Chapter 6 Power Flow Analysis Part IV 6 8 Power Flow Programs Read carefully Note that if tap changing transformers are not used instead of tap setting the author uses 1 See for example the last column on page 227 Note that these Programs must be run in a very specific order see p 222 6 10 Newton Raphson Power Flow Solution Due to its quadratic convergence the N R method has rapid convergence independent from system size Thus the method usually converges in less than 10 iterations regardless of system size However as the system size number of equations increases the function evaluations at each iteration increases For the typical node shown on the right we can write yi1 Vi n Ii YijV j This equation includes bus i under the V1 V2 y i2 Ii j 1 yin summation Using polar form this equation is written n as I i Yij Vi ij j where Yij Yij ij and j 1 y i0 V j Vj j The conjugate of the complex power at Bus i bus i is given by Si Pi jQi Vi I i and using the expression for current we have n Pi jQi Vi i Yij V j ij j j 1 Solving for real and imaginary parts we have n Pi Vi Vj Yij cos ij i j 1 j and n Qi Vi Vj Yij sin ij i j 1 j If these equations are linearized as before around the point i k and Vi k using the Taylor series expansion around these points we have Vn P2 k P2 k P2 k P2 k L L n V2 Vn 2 2 k M O M O M P2 k M k k k P k M P P P M n n n n L L k n V2 Vn n Pn k 2 k k k k Q2 k Q2 k Q2 Q2 V2 Q 2 L V2 Vn M L n M 2 Q k O M M O M Vn k n M k k k k L Qn Qn Qn Qn L 2 n V2 Vn Note that P1 and Q1 for the slack bus can be computed since for this bus we know V1 and 1 hence they are not included above All quantities are assumed in per unit and the angles are in radians If some bus is a generator bus P V bus then we know P and V for that bus hence the equation in Q and V for that bus is not needed Thus we can eliminate that row and the corresponding column for the equations above Thus for each generator bus we have one less equation one less unknown to solve for If there are n buses then we have a maximum of 2 n 1 equations Let us assume we have also m n 1 generator buses then the number of equations becomes 2n 2 m equations Note also that the Jacobian matrix is evaluated at every iteration step The above equation can be put into the more compact form P J 1 J 2 Q J J V 3 4 where J 1 n 1 n 1 is the derivatives of P with respect to J 2 n 1 n 1 m is the derivative of P with respect to V J 3 n 1 m n 1 is the derivative of Q with respect to and J 4 n 1 m n 1 m is the derivative of Q with respect to V All these derivatives are the partial derivatives Starting with some initial value for V and and solving these equations for and V we can proceed to the next iteration thus i k 1 i k i k Vi k 1 Vi k Vi k 2 We can stop the iteration process when the power residuals are smaller than a prespecified value These are Pi k Pi sch Pi k and Qi k Qisch Qi k These are also known as the power mismatch The general form for the Jacobian elements are as follows For J 1 the diagonal and off diagonal elements are Pi Vi Vj Yij sin ij i j i j i Pi Vi Vj Yij sin ij i j j j i and the diagonal and off diagonal elements for J 2 are Pi 2 Vi Yii cos ii V j Yij cos ij i j Vi j i Pi Vi Yij cos ij i j Vj j i and the diagonal and off diagonal elements for J 3 are Qi Vi Vj Yij cos ij i j i j i Qi Vi Vj Yij cos ij i j j j i and the diagonal and off diagonal elements for J 4 are Qi 2 Vi Yii sin ii V j Yij sin ij i j Vi j i Qi j i Vi Yij sin ij i j Vj The procedure for the power flow solution using the Newton Raphson method is as follows 1 For load buses where Pi sch and Qisch are specified the initial voltage magnitudes and angles are set equal to that of the slack bus often unity at zero angle i e Vi 0 1 0 and i 0 0 0 For generator buses voltage regulated buses where Vi and Pi sch are specified set the voltage angle i equal to that of the slack bus usually zero i e i 0 0 0 2 For load buses Pi k and Qi k are computed using the equations for Pi and Qi derived earlier Now compute the power mismatch Pi k Pi sch Pi k and Qi k Qisch Qi k 3 3 For voltage controlled buses generator buses Pi k is computed directly hence Pi k Pi sch Pi is computed next The four elements of the Jacobian matrix are computed Now the equations involving the Jacobian are solved in an optimal manner for the change in voltage magnitudes and angles The new voltage magnitudes and angles are found thus completing one iteration The process is repeated till the residuals are less than some specified value i e k 4 5 6 7 Pi k and Qi k for all buses The method will converge at a small number of iterations usually less than 10 regardless of the size of the system In large systems the Jacobian matrix is not updated at each iteration Instead it is kept for a few iterations then updated This saves a lot of time in the larger power systems Go over example 6 10 page 235 in detail Example 6 10 Newton Raphson method V 1 05 1 0 1 04 d 0 0 0 Ps 4 2 0 Qs 2 5 YB 20 j 50 10 j 20 10 j 30 10 j 20 26 j 52 16 j 32 10 j 30 16 j 32 26 j 62 Y abs YB t angle YB iter 0 pwracur 0 00025 Power accuracy DC 10 Set the maximum power residual to a high value while max abs DC pwracur iter iter 1 P V 2 V 1 Y 2 …