Introduction Of Power System Planning
Power system planning and operation on a large network is a complicated engineering problem. Recently, the growing concerns about economics and the environment have made the problem interdisciplinary and more complicated. For example, the Engineering, Economic, and Environmental Electricity Simulation Tool (E4ST) group of Cornell is focusing on the power system planning problem which not only considers the grid reliability but also pricing of the energy and emission of the NOx and SOx gases. Such problems, are complex to solve for a large-scale bulk system and require exponentially increasing large amounts of memory and computation time. Thus there exist the need to reduce the computational burden.
Two main paths to reducing the computational burden are to 1) generate small equivalent networks to replace the original large networks (network reduction) and 2) simplify the problem by making assumptions or improving the algorithms.
Network reduction methods are widely used in different studies. In this work, we focus on the reduction methods used in static analysis. Three major categories of network reduction methods have been historically used: 1) the Ward-type methods, 2) the REI-type methods and 3) the PTDF-based methods.
The Ward reduction was first proposed by J.B. Ward. A Ward-type reduced network is generated by performing partial matrix factorization. During partial matrix factorization, the external buses are eliminated by Gauss elimination. The equivalent model thus produced is often very dense and with high impedance branches. This is because the nonzero fills in the factorized matrix create fictitious (or equivalent) branches in the reduced model and the value of the fills are equal to the branch admittances. Some of the fills have extremely small values, which implies that the corresponding equivalent branches have extremely high impedances. These high-impedance branches can be discarded with minimal impact on accuracy in order to reduce the density of the equivalent model proved the impedance threshold selected is sufficiently high. When PV buses are eliminated from the model, the predictions of the Ward equivalent may deviate far from the original model when the operating point changes. This is mainly due to the elimination of external PV buses.
The voltage magnitude of the PV buses are given in the power-flow formulation and constrained in the calculations. To maintain the voltage magnitude at the specified value, reactive power is generated to support the bus voltage. When the operating point changes, the reactive power support from the PV buses is hard to approximate accurately which leads to the degradation in performance of the models produced by Ward-type and other conventional reduction methods. Two improved reduction methods were proposed to deal with the problem. The first improvement was the Ward-PV method, which retains all the external PV buses and eliminates the external load buses (PQ buses) only. This method bypasses all of the problems of eliminating PV buses and can accurately approximate the original model performance over a broader range of operating conditions.
However, when the number of PV buses is large, the reduced model is not small enough to reduce the computational burden significantly. Another method, the extended-Ward method, was derived to approximate the Ward-PV method. The main idea is to make the incremental response of VAr support close to the Ward PV method. This method adds one fictitious PV bus to every boundary PQ bus to provide reactive support.
The fictitious PV buses are radially connected to the boundary PQ buses. Though Ward reduction can be performed in a relative simple way, one drawback is that it has to split the external generators and distribute them across the boundary buses.
Two problems arise. First, for a boundary PQ bus, if fractions of external generators (PV buses) are distributed to it, the bus type is strictly neither PV nor PQ. Second, for optimal power-flow (OPF) studies, the generator fractions make the equivalents unusable. One way to solve the problems is to use to the REI reduction method and the other way is to keep the generators whole and to move/place the external generators at “appropriate” boundary buses.
The REI (radial equivalent independent) was first introduced by P. Dimo in 1975. It has been implemented and improved by many researchers. The REI equivalence is a bus-aggregation-based equivalencing technique. The general steps of REI require one to:
1. Define the essential buses and non-essential buses. The non-essential buses are to be equivalenced.
2. Group the non-essential buses into different study areas.
3. Create a zero power balance network for each study area.
4. Eliminate all zero injection buses in all zero-power-balance networks via Gaussian elimination.
The REI method groups the external buses instead of splitting them by constructing the zero-power balance network. It can avoid the problem of assigning boundary bus types.
Unlike the Ward reduction, which generates a reduced network (topology and branch reactances) independent to the different operating conditions. The REI method is a hot start method, which needs the power-flow solution of the base case. As a result, the REI reduction has following two properties:
1. The REI reduction is case dependent. The equivalent is created based on the base-case power-flow solution.
2. At the base case the REI equivalent can perform exactly the same as the original system; however when the operating condition changes, the predictions become approximate, with the approximation growing worse the further the operating point moves from the base case.
The property 2 above motivated researchers to develop the online calibrating methods so as to make the REI equivalent perform as close to the original system as possible in different operating conditions. An X-REI method was proposed in and an SREI method was proposed in. Both X-REI and S-REI methods enable online calibrating.
The X-REI adds one calibrating bus to each of the zero–power-balance networks which updates the boundary bus power injection in accordance with the changes of the operating condition. The S-REI method solves an overdetermined problem (obtained from redundant real-time measurements, i.e., state estimation) to update the boundary power injections and applies system identification techniques to update the equivalent network parameters.
In addition, a critical factor of the REI method is the criteria used for grouping the external buses. In and, theoretical studies were performed and strict criteria on bus grouping were proposed. However, the theoretical criteria are too strict and hard to implement in practical analysis. Some heuristic criteria are purposed in and.
One new group of reduction methods, strictly applicable to only dc models, was proposed which are based on the power transfer distribution factors (PTDF).
These methods focus on approximating the original system’s interactions between areas and generate the network equivalents using the following steps:
1. Group all buses into different areas.
2. Calculate the area PTDF of the original system.
3. Represent each area as a fictitious bus and connect the adjacent areas with fictitious branches. Calculate the fictitious branch admittance.
This method has proved to be useful in planning studies. However, several challenges need to be dealt with in the implementation. First, calculating the fictitious branch admittances involves solving an overdetermined homogeneous system. One can always find the trivial “all zeros” solution to the problem. To find a non-trivial solution, some techniques must be applied. In, the author iteratively found the cut nodes of the system and divided the original large system into several sub-systems. Then the problem was formulated based on the subsystems and each individual subsystem was solved separately.
In, the author used the QR factorization method to solve the problem. It turns out the QR factorization is an effective way which not only can find the non-trivial solution but also reduces the computational memory requirements because it can find the most linearly independent rows and columns in the original problem.
For the studies based on dc-modeling assumptions, all three groups of equivalent methods can be applied. However, for ac-type studies, the fundamental assumptions in the derivation of the PTDF-based methods are violated. The reason is simple: in the ac scenario, the PTDF matrix is a function of operating point which means that at different operating points the PTDF matrices are different. This is due to the changes in the line losses.
The Need for ac Model Reductions
As introduced earlier, it is impractical to solve the complicated planning problem over a large system due to the high computational demand. Though the dc-assumption-based network reduction is desirable in many applications because of its simplicity, a nonlinear ac model reduction, which is more computationally complex and more accurately models the system, is needed when the nonlinear features of the power system become important, such as for reactive power planning (RPP). For the RPP problem, which studies the placement and size of reactive power sources in the network to maintain voltage levels within appropriate ranges, the dc assumption, which assumes all voltage to be 1.0 pu, renders the formulation useless. Further, an RPP problem is a mixed-integer nonlinear programming (MINLP) problem which is of high computational complexity when applied to a large system; thus a reduced ac model is typically necessary. For example, in, and, a 17-bus equivalent model of the New Zealand power system was used to solve the RPP problem while incorporating the voltage stability constraints. In, the New England 39-bus system and a 2069-bus equivalent of the eastern-interconnection were used.
Reduced-order ac models are also found to be useful for investment studies. For example, in a 46-bus and an 87-bus ac equivalent model of the Brazilian power system were applied to the optimal investment problem. In, the authors show that the ac reduced model is also applied in online operations when the data of the external network is not available.
The network reduction work reported here focuses on three objectives: 1) Improve flexibility, 2) Improve robustness and 3) Improve accuracy.
To improve flexibility, an optimization-based network reduction method (OPNR) is proposed. This method formulates an optimization problem which can be treated as a framework for a class of PTDF-based reductions. The objective function and constraints can be modified to generate different equivalents for different studies. It is shown in this work that OPNR can replicate the Ward reduction on large test systems (IEEE 118-bus system, IEEE 300-bus system). In addition, the method can improve the accuracy and sparsity pattern of the Ward reduction by appropriately compensating for the elimination of high impedance equivalent branches.
Planning studies require running OPFs. Ward reduction splits the external generators into fractions and distributes them across (typically) a large fraction of the boundary buses, which adds computational complexity to OPF-type algorithms, which already have a high order of complexity. One way to solve the problem (albeit an approximate technique) is to move each of the external generators to one “appropriate” boundary buses. In this work, different generator placement methods are investigated. An important metric which can be used to evaluate a generator method is its robustness (defined more precisely later.) After placing the external generators in the reduced network, the reduced-model OPF may not be feasible under certain operating conditions for which the full model OPF is feasible. The robustness is measured by the frequency of the occurrence of infeasibility on the reduced model. Compared to finding the optimal solution, achieving a feasible solution is more fundamental. Naturally, a basic requirement of the equivalent model is that a feasible solution exists when the original model has a feasible solution. It is shown in this work that the generator placement in the network reduction process can significantly affect powerflow robustness. Three generator placement methods are proposed and: 1) the shortestelectrical– distance-based method (SED), 2) the optimization-based generator placement method (OGP), 3) the minimum-shift-factor-change-based method (MIN-SF). Results of tests on large and existing systems (IEEE 118, ERCOT, WECC) are shown and discussed.
One of the most fundamental and critical problems in power system analysis is solving the ac power-flow problem using reduced network equivalents. It is shown that the traditional reduction methods (e.g., Ward, REI) fail to yield accurate results when the operating condition changes. This is mainly due to two reasons. One is that the approximation to real and reactive power losses is inaccurate and the other one is that the external controlled reactive power generation is hard to approximate. In this work, a novel network reduction method, taking advantage of the holomorphic embedding (HE) technique is proposed. Results show that the HE equivalent yields superior results compared to the Ward-type or the REI-type methods.