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MODULE III
LOAD FLOW STUDIES
Load flow studies are important in planning and designing future expansion of power systems. The load flow gives us the sinusoidal steady state of the entire system voltages, real and reactive power generated and absorbed and line losses. Generally, load flow studies are limited to the transmission system, which involves bulk power transmission.
Through the load flow studies we can obtain the voltage magnitudes and angles at each bus in the steady state. This is rather important as the magnitudes of the bus voltages are required to be held within a specified limit. Once the bus voltage magnitudes and their angles are computed using the load flow, the real and reactive power flow through each line can be computed. Also based on the difference between power flow in the sending and receiving ends, the losses in a particular line can also be computed. Furthermore, from the line flow we can also determine the over and under load conditions. Load flow studies throw light on some of the important aspects of the system operation, such as: violation of voltage magnitudes at the buses, overloading of lines, overloading of generators, stability margin reduction, indicated by power angle differences between buses linked by a line, effect of contingencies like line voltages, emergency shutdown of generators, etc. Load flow studies are required for deciding the economic operation of the power system. They are also required in transient stability studies. Hence, load flow studies play a vital role in power system studies.
CLASSIFICATION OF BUSES
For load flow studies it is assumed that the loads are constant and they are defined by their real and reactive power consumption. It is further assumed that the generator terminal voltages are tightly regulated and therefore are constant. The main objective of the load flow is to find the voltage magnitude of each bus and its angle when the powers generated and loads are pre- specified. To facilitate this we classify the different buses of the power system as listed below.
- Load Buses : In these buses no generators are connected and hence the generated real power PGi and reactive power QGi are taken as zero. The load drawn by these buses are defined by real power P (^) Li and reactive power Q (^) Li in which the negative sign accommodates for the
power flowing out of the bus. This is why these buses are sometimes referred to as P-Q bus. The objective of the load flow is to find the bus voltage magnitude Vi and its angle (^) i.
- Voltage Controlled Buses : These are the buses where generators are connected. Therefore the power generation in such buses is controlled through a prime mover while the terminal voltage is controlled through the generator excitation. Keeping the input power constant through turbine-governor control and keeping the bus voltage constant using automatic voltage regulator, we can specify constant PGi and Vi for these buses. This is why such buses are also referred to as P-V buses.
- Slack or Swing Bus : Usually this bus is numbered 1 for the load flow studies. This bus sets the angular reference for all the other buses. Since it is the angle difference between two voltage sources that dictates the real and reactive power flow between them, the particular angle of the slack bus is not important. However it sets the reference against which angles of all the other bus voltages are measured. For this reason the angle of this bus is usually chosen as 0. Furthermore it is assumed that the magnitude of the voltage of this bus is known.
Now consider a typical load flow problem in which all the load demands are known. Even if the generation matches the sum total of these demands exactly, the mismatch between generation and load will persist because of the line I^2 R losses. Since the I^2 R loss of a line depends on the line current which, in turn, depends on the magnitudes and angles of voltages of the two buses connected to the line, it is rather difficult to estimate the loss without calculating the voltages and angles. For this reason a generator bus is usually chosen as the slack bus without specifying its real power. It is assumed that the generator connected to this bus will supply the balance of the real power required and the line losses.
REAL AND REACTIVE POWER INJECTED IN A BUS
For the formulation of the real and reactive power entering a bus, we need to define the following quantities. Let the voltage at the i th^ bus be denoted by
^
n i (^) k ik i k ik k i
P YVV
1
cos (3.6)
^ ^
n i (^) k ik i k ik k i
Q YVV
1
sin (3.7)
PREPARATION OF DATA FOR LOAD FLOW
Let real and reactive power generated at bus- i be denoted by P (^) Gi and QGi respectively. Also let us denote the real and reactive power consumed at the i th^ bus by P (^) Li and QLi respectively. Then the net real power injected in bus- i is
Pi , inj PGi P Li (3.8)
Let the injected power calculated by the load flow program be P (^) i , calc. Then the mismatch between the actual injected and calculated values is given by
P (^) i Pi , inj Pi , calc PGi PLi Pi , calc (3.9)
In a similar way the mismatch between the reactive power injected and calculated values is given by
Q (^) i Qi , inj Qi , calc QGi QLi Qi , calc (3.10)
The purpose of the load flow is to minimize the above two mismatches. It is to be noted that (3.6) and (3.7) are used for the calculation of real and reactive power in (3.9) and (3.10). However since the magnitudes of all the voltages and their angles are not known a priori, an iterative procedure must be used to estimate the bus voltages and their angles in order to calculate the mismatches. It is expected that mismatches Pi and Qi reduce with each iteration and the load flow is said to have converged when the mismatches of all the buses become less than a very small number.
For the load flow studies we shall consider the system of Fig. 3.1, which has 2 generator and 3 load buses. We define bus-1 as the slack bus while taking bus-5 as the P-V bus. Buses 2, 3 and 4 are P-Q buses. The line impedances and the line charging admittances are given in Table 3.1. Based on this data the Y (^) bus matrix is given in Table 3.2. This matrix is formed using the same
procedure as given in Section 3.1.3. It is to be noted here that the sources and their internal impedances are not considered while forming the Y (^) bus matrix for load flow studies which deal only with the bus voltages.
Fig. 3.1 The simple power system used for load flow studies. Table 3.1 Line impedance and line charging data of the system of Fig. 3.1. Line (bus to bus) Impedance Line charging ( Y /2) 1-2 0.02 + j 0.10 j 0. 1-5 0.05 + j 0.25 j 0. 2-3 0.04 + j 0.20 j 0. 2-5 0.05 + j 0.25 j 0. 3-4 0.05 + j 0.25 j 0. 3-5 0.08 + j 0.40 j 0. 4-5 0.10 + j 0.50 j 0.
Table 3.3 Bus voltages, power generated and load – initial data.
Bus no.
Bus voltage Power generated Load Magnitude (pu)
Angle (deg)
P (MW) Q
(MVAr)
P (MW) P
(MVAr) 1 1.05 (^0) 0 0 (^2) 1 0 0 0 96 62 3 1 0 0 0 35 14 4 1 0 0 0 16 8 5 1.02 0 (^48) 24 11
LOAD FLOW BY GAUSS-SEIDEL METHOD
The basic power flow equations (3.6) and (3.7) are nonlinear. In an n -bus power system, let the number of P-Q buses be n (^) p and the number of P-V (generator) buses be n (^) g such that n = n (^) p + n (^) g +
- Both voltage magnitudes and angles of the P-Q buses and voltage angles of the P-V buses are unknown making a total number of 2 n (^) p + n (^) g quantities to be determined. Amongst the known quantities are 2 n (^) p numbers of real and reactive powers of the P-Q buses, 2 n (^) g numbers of real powers and voltage magnitudes of the P-V buses and voltage magnitude and angle of the slack bus. Therefore there are sufficient numbers of known quantities to obtain a solution of the load flow problem. However, it is rather difficult to obtain a set of closed form equations from (3.6) and (3.7). We therefore have to resort to obtain iterative solutions of the load flow problem.
In the Gauss-Seidel load flow we denote the initial voltage of the i th^ bus by Vi (0)^ , i = 2, , n. This should read as the voltage of the i th^ bus at the 0 th^ iteration, or initial guess. Similarly this voltage after the first iteration will be denoted by V (^) i (1)^. In this Gauss-Seidel load flow the load buses and voltage controlled buses are treated differently. However in both these type of buses we use the complex power equation given in (3.5) for updating the voltages. Knowing the real and reactive power injected at any bus we can expand (3.5) as
i ^ i i ii i in n
n i inj iinj ik ik k P jQ V YV V YV YV YV YV
11 2 2 , , 1
We can rewrite (3.11) as
^
i ^ i i in n
iinj iinj ii i (^) V YV Y V Y V
P jQ Y
V^1 , , 1 1 2 2 (3.12)
In this fashion the voltages of all the buses are updated.
Algorithm for GS method
- Prepare data for the given system as required.
- Formulate the bus admittance matrix YBUS. This is generally done by the rule of inspection.
- Assume initial voltages for all buses, 2,3,…n. In practical power systems, the magnitude of the bus voltages is close to 1.0 p.u. Hence, the complex bus voltages at all (n-1) buses (except slack bus) are taken to be 1.0<0.This is normally referred as the flat start solution.
- Update the voltages. In any k+1 iteration, the voltages are given by
^
n ji
i jk ij jk iK j ij
iinj iinj i (^) ii
YV YV
V
P jQ Y
V
1
(^11)
, , ( )
(^1) for i=2,3,......n
Here note that when computation is carried out for bus-i, updated values are already available for buses 2,3…....(i-1) in the current (k+1) iteration. Hence these values are used. For buses (i+1)…..n, values from previous, kth^ iteration are used.
- Continue the iteration till Vi k ^1 Vik ^1 Vik for i=2,3,......n Where,ε is the tolerance
value. Generally it is customary to use a value of 0.0001 pu.
- Compute slack bus power after voltages have converged [assuming bus 1 is slack bus].
( ) 1 1 1 * 1 1 1 *
n j j^ j
S P jQ V Y V
Updating P-V Bus Voltages
It can be seen from Table 3.3 that even though the real power is specified for the P-V bus-5, its reactive power is unknown. Therefore to update the voltage of this bus, we must first estimate the reactive power of this bus. Note from Fig. 3.11 that
i i i ii i in n
n i inj i k ik k
Q V Y V V YV Y V YV Y V
, (^1) Im Im (3.16)
And hence we can write the k th^ iteration values as
Qi , inj^ ^ k ^ Im V i ^ k ^1 ^ Y i 1 V 1 Yi 2 V 2 ^ k ^ YiiVi ^ k ^1 ^ YinVn ^ k ^1 (3.17)
For the system of Fig. 3.1 we have
Q 5 , inj^ ^1 Im V 1 ^ ^0 ^ Y 51 V 1 Y 52 V 2 ^1 Y 53 V 3 ^1 Y 54 V 4 ^1 Y 55 V 5 ^0 (3.18)
This is computed as 0.0899 per unit. Once the reactive power is estimated, the bus-5 voltage is updated as
(^)
5
5 , 5 ,^1 55
51 1 YV YV YV YV
V
P jQ V (^) Y inj inj (3.19)
It is to be noted that even though the power generation in bus-5 is 48 MW, there is a local load that is consuming half that amount. Therefore the net power injected by this bus is 24 MW and consequently the injected power P (^) 5, inj in this case is taken as 0.24 per unit. The voltage is calculated as V (^) 4 (1)^ = 1.0169 0.8894. Unfortunately however the magnitude of the voltage obtained above is not equal to the magnitude given in Table 3.3. We must therefore force this voltage magnitude to be equal to that specified. This is accomplished by
5 ^1
51 5 ,^15 V
V V V
corr ^ (3.20)
This will fix the voltage magnitude to be 1.02 per unit while retaining the phase of 0.8894. The corrected voltage is used in the next iteration.
Convergence of the Algorithm
As can be seen from Table 3.3 that a total number of 4 real and 3 reactive powers are known to us. We must then calculate each of these from (3.6) and (3.7) using the values of the voltage magnitudes and their angle obtained after each iteration. The power mismatches are then calculated from (3.9) and (3.10). The process is assumed to have converged when each of P (^) 2 ,
P 3 , P (^) 4 , P (^) 5 , Q (^) 2 , Q (^) 3 and Q 4 is below a small pre-specified value. At this point the process is terminated.
Sometimes to accelerate computation in the P-Q buses the voltages obtained from (3.12) is multiplied by a constant. The voltage update of bus- i is then given by
(^) (^) 1 , 1 , 1 , 1 , ^ k iacc k i k iacc k i k iacc k Vi acc V V V V V (3.21)
where is a constant that is known as the acceleration factor. The value of has to be below 2. for the convergence to occur. Table 3.4 lists the values of the bus voltages after the 1 st^ iteration and number of iterations required for the algorithm to converge for different values of . It can be seen that the algorithm converges in the least number of iterations when is 1.4 and the
maximum number of iterations are required when is 2. In fact the algorithm will start to diverge if larger values of acceleration factor are chosen. The system data after the convergence of the algorithm will be discussed later.
where f (^) 1 , , fn are functions of the variables x (^) 1 , x (^) 2 , , x (^) n. We can then define another set of
functions g (^) 1 , , g (^) n as given below
, , , , 0
1 1
2 1 2 1 2
1 1 1 1 1
n n n n n
n n
n n
g x x f x x
g x x f x x
g x x f x x
Let us assume that the initial estimates of the n variables are x (^) 1 (0)^ , x (^) 2 (0)^ , , x (^) n (0)^. Let us add
corrections x (^) 1 (0)^ , x (^) 2 (0)^ , , x (^) n (0)^ to these variables such that we get the correct solution of these variables defined by
0 0
2 20 20
1 10 10
xn xn x n
x x x
x x x
The functions in (3.23) then can be written in terms of the variables given in (3.24) as
g (^) k x 1 , , xn gk x 1 ^0 ^ x 1 ^0 ^ ,, xn ^0 ^ xn ^0 , k 1 ,, n (3.25)
We can then expand the above equation in Taylor’s series around the nominal values of x (^) 1 (0)^ , x (^) 2 (0)^ , , x (^) n (0)^. Neglecting the second and higher order terms of the series, the expansion of g (^) k , k = 1, , n is given as
^ ^ ^ ^
0 0 0 2 20
0 1
1 , , 10 , ,^010
n k n k n k k n k x x g x x g x g x x g x x x g
where g (^) k x i ^0 is the partial derivative of g (^) k evaluated at x (^) 2 (0)^ , , x (^) n (0).
Equation (3.26) can be written in vector-matrix form as
(^)
^ ^ ^ ^
^ ^
10 0
2 10 0
1 10 0
0
20
(^010)
1 2
2 1 2 2 2
1 1 1 2 1
n n
n
n
n n n n n
n
n
g x x
g x x
g x x
x
x
x
g x g x g x
g x g x g x
g x g x g x
The square matrix of partial derivatives is called the Jacobian matrix J with J (0)^ indicating that the matrix is evaluated for the initial values of x (^) 2 (0)^ , , x (^) n (0)^. We can then write the solution of (3.27) as
^
(^)
0
20
10 0 1 0
20
10
n g n
g
g J x
x
x
Since the Taylor’s series is truncated by neglecting the 2 nd^ and higher order terms, we cannot expect to find the correct solution at the end of first iteration. We shall then have
1 0 0
21 20 20
11 10 10
xn xn x n
x x x
x x x
These are then used to find J (1)^ and g (^) k (1)^ , k = 1, , n. We can then find x (^) 2 (1)^ , , x (^) n (1)^ from an
equation like (3.28) and subsequently calculate x (^) 2 (1)^ , , x (^) n (1). The process continues till g (^) k , k = 1, , n becomes less than a small quantity.
LOAD FLOW BY NEWTON-RAPHSON METHOD
Let us assume that an n -bus power system contains a total number of n (^) p P-Q buses while the number of P-V (generator) buses be n (^) g such that n = n (^) p + n (^) g + 1. Bus-1 is assumed to be the slack bus. We shall further use the mismatch equations of Pi and Qi given in (3.9) and (3.10) respectively. The approach to Newton-Raphson load flow is similar to that of solving a system of nonlinear equations using the Newton-Raphson method: at each iteration we have to form a
p
p
p
p
n
n n n
n n
V
V P
V
V P
V
V P
V
V P
J
1
2 2 1
1 1 2 2 2 2 12
n
n n
n Q p Q p
Q Q
J
1 2
1
2 2
2 21
p
p p p
p p
n
n n n
n
n
V
Q
V V
Q
V
V
V Q
V
V Q
J
1
1 2 1
1 2
1
1 2 2 2 2 22
Load Flow Algorithm
The Newton-Raphson procedure is as follows:
Step-1: Choose the initial values of the voltage magnitudes V (0)^ of all n (^) p load buses and n 1 angles (0)^ of the voltages of all the buses except the slack bus.
Step-2: Use the estimated V (0)^ and (0)^ to calculate a total n 1 number of injected real power
Pcalc (0)^ and equal number of real power mismatch P (0)^.
Step-3: Use the estimated V (0)^ and (0)^ to calculate a total n (^) p number of injected reactive power Qcalc (0)^ and equal number of reactive power mismatch Q (0).
Step-3: Use the estimated V (0)^ and (0)^ to formulate the Jacobian matrix J (0)^.
Step-4: Solve (3.30) for (0)^ and V (0)^ V (0)^.
Step-5: Obtain the updates from
^1 ^ ^0 ^ ^0 (3.36)
^ (^)
1 0 0 1 V
V V V (3.37)
Step-6: Check if all the mismatches are below a small number. Terminate the process if yes. Otherwise go back to step-1 to start the next iteration with the updates given by (3.36) and (3.37).
Formation of the Jacobian Matrix
We shall now discuss the formation of the submatrices of the Jacobian matrix. To do that we shall use the real and reactive power equations of (3.6) and (3.7). Let us rewrite them with the help of (3.2) as
^ ^
n kk i
Pi Vi Gii 1 YikViVk ik k i (^2) cos (3.38)
^ ^
n kk i
Qi Vi Bii 1 YikViVk ik k i (^2) sin (3.39)
A. Formation of J (^) 11
Let us define J (^) 11 as
n nn
n
L L
L L
J
2
22 2 11 (3.40)
It can be seen from (3.32) that M (^) ik ’s are the partial derivatives of P (^) i with respect to k. The derivative P (^) i (3.38) with respect to k for i k is given by
L P YikViVk (^) ik k i i k k ik i sin ,
Similarly the derivative Pi with respect to k for i = k is given by
As evident from (3.33), the elements of J (^) 21 involve the derivatives of real power P with respect to magnitude of bus voltage V . For i k , we can write from (3.38)
N V VP YikViVk (^) ik k i Mik i k k ik k i cos (3.47)
For i = k we have
(^) i ii ii
n kki i ii ik i k ik k i
n kki
ii i ii i i ii ik k ik k i
V G YVV V G M
N V VP V VG YV
2 1
2
1
2 cos 2
2 cos
D. Formation of J (^) 22
For the formation of J (^) 22 let us define
p p p
p
n nn
n
O O
O O
J
2
22 2 22 (3.49)
For i k we can write from (3.39)
O V VQ ViYikViVk (^) ik k i Lik i k k ik i i sin , (3.50)
Finally for i = k we have
(^) i ii ii
n kki i ii ik i k ik k i
n kki
ii i ki i i ii ik k ik k i
V B YVV V B L
O V VQ V VB YV
2 1
2
1
2 sin 2
2 sin
We therefore see that once the submatrices J (^) 11 and J (^) 21 are computed, the formation of the submatrices J (^) 12 and J (^) 22 is fairly straightforward. For large system this will result in considerable saving in the computation time.
TAP-CHANGING AND REGULATING TRANSFORMERS
Transformers which provide a small adjustment of voltage magnitude, usually in the range o f ± 10%, and others which shift the phase angle of the line voltages are important components of a power system. Some transformers regulate both the magnitude and phase angle.
Almost all transformers provide taps on windings to adjust the ratio of transformation by changing taps when the transformer is deenergized. A change in tap can be made while the transformer is energized and such transformers a recalled load-tap-changing (LTC) transformers or tap-changing-under-load (TCUL) transformers. The tap changing is automatic and operated by motors which respond to relays set to hold the voltage at the prescribed level. Special circuits allow the change to be made without interrupting the current.
A type of transformer designed for small adjustments of voltage rather than large changes in voltage levels is called a regulating transformer Each of the three windings to which taps are made is on the same magnetic core as the phase winding whose voltage is 90° out of phase with the voltage from neutral to the point connected to the center of the tapped winding. For instance, the voltage
to neutral is increased by a component ∆ which is in phase or 180° out
of phase with ∆ .
