Application of Monte Carlo Simulation to Multi-Area Reliability Calculations. The NARP Model

Transcription

1 Application of Monte Carlo Simulation to Multi-Area Reliability Calculations The NARP Model Any power system reliability model using Monte Carlo simulation consists of at least the following steps: 1. Sampling of States The states may be sampled using random sampling or sequential simulation. The sampled state is defined by the status of all components comprising the system and the magnitude of load at various buses. 2. Evaluation of States This step consists of determining whether the load of all buses can be satisfied given the status of generators and transmission lines. 3. Estimation of Indices Reliability indices are estimated from the repeated use of steps 1 and 2. The stopping criterion is based on the coefficient of variation being less than a specified value. The NARP model was developed by Associated Power Analysts, Inc. and is being used by ERCOT (Electric Reliability Council of Texas). The basic model is described in the following section. In the NARP model simulation proceeds sequentially through time in hourly steps. 1. Each hour the status of every generator and transmission link is randomly and independently drawn according to the probability distribution of generating unit and transmission link states. The available capacities of individual units can be added to obtain the area generating capacities. 2. The load of each area is updated to the current hour. 210

2 3. If no area has a negative margin (capacity - load), then the simulation proceeds to the next hour, otherwise the state evaluation module is called. 4. If all area loads are satisfied, then the simulation proceeds to the next hour. If there is a loss of load in one or more areas then this is counted as loss of load for those areas and the system and area loss of load magnitudes are computed. 5. Simulation is performed till the end of the year and statistics of number of loss of load hours per year are collected. 6. The simulation process is continued until the specified convergence criterion is reached. Convergence of results: An important issue in Monte Carlo simulation programs is number of years of artificial history that must be created to achieve an acceptable level of statistical convergence in reliability indices of interest. Here the degree of statistical convergence is measured by the standard deviation of the estimate of the reliability index obtained from simulation data. Let I i = value of reliability index obtained from simulation data for year i N = number of years of simulated data available Then the estimate of index I _ N I = I i / N i=1 and S I = (S 2 /N) = standard dev of estimate of I 211

3 where N _ S 2 = ( I i - I ) 2 /N i=1 _ Note that standard dev of I, varies as the inverse of square root of N. Clearly, S I can never be made zero in practice and so the computed value of reliability indices (mean or expected values) will always contain some uncertainty. The goal here is to reduce the uncertainty in the computed reliability indices to an acceptable level and to understand the degree of uncertainty that remains. The NARP model stops until one of the following criteria is satisfied. 1. Max number of years specified. _ 2. S I / I is less than a specified fraction. The NARP model is so structured that if criterion 1 is satisfied and the computations terminate, you could restart the computations from the year of termination so that simulated years are not lost. State evaluation module: The following description assumes no loss sharing policy between areas, that is, an area will provide emergency assistance to other areas only to the extent of its surplus capacity. A loss sharing policy has also been implemented but is not described here to keep the discussion simple. 1. The scheduled transfers due to firm contracts and jointly owned units are algebraically added to determine the net scheduled transfers. These transfers are then input to the network flow module (described later) to determine feasible transfers and line flows. 2. If the load in each area can be satisfied by the capacity in each area together with the feasible scheduled transfers determined in step 1, then further computation is not required, otherwise go to step

4 3.First, the net injections are assigned to each area: M i = injection at bus i = margin in area i = capacity in area i - load in area i Now the feasible scheduled transfers calculated in step 1 are subtracted from the net injections, and line capacity limits are modified by the flows due to scheduled transfers. The network flow model is called to determine the loss of load. The line flows calculated in this step are algebraically added to line flows calculated in step 1. Network flow calculations: The underlying model in NARP is a DC load flow model. This model needs both tie capacity limits and line admittances. This can be, however, easily replaced by capacity flow model if preferred. The following description is based on DC flow model. The state evaluation model in NARP proceeds in two steps. 1. Stage 1: Heuristic method If the sum of positive injections is greater than the sum of negative injections, all positive injections are scaled down in the same ratio so as to make these sums equal. If the sum of negative injections is greater than the sum of positive injections, all negative injections are scaled down in the same ratio to make these sums equal. Then DC load flow model is called to make flow calculations. This model is usually expressed by the equation B θ = M 213

5 where B matrix is such that b ij = ijth element of B =-(susceptance between nodes i and j) if i j b ii = sum of susceptances connected to node i. θ = node voltage angle vector and M = bus injection vector The line flow from node i to node j is given by f ij = (θ j - θ i ) b ij If the flows are within the tie capacity constraints, then a feasible flow has been found otherwise the program proceeds to the next stage to find a feasible flow. Stage 2: Linear programming If a feasible solution is not found in stage 1 then network flow module enters the optimization phase. The optimization procedure is based on LP and it assigns positive margins or curtails negative margins so as to minimize the pool loss of load. Mathematically the formulation is Loss of load = Min C i subject to: B θ + G +C =D 214

6 G G max C D F F f -F F r S S max where G = vector of positive injections D = vector of negative injections C = vector of negative injection curtailments C i = ith element of C S = vector of sum of flows at nodes F= vector of flows G max = vector of max available net positive injections S max = max flow values of flows at node s F f,f r = forward and reverse tie capacities F and S are related to tie line susceptances by the equation f ij = (θ j - θ I ) b ij 215