Load flow studies are one of the most important aspects of power system planning and operation. The load flow gives us the sinusoidal steady state of the entire system - voltages, real and reactive power generated and absorbed and line losses. Since the load is a static quantity and it is the power that flows through transmission lines, the purists prefer to call this Power Flow studies rather than load flow studies. We shall however stick to the original nomenclature of load flow.

LOAD FLOW STUDY

Overview

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.

The steady state power and reactive power supplied by a bus in a power network are expressed in terms of nonlinear algebraic equations. We therefore would require iterative methods for solving these equations. In this chapter we shall discuss two of the load flow methods. We shall also delineate how to interpret the load flow results.

Section I: Real And Reactive Power Injected in a Bus

For the formulation of the real and reactive power entering a b us, we need to define the following quantities. Let the voltage at the i ^th bus be denoted by

(4.1)

Also let us define the self admittance at bus- i as

(4.2)

Similarly the mutual admittance between the buses i and j can be written as

(4.3)

Let the power system contains a total number of n buses. The current injected at bus- i is given as

(4.4)

It is to be noted we shall assume the current entering a bus to be positive and that leaving the bus to be negative. As a consequence the power and reactive power entering a bus will also be assumed to be positive. The complex power at bus- i is then given by

(4.5)

Note that

Therefore substituting in (4.5) we get the real and reactive power as

	(4.6)
	(4.7)

Section II: 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 shown in the chart below.

Load Buses : In these buses no generators are connected and hence the generated real power P_Gi and reactive power Q_Gi 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 |V_i| and its angle δ_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 P_Gi and | V_i | for these buses. This is why such buses are also referred to as P-V buses. It is to be noted that the reactive power supplied by the generator Q_Gi depends on the system configuration and cannot be specified in advance. Furthermore we have to find the unknown angle δ_i of the bus voltage.

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 ²R losses. Since the I ²R 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.

Section III: Preparation Of Data For Load Flow

Let real and reactive power generated at bus- i be denoted by P_Gi and Q_Gi respectively. Also let us denote the real and reactive power consumed at the i ^th th bus by P_Li and Q_Li respectively. Then the net real power injected in bus- i is

(4.8)

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

(4.9)

In a similar way the mismatch between the reactive power injected and calculated values is given by

(4.10)

The purpose of the load flow is to minimize the above two mismatches. It is to be noted that (4.6) and (4.7) are used for the calculation of real and reactive power in (4.9) and (4.10). 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 ΔP_i and ΔQ_i 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. 4.1, 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 4.1. Based on this data the Y bus matrix is given in Table 4.2. 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. 4.1 The simple power system used for load flow studies.

Table 4.1 Line impedance and line charging data of the system of Fig. 4.1.

Line (bus to bus)	Impedance	Line charging ( Y /2)
1-2	0.02 + j 0.10	j 0.030
1-5	0.05 + j 0.25	j 0.020
2-3	0.04 + j 0.20	j 0.025
2-5	0.05 + j 0.25	j 0.020
3-4	0.05 + j 0.25	j 0.020
3-5	0.08 + j 0.40	j 0.010
4-5	0.10 + j 0.50	j 0.075

Table 4.2 Y_bus matrix of the system of Fig. 4.1.

	1	2	3	4	5
1	2.6923 - j 13.4115	- 1.9231 + j 9.6154	0	0	- 0.7692 + j 3.8462
2	- 1.9231 + j 9.6154	3.6538 - j 18.1942	- 0.9615 + j 4.8077	0	- 0.7692 + j 3.8462
3	0	- 0.9615 + j 4.8077	2.2115 - j 11.0027	- 0.7692 + j 3.8462	- 0.4808 + j 2.4038
4	0	0	- 0.7692 + j 3.8462	1.1538 - j 5.6742	- 0.3846 + j 1.9231
5	- 0.7692 + j 3.8462	- 0.7692 + j 3.8462	- 0.4808 + j 2.4038	- 0.3846 + j 1.9231	2.4038 - j 11.8942

The bus voltage magnitudes, their angles, the power generated and consumed at each bus are given in Table 4.3. In this table some of the voltages and their angles are given in boldface letters. This indicates that these are initial data used for starting the load flow program. The power and reactive power generated at the slack bus and the reactive power generated at the P-V bus are unknown. Therefore each of these quantities are indicated by a dash ( - ). Since we do not need these quantities for our load flow calculations, their initial estimates are not required. Also note from Fig. 4.1 that the slack bus does not contain any load while the P-V bus 5 has a local load and this is indicated in the load column.

Table 4.3 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