Power Flow Analysis
Here, Yij are the elements of the bus admittance matrix, Vi are the bus voltages, and Ii are the currents injected at each node. The node equation at bus i can be written as
Relationship between per-unit real and reactive power supplied to the system at bus i and the per-unit current injected into the system at that bus:
Here, Vi is the per-unit voltage at the bus; Ii* - complex conjugate of the per-unit current injected at the bus; Pi and Qi are per-unit real and reactive powers. Therefore,
Each bus is associated with following 4 variables
There are two power flow equations associated with each bus. Two of the four variables are defined an the other two are unknown in a power flow study, . That way, we have the same number of equations as the number of unknown. The known and unknown variables depend on the type of bus.
In a power system, each bus can be classified as one of three types:
- Load bus (P-Q bus): The real and reactive power are specified in this type of bus, and for which the bus voltage will be calculated. All buses having no generators are load buses. In this case, V and δ are unknown.
- Generator bus (P-V bus): In this type of bus, the magnitude of the voltage is defined and is kept constant by adjusting the field current of a synchronous generator. According to the economic dispatch, real power generation for each generator are assigned. In this case, Q and δ are unknown.
- Slack bus (swing bus): This is a special generator bus which serve as the reference bus. Its voltage is assumed to be fixed in both magnitude and phase (for instance, 1 angle ∠0° pu). In this case, P and Q are unknown.
The power flow equations are non-linear, thus cannot be solved analytically. A numerical iterative algorithm is required to solve such equations. A standard procedure follows:
- Creating a bus admittance matrix Ybus for the power system;
- Making an initial estimate for the voltages (both magnitude
- and phase angle) at each bus in the system;
- Substituting in the power flow equations and determine the deviations from the solution.
- Updating the estimated voltages based on some commonly known numerical algorithms (e.g., Newton-Raphson or Gauss-Seidel). Repeating the above process until the deviations from the solution are minimal.
Lets consider a 4 bus power system below. Assume that
- bus 1 is the slack bus and that it has a voltage V1 = 1.0∠0° pu.
- The generator at bus 3 is supplying a real power P3 = 0.3 pu to the system with a voltage magnitude 1 pu.
- The per-unit real and reactive power loads at buses 2 and 4 are P2= 0.3 pu, Q2 = 0.2 pu, P4 = 0.2 pu, Q4 = 0.15 pu.
Y-bus matrix Example:
Power Flow Solution:
We can easily calculate the power flow (both active and reactive) in each branch of the circuit by knowing the node voltages.