn k k k fc fc fc C ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ = )()( 2 )( 1 )( 2 )( 2 2 )( 1 2 )( 1 )( 2 1 )( 1 1 )( k n n k n k n k n kk k n kk k x f x f x f x f x f x f x f x f x f J ∆ ∆ ∆ =∆ )( )( 2 )( 1 )( k n k k k x x x X
- 25. Energy Conversion Lab NEWTONRAPHSON METHOD FOR n VARIABLES The Newton-Raphson algorithm J(k) is called the Jacobian matrix solution to X(k+1) is inefficient because it involves inverse of J(k) , a triangular factorization is used to facilitate the computation in MATLAB, the operator “” (i.e., ΔX=JΔC) is used to apply the triangular factorization Newton-Raphson method converge to solution quadratically when near a root The limitation is that it does not generally converge to a solution from an arbitrary starting point
- 26. Energy Conversion Lab LINEFLOWS AND LOSSES Complex power flow between bus i,j for line model, see Fig. 6.8 current flow from bus i to bus j current flow from bus j to bus i complex power Sij from bus i to j and Sji from j to i power loss in the line i-j for more Gauss-Seidel method examples, see Ex. (6.7) and Ex. (6.8) iijiijilij VyVVyIII 00 )( +−=+= jjijijjlji VyVVyIII 00 )( +−=+−= ** jijjiijiij IVSIVS == jiijjiL SSS +=− )(
- 27. Energy Conversion Lab NEWTON-RAPHSONPOWER FLOW Real power flow in terms of Vi , ∠δ, and Yij Reactive power flow Newton-Raphson matrix form: ΔC(k) = J(k) ΔX(k) diagonal and off-diagonal elements of J1 ( )∑= +−= n j jiijijjii YVVP 1 cos δδθ ( )∑= +−−= n j jiijijjii YVVQ 1 sin δδθ ∆ ∆ = ∆ ∆ VJJ JJ Q P δ 43 21 ( ) ( ) ijsin sin ≠+−−= ∂ ∂ +−= ∂ ∂ ∑≠ jiijijji j i ij jiijijji i i YVV P YVV P δδθ δ δδθ δ
- 28. Energy Conversion Lab NEWTON-RAPHSONPOWER FLOW Newton-Raphson matrix form: ΔC(k) = J(k) ΔX(k) diagonal and off-diagonal elements of J2 diagonal and off-diagonal elements of J3 ∆ ∆ = ∆ ∆ VJJ JJ Q P δ 43 21 ( ) ( ) ijcos coscos2 ≠+−= ∂ ∂ +−+= ∂ ∂ ∑≠ jiijiji j i ij jiijijjiiiii i i YV V P YVYV V P δδθ δδθθ ( ) ( ) ijcos cos ≠+−−= ∂ ∂ +−= ∂ ∂ ∑≠ jiijijji j i ij jiijijji i i YVV Q YVV Q δδθ δ δδθ δ
- 29. Energy Conversion Lab NEWTON-RAPHSONPOWER FLOW Newton-Raphson matrix form: ΔC(k) = J(k) ΔX(k) diagonal and off-diagonal elements of J4 power residuals ΔPi (k) ΔQi (k) new estimates for bus voltages ∆ ∆ = ∆ ∆ VJJ JJ Q P δ 43 21 ( ) ( ) ijsin sinsin2 ≠+−−= ∂ ∂ +−−−= ∂ ∂ ∑≠ jiijiji j i ij jiijijjiiiii i i YV V Q YVYV V Q δδθ δδθθ )()()()( , k i sch i k i k i sch i k i QQQPPP −=∆−=∆ )()()1()()()1( , k i k i k i k i k i k i VVV ∆+=∆+= ++ δδδ
- 30. Energy Conversion Lab NEWTON-RAPHSONPOWER FLOW Procedure for Newton-Raphson method: PQ bus: set |Vi (0)|=1.0, δi (0)=0.0 PV bus: set δi (0)=0.0 set PQ bus equation for J matrix elements: set PV bus equation for J matrix elements: )()()()( , k i sch i k i k i sch i k i QQQPPP −=∆−=∆ ( )∑= +−= n j jiijijjii YVVP 1 cos δδθ ( )∑= +−−= n j jiijijjii YVVQ 1 sin δδθ ( )∑= +−= n j jiijijjii YVVP 1 cos δδθ )()( k i sch i k i PPP −=∆
- 31. Energy Conversion Lab NEWTON-RAPHSONPOWER FLOW Procedure for Newton-Raphson method: use above equation to calculate Jacobian matrix (J1, J2, J3, J4) solve Δ|V| and Δδ using Newton-Raphson matrix update Δ|V| and Δδ by repeat the calculation until for example: see Ex.(6.10) εε ≤∆≤∆ )()( , k i k i QP ∆ ∆ = ∆ ∆ VJJ JJ Q P δ 43 21 )()()1()()()1( , k i k i k i k i k i k i VVV ∆+=∆+= ++ δδδ
- 32. Energy Conversion Lab FASTDECOUPLED POWER FLOW Fast decoupled power flow solution: the algorithm is based on Newton-Raphson method when transmission lines has a high X/R ratio, the Newton-Raphson method could be further simplified Consider the Newton-Raphson power flow equation ΔP are less sensitive to |V| and most sensitive to Δδ ΔQ is less sensitive to Δδ and most sensitive to |V| we can reasonably eliminate J2 and J3 elements in Jacobian matrix ∆ ∆ = ∆ ∆ VJJ JJ Q P δ 43 21
- 33. Energy Conversion Lab FASTDECOUPLED POWER FLOW Consider the Newton-Raphson power flow equation the power flow equation reduces to ΔP = J1Δδ = [∂P/∂δ]Δδ, ΔQ = J4Δ|V| = [∂Q/∂|V|]Δ|V| ∂Pi/∂δi = -Qi - |Vi|2Bii, Bii = |Yii|sinθii is the imaginary part of the diagonal elements since Bii >> Qi, ∂Pi/∂δi (diagonal elements of J1) can be further reduced to ∂Pi/∂δi = - |Vi|Bii (|Vi|2 ≈|Vi| ) off diagonal element of J1: ∂Pi/∂δi = - |Vi||Vj|Yijsin(θij-δi+δj), since δj-δi is quite small, θij-δi+δj = θij, J1 = ∂Pi/∂δj = - |Vi||Vj|Bij since |Vj|≈1, off diagonal elements of J1 = ∂Pi/∂δj = - |Vi|Bij ∆ ∆ = ∆ ∆ VJ J Q P δ 4 1 0 0
- 34. Energy Conversion Lab FASTDECOUPLED POWER FLOW Consider the Newton-Raphson power flow equation similarly, diagonal elements of J4: ∂Qi/∂|Vi| = - |Vi|Bii off diagonal elements of J4: ∂Qi/∂|Vj| = - |Vi|Bij therefore, ΔP and ΔQ has the following forms B’ and B” are the imaginary part of Ybus the updated Δδ and Δ|V| can be obtained from to calculate PQ bus, use simplified J1 and J4 to obtain solution to calculate PV bus, J4 can be further eliminated, only J1 is used to obtain solution VB V Q B V P ii ∆−= ∆ ∆−= ∆ ''' ,δ [ ] [ ] V Q BV V P B ∆ −=∆ ∆ −=∆ −− 11 ",'δ
- 35. Energy Conversion Lab FASTDECOUPLED POWER FLOW Comparison between fast decouple power flow solution and Newton Raphson power flow solution fast decoupled solution requires more iterations than Newton Raphson solution fast decoupled solution requires less time per iteration since decoupled solution needs less time for iteration, the overall computation time may be less than using the Newton Raphson method fast decoupled solution often used in fast computation of power flow, for example, contingency analysis or on- line control of power flow see Ex. 6.12