PART III ADVANCED TOPICS IN POWER SYSTEM DYNAMICS
13.4 Comparison Between the Methods
The simultaneous solution methods allow rotor saliency and nonlinear loads to be readily included and are especially attractive for simulations that cover a long time period. Newton’s method, together with implicit integration formulae, allow the integration step length to be increased when the changes in the variables are not very steep. The dishonest Newton method can be used to speed up the calculations. Interfacing problems between the algebraic and differential equations do not exist.
In contrast, the partitioned solution methods are attractive for simulations that cover a shorter time interval. They are more flexible, easier to organize and allow a number of simplifications to be introduced that speed up the solution. However, unless care is taken, these simplifications may cause large interfacing errors. The majority of dynamic simulation programs described in the literature are based on partitioned solution methods.
The main characteristics of the partitioned solution methods relate to the way in which the network equations are solved. Partial matrix inversion is only attractive for simplified systems because the submatrices of the partially inverted nodal matrix are dense. If the nodal matrix is large these submatrices take up a lot of computer memory. Additionally, because of the large number of non-zero elements in these submatrices, the number of arithmetic operations needed to solve the network equations is also large. The speed of solution can be improved by assuming that the loads are linear (constant admittances) and by calculating the voltages at only a small number of load nodes thereby limiting the size of the relevant inverted submatrices. This method becomes particularly attractive when model reduction is employed based on the aggregation of coherent generators as discussed in Chapter 14. In this case, when the algorithm is reorganized, the transfer matrix that is used to predict groups of coherent generators (after certain transformations corresponding to aggregation) can also be used to solve the equations of the reduced network.
If nonlinear loads are included, or the voltage change at a certain number of loads is required, then triangular factorization is superior to partial inversion because the factor matrices remain sparse after factorization. For a typical power network the factor matrices only contain about 50 % more elements than the original admittance matrix and the number of arithmetic operations required to solve the network is not very high. If additional modifications that limit the number of iterations due to rotor saliency and nonlinear loads are included, then triangular factorization becomes by far the fastest solution method.
The properties of the computer algorithms that use Newton’s method are similar to those for the simultaneous solution method. Compared with triangular factorization, Newton’s method requires a larger computer memory and more arithmetic calculations per integration step. However, due to good convergence, Newton’s method can use a longer integration step than the factorization method, which partially compensates for the greater number of computations per step. The use of the dishonest Newton method speeds up the calculations quite considerably. Moreover, rotor saliency and nonlinear loads can be included more easily than is the case with triangular factorization.
It is worth adding that fairly recently, with the ever-increasing power of computers, there has been a tendency to developreal-time simulatorsto train operators for dispatch and security monitoring and which can also be used as the core of an online dynamic security assessment system. To make these simulators operate in real time, it is often required to split the program into independent tasks to be executed in parallel (Chai and Bose, 1993; Bialek, 1996).
14
Power System Model
Reduction – Equivalents
Because contemporary power systems are so large, power system analysis programs do not usually model the complete system in detail. This problem of modelling a large system arises for a number of reasons including:
rPractical limitations on the size of computer memory.
rThe excessive computing time required by large power systems, particularly when running dynamic simulation and stability programs.
rParts of the system far away from a disturbance have little effect on the system dynamics and it is therefore unnecessary to model them with great accuracy.
rOften parts of large interconnected systems belong to different utilities, each having its own control centre which treats the other parts of the system as external subsystems.
rIn some countries private utilities compete with each other and are reluctant to disclose detailed information about their business. This means that vital data regarding the whole system may not be available.
rEven assuming that full system data are available, maintaining the relevant databases would be very difficult and expensive.
To avoid all these problems, only a part of the system, called theinternal subsystem, is modelled in detail. The remainder of the system, called theexternal subsystem, is represented by simple models referred to as theequivalent systemor simply as theequivalent.