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.
Rearranging the elements as a function of voltages, the current equation becomes as follows:. And the current I i can be written as a function of the power as follows:. For the below 4-bus system in Figure 2 , the admittance matrix is constructed by converting all impedances in the system into admittances as shown in Figure 3. Then, diagonal and off-diagonal elements are calculated using Eqs.
Impedance diagram. Admittance diagram. The Gauss-Seidel GS method, also known as the method of successive displacement, is the simplest iterative technique used to solve power flow problems. Calculate the value g x 0 based on initial estimates x 0. In the context of a power flow problem, the unknown variables are voltages at all buses, but the slack.
The voltage V i at bus i can be calculated using either equations:. The net injected quantities are the sum of the generation minus load. Both real and reactive powers are scheduled for the load buses, and Eq. For regulated buses, only real power is scheduled. The new real part Re i new can be calculated from the specified magnitude V i and the iterative imaginary part.
Figure 4 below shows a 3-bus system. Perform 2 iterations to obtain the voltage magnitude and angles at buses 2 and 3. Impedances are given on MVA base. The admittance values of the transmission network and the injected power in per unit at buses 2 and 3 are calculated as shown in Figure 5. Note that net injected power at the load bus is negative while that of the PV bus is positive. Power flow input data. As Q 3 is not given, it is calculated based on the latest available information using Eqs.
Now that Q 3 1 is calculated, the voltage V 3 1 can be calculated:. Since the magnitude of V 3 is specified, we retain the imaginary part of V 3 1 and calculate the real part using Eq. Q 3 2 calculation is given below:. The voltage V 3 2 is calculated as follows:.
Only imaginary value of the calculated V 3 2 is retained and the real part is calculated based on the retained imaginary values and the scheduled V 3. The iterative solution is presented in Table 2. Based on x 0 , the deviation from the correct solution can be iteratively calculated. The scheduled specified quantities c are both net real P i sch and reactive power j Q i sch values at load buses and real power at generation buses as shown in Table 1.
The iterative values of reactive power are calculated using Eqs. Similarly, the iterative values of real power are calculated using Eqs. The partial derivative matrix is called the Jacobian matrix. Therefore, the Newton-Raphson power flow formulation can be solved using the below equation:. To solve for the deviation, the inverse of the Jacobian matrix is required for every iteration. Solve the power flow problem shown in Figure 3 using the Newton-Raphson technique.
Perform two iterations. To calculate the Jacobian matrix elements, P 2 , P 3 , and Q 2 equations are obtained using 4 and 5. It is worth noting that the mismatch between calculated and scheduled quantities diminishes very quickly. The iterative solution is presented in Table 3. In high voltage transmission systems, the voltage angles between adjacent buses are relatively small. These two properties result in a strong coupling between real power and voltage angle and between reactive power and voltage magnitude.
In contrary, the coupling between real power and voltage magnitude, as well as reactive power and voltage angle, is weak. Considering adjacent buses, real power flows from the bus with a higher voltage angle to the bus with a lower voltage angle.
Similarly, reactive power flows from the bus with a higher voltage magnitude to the bus with a lower voltage magnitude. Fast-decoupled power flow technique includes two steps: 1 decoupling real and reactive power calculations; 2 obtaining of the Jacobian matrix elements directly from the Y-bus.
Another advantage of this method is that the Jacobian matrix has constant term elements which are obtained and inverted once at the beginning of the iterative process. Download preview PDF. Skip to main content. This service is more advanced with JavaScript available. Advertisement Hide. Chapter First Online: 03 July This process is experimental and the keywords may be updated as the learning algorithm improves.
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