**3. Consumed load model**

The load forecasting art is in selecting the most appropriate way and model for and the closest ones to the existing reality of the network among different methods and models of load forecasting, by studying and analyzing the last procedure of load and recognizing the effective factors sufficiently and maximizing each of them, and then in this way it forecasts different time periods required for the network with an acceptable approximation. It should be accepted that there is always some error in load forecasting due to the accidental load behavior but never this error should go further than the acceptable and tolerable limit. Relative accuracy has a particular importance in load forecasting in power industry. Especially when load forecasting is the basis of network development planning and power plant capacity. Since, any forecasting with open hand causes extra investment and the installation capacity to be useless and vice versa any forecasting less than real needs, faces the network with shortage in production and damages the instruments due to extra load.

Consumed load model is influenced by different parameters such as weather, vacations or holidays, working days of week and etc. in order to build a short-term load forecasting system, we should consider the influence of different parameters in load forecasting, which it can be full field by a correct selection of system entries. Selection of these parameters depends on experimental observations and is influenced by the environment conditions and is determined by trial and error.

A Multi Adaptive Neuro Fuzzy Inference System for

rates given by the Lyapunov exponents.

will be arranged such that

exponents, the sum across the entire spectrum is negative.

Kolmogorov entropy (K) or mean rate of information gain [58]:

reconstruction, e.g., method of delays [60], singular value decomposition.

Short Term Load Forecasting by Using Previous Day Features 343

n-dimensional sphere of initial conditions, where n is the number of equations (or, equivalently, the number of state variables) used to describe the system. As time (t) progresses, the sphere evolves into an ellipsoid whose principal axes expand (or contract) at

The presence of a positive exponent is sufficient for diagnosing chaos and represents local instability in a particular direction. Note that for the existence of an attractor, the overall dynamics must be dissipative, i.e., globally stable, and the total rate of contraction must outweigh the total rate of expansion. Thus, even when there are several positive Lyapunov

Wolf et al. [59] explain the Lyapunov spectrum by providing the following geometrical interpretation. First, arrange the n principal axes of the ellipsoid in the order of most rapidly expanding to most rapidly contracting. It follows that the associated Lyapunov exponents

��>��>…..>�� where �� and �� correspond to the most rapidly expanding and contracting principal axes, respectively. Next, recognize that the length of the first principal axis is proportional to ����; the area determined by the first two principal axes is proportional to ���������; and the volume determined by the first k principal axes is proportional to ��������������. Thus, the Lyapunov spectrum can be defined such that the exponential growth of a k-volume element is given by the sum of the k largest Lyapunov exponents. Note that information created by the system is represented as a change in the volume defined by the expanding principal axes. The sum of the corresponding exponents, i.e., the positive exponents, equals the

K=∑ �� ���� When the equations describing the dynamical system are available, one can calculate the entire Lyapunov spectrum. The approach involves numerically solving the system's n equations for n+1 nearby initial conditions. The growth of a corresponding set of vectors is measured, and as the system evolves, the vectors are repeatedly reorthonormalized using the Gram-Schmidt procedure. This guarantees that only one vector has a component in the direction of most rapid expansion, i.e., the vectors maintain a proper phase space orientation. In experimental settings, however, the equations of motion are usually unknown and this approach is not applicable. Furthermore, experimental data often consist of time series from a single observable, and one must employ a technique for attractor

As suggested above, one cannot calculate the entire Lyapunov spectrum by choosing arbitrary directions for measuring the separation of nearby initial conditions. One must measure the separation along the Lyapunov directions which correspond to the principal axes of the ellipsoid previously considered. These Lyapunov directions are dependent upon the system flow and are defined using the Jacobian matrix, i.e., the tangent map, at each point of interest along the flow [58]. Hence, one must preserve the proper phase space orientation by using a suitable approximation of the tangent map. This requirement, however, becomes unnecessary when calculating only the largest Lyapunov exponent.
