Formulation of Load Flow Equations | Power System | Electrical Engineering

In this article we will discuss about the formulation of load flow equations to determine load flow in the power system.

The complex power injected by the generating source into the i^th bus of a power system is given as:

S_i = P_i + j Q_i = V_i I_i*                                         i = 1, 2, …, n                  …(6.56)

where V_i is the voltage at the i^th bus with respect to ground and I_i* is the complex conjugate of source current I_i injected into the bus.

It is convenient to handle load flow problem by using I_i rather than I_i*. So, taking the complex conjugate of Eq. (6.56), we have –

S_i* = P_i – j Q_i = V_i*I_i ; n = 1, 2, 3, …., n                          …(6.57a)

 

 

 

 

Equating real and imaginary parts, we have –

 

 

 

 

 

 

 

 

So real and reactive power can now be expressed as –

 

 

 

Above Eqs. (6.59) and (6.60) are known as static load flow equations. (SLFE). These equations are nonlinear equations and, therefore, only a numerical solution is possible. For each of the n system buses we have two such equations giving a total of 2n equations (n real flow power equations and n reactive power flow equations).

Each bus is characterized by four variables P_i, Q_i, V_i and δ_i giving a total of 4n variables. To obtain a solution it is necessary to specify two variables at each bus so that the number of unknowns is reduced to 2n.

Evidently we should specify the variables over which we have physical control. The choice is influenced somewhat by the devices which are connected to a particular bus. Depending upon the quantities specified, the buses can be classified into three types – generation bus, load bus and slack bus.

The load buses are most common in power system. At these buses P_i and Q_i are known because P_Di, Q_Di are known from the load forecast data and P_Gi and Q_Gi are either zero (no generation at these buses) or specified. At the generator buses P_Gi and V_i are specified as these can be controlled by governor control and excitation control. P_i is known because P_Di is known from load forecast data.

Voltage at the slack bus is usually specified to be equal to 1 pu as the voltages throughout the system must be close to 1 pu. From elementary ac theory we know that any one phasor can be taken as reference and so the voltage of the slack or swing bus may be taken as reference phasor making its angle δ_i zero.

The solution to the power flow problem consists in assuming a certain initial bus load configuration, specifying the 2n known variables as discussed above and using some numerical method of determination of remaining 2n variables for the system with known Y_bus matrix.

The final solution must satisfy some constraints such as – (i) voltage magnitude at different buses must be within limits (ii) the real and reactive generator power at different buses must be within the minimum and maximum limits (iii) total generation must be equal to total load demand plus losses. In addition, from the point of view of system stability, δ_i cannot exceed a certain magnitude.

It has been demonstrated that load flow equations being essentially nonlinear algebraic equations, have to be solved through iterative numerical techniques. At the cost of solution accuracy, it is possible to linearize power flow equations by making suitable assumptions and approximations so that fast and explicit solutions become possible.

Approximate Load Flow Study:

Let us make the following assumptions and approximations in load flow Eqs. (6.59) and (6.60):

1. Line resistances being smaller are neglected. Shunt conductances of the overhead lines are always negligible. By making this assumption active power loss in the lines becomes zero and complexity of the equations is reduced because active power generation becomes equal to total active power demand. The effect of this assumption on Eqs. (6.59) and (6.60) is that θ_ik ≃ 90° and θ_ii = – 90°.

2. The angle δ_i is so small that sin δ_i = δ_i. This approximation converts the nonlinear power flow equations into linear ones therefore, makes analytical solution possible.

3. All buses leaving the slack or swing bus (numbered as bus 1) are voltage controlled buses, i.e., voltage magnitudes at all the buses including the slack bus are specified.

With the above assumptions and approximations, Eqs. (6.59) and (6.60) are modified to –

Since V_is are specified, Eq. (6.61) represents a set of linear algebraic equations. The only unknowns are angles 8. For slack or swing bus (bus no. 1) δ_i = 0, therefore, we have (n – 1) linear equations from which the values of δ at all buses can be determined. These values of δ when put in set of Eq. (6.62) provide the values of Q_i at all buses.

It is noteworthy that by making assumptions and approximations the Eqs. (6.61) and (6.62) have decoupled and there is no need of solving them simultaneously but can be solved sequentially solution of Eq. (6.62) follows immediately upon simultaneous solution of Eq. (6.61). Since the solution is non-iterative and the dimension is reduced to (n – 1) from 2n, it is computationally highly economical.