Load Flow (Power Flow)
The need to know the flow patterns and voltage profiles in a network was the driving force behind the development of load flow programs. Although the network is linear, load flow analysis is iterative because of nodal (busbar) constraints.
At most busbars the active and reactive powers being delivered to customers are known but the voltage level is not. As far as the load flow analysis is concerned, these busbars are referred to as PQ buses. The generators are scheduled to deliver a specific active power to the system and usually the voltage magnitude of the generator terminals is fixed by automatic voltage regulation.
These busbars are known as PV buses. As losses in the system cannot be determined before the load flow solution, one generator busbar only has its voltage magnitude specified. In order to give the required two specifications per node, this bus also has its voltage angle defined to some arbitrary value, usually zero.
This busbar is known as the slack bus. The slack bus is a mathematical requirement for the program and has no exact equivalent in reality. However, in operating practice, the total load plus the losses are not known. When a system is not in power balance, i.e., when the input power does not equal the load power plus losses, the imbalance modifies the rotational energy stored in the system.
The system frequency thus rises if the input power is too large and falls if the input power is too little. Usually a generating station and probably one machine is given the task of keeping the frequency constant by varying the input power. This control of the power entering a node can be seen to be similar to the slack bus.
The algorithms first adopted had the advantages of simple programming and minimum storage but were slow to converge requiring many iterations. The introduction of ordered elimination, which gives implicit inversion of the network matrix, and sparsity programming techniques, which reduces storage requirements, allowed much better algorithms to be used.
The Newton-Raphson method gave convergence to the solution in only a few iterations. Using Newtonian methods of specifying the problem, a Jacobian matrix containing the partial derivatives of the system at each node can be constructed. The solution by this method has quadratic convergence. This method was followed quite quickly by the Fast Decoupled Newton-Raphson method.
This exploited the fact that under normal operating conditions, and providing that the network is predominately reactive, the voltage angles are not affected by reactive power flow and voltage magnitudes are not affected by real power flow.
The Fast Decoupled method requires more iterations to converge but each iteration uses less computational effort than the Newton Raphson method. A further advantage of this method is the robustness of the algorithm.
Further refinements can be added to a load flow program to make it give more realistic results. Transformer on-load tap changers, voltage limits, active and reactive power limits, plus control of the voltage magnitudes at buses other than the local bus help to bring the results close to reality. Application of these limits can slow down convergence.
The problem of obtaining an accurate, load flow solution, with a guaranteed and fast convergence has resulted in more technical papers than any other analysis topic. This is understandable when it is realized that the load flow solution is required during the running of many other types of power system analyses.
While improvements have been made, there has been no major breakthrough in performance. It is doubtful if such an achievement is possible as the time required to prepare the data and process the results represents a significant part of the overall time of the analysis.