**Estimation of Time Series Prediction:**

Estimation of Time Series Prediction – If y_{d}(k) is subtracted from the sequence y(k), the result would be a sequence of data for the stochastic part of the load. We have to identify the model for y_{s} (k) and then use it to make the prediction y_{s}(k+j). A convenient method for this is based on the use of the stochastic time series models. The simplest form of such a model is the so called auto-regressive model which has been widely used to represent the behaviour of a zero mean stationary stochastic sequence. The method proposed for the generation of data for y_{s}(k) ensures that the sequence y_{s}(k) will have a zero mean. If it is also assumed to be stationary, it may be possible to identify a suitable autoregressive model for this sequence.

**Auto-regressive Models**

The sequence y_{s}(k) is said to satisfy an AR model of order n i.e. it is [AR(n)], if it can be expressed as

where a_{i} are the model parameters and w(k) is a zero mean white sequence. In order that the solution of this equation may represent a stationary process, it is required that the coefficients a_{i} make the roots of the characteristics equation

lie inside the unit circle in the z-plane.

The problem in estimating the value of n is referred to as the problem of structural identification, while the problem of estimation of the parameters a_{i} is referred to as the problem of parameter estimation. An advantage of the AR model is that both these problems are solved relatively easily if the autocorrelation functions are first computed using the given data. Once the model order n and the parameter vector a have been estimated, the next problem is that of estimating the statistics of the noise process w(k). The best that can be done, is based on the assumption that an estimate of w(k) is provided by the residual e(k) = y_{s}(k) — ŷ_{s}(k), the estimate ŷ_{s}(k) having been determined from

The variance σ^{2} of w(k) is then estimated using the relation

**Auto-Regressive Moving-Average Models**

In some cases, the AR model may not be adequate to represent the observed load behaviour unless the order n of the model is made very high. In such a case ARMA (n, m) model is used.

Estimation of two structural parameters n and m as well as model parameters a_{i,}

b_{j} and the variance σ^{2} of the noise term w(k) is required. More complex behaviour can be represented. The identification problem is solved off-line. The acceptable load model is then utilized on-line for obtaining on-line load forecasts. ARMA model can easily be modified to incorporate the temperature, rainfall, wind velocity and humidity data. In some cases, it is desirable to show the dependence of the load demand on the weather variables in an explicit manner. The time series models are easily generalized in order to reflect the dependence of the load demand on one or more of the weather variables.