Comparative Analysis for Prediction of Electrical Power Demand of Bayelsa State (Nigeria) by Year 2025

study, the statistical analyses study ABSTRACT This work predicts the peak electrical load demand consumption of Bayelsa State by the year 2025. The data of electrical load allocation and utilization of power consumption of the State in the past five (5) years (2006-2010) were extracted from control room log sheets of Power Holding Company of Nigeria (PHCN) and Kolo Creek Gas Turbine Power Station (Glory Power Company Ltd) for the comparative analysis. Population growth rate for the state as well as Master plan by the Capital City Development Authority were also used. The loads for the analysis were divided into the following categories: domestic, commercial, industrial and electrical load consumed by government agencies. Various improved engineering analysis models and applications including calculations, diagrams and graphs were reviewed and applied to justify the study; the Study revealed that Second-Order Polynomial Model gives the best fit because it gives the least error of 0.66 and provides a good approximation to the shape of ) ( x f function which describes the trend of the electrical power demand distribution in Bayelsa State.

revealed that Second-Order Polynomial Model gives the best fit because it gives the least error of 0.66 and provides a good approximation to the shape of

INTRODUCTION
Electrical load is the power that an electrical utility needs to supply in order to meet the demands of its customers. It is therefore very important to the utilities to have advance knowledge of their electrical load so that they may ensure that this load is met and interruptions are minimized to their services; this will avert an unstable power system normally created when power demand exceeds supply. This will be possible if we know the expected peak load demand in each hour of the day, each day of the week, each week of the month, each month of the year and each year of the period under investigation. Prediction is the act of making known before-hand and foretelling what will happen in the future [1]. Bayelsa state which is located in the Southern Niger Delta of Nigeria has a difficult mangrove terrain with a rural setting. This generates a lot of problems in effective power distribution to end-users. The state has a growth rate of 2.9% [2] and an expected population of 3.517million in 2025.
Several methods have been used in the past to predict electrical load. Leonardo et al. [3] studied the electricity load behaviour in Brazil. They adopted a Generalized Long Memory to model seasonal behaviour of the load. Their model involves statistical linear methods. Bhardwaj et al. [4] in their work employed key energy indicators to predict the load demand by kaval cities in India by year 2023 using mean temperature and population growth rate of the cities as dependent variables. They also analyzed the electricity power demand of Lucknow City [5]. Artificial Neural Network (ANNs), a 3-layer network has been used in the past [6] to carry out short-term forecast of load demand in the Nigerian Electrical system. Peng [7] also used this method with deficit management to carry out load forecasting. Bayasian Statistical Technique on dynamic linear model has also been used [8] to predict residential electricity consumption of a rapidly developing industrial nation. The Bayesian Statistical Technique has the flexibility to solve multi-dimensional time-series models and update estimated parameters as demand changes over time.
Fuzzy back propagation network has been used to predict short term electric load. This involves using an algorithm that is capable of predicting information and fluctuating data patterns. Forecast errors can be evaluated using Mean Absolute Percentage Error (MAPE). The importance of Mean Absolute Percentage Error was cited by Park [9] and Rajasenkaran [10], because it measures proportionality between error and load. The least square model was used by Odubo [11] to forecast electrical load in Bayelsa state by year 2020. This method gives good result although it doesn't take into account variation in load demand due to industrialization and population increase.
Bayelsa State was created on 1 st October, 1996 and the only source of Electric Power is 2x20MW gas turbine plant installed in 1988. The transmission and distribution system of the state is that of a rural setting until 2005 when industries started being constructed and the state. Only the state capital (Yenagoa) has been connected to the Nigerian National grid in 2006. In this work, several methods of load prediction shall be briefly examined and compared with the Second-Order Polynomial model as applied to Bayelsa State.

MATERIALS AND METHODS
The research methodology involved data collection and analysis. The data collection was carried out as follows:

THEORETICAL ANALYSIS
In this work data which comprised of hourly electricity load demand from the area covered by the utilities from January 2006 to December 2010 shall be used. Power generation data from the Bayelsa Ministry of Power [13] shall also be used. A variety of models and methods were employed in the study; in particular, Numerical Methods, Linear Statistical Methods were adopted in this research. The errors in each of the methods were evaluated and the model and method with the least error was accepted as the best for the forecast.

LEAST SQUARE METHOD
The least square method can be used in approximating the relationship between two variables using polynomial of any degree; and in this research since the study involves the forecast between two variables (Load and Time), the method of least square will give a good approximation as shown in Fig. 1.
In the standard formulation, a set of N pair of were used to find a function given the value of the dependent variable ( )

Fig. 1. Fitting a curve with an approximate line
where ε is the "error" quantity to be minimized.
Taking the derivative of ε with respect to a and b and setting them to zero gives the following set of equation (called the normal equation): Solving (3) and (4) simultaneously, we get the estimates of a and b as: The method of least squares can be extended to more than one independent variable (using matrix algebra) as stated below: However, the research will be limited to a 2 2 × matrix since the forecast is a relationship between load and time. For a 2 2 × matrix, the following expression will be employed: and using Crammer's rule we obtain; The results using the least square method shows that power demand by Bayelsa State in 2025 is 77.11 MW. The method also gives an error of 8.14MW (indicating the extent with which the forecast values differ from the real values). Table  1 shows the forecast values for Bayelsa State using least square method while Table 2 shows the error values for this method.
The results of these test calculations show that Least Square Method can be very efficient if it is used in an appropriate way, however several reasons are responsible for the high error experienced using the least square method; outliers, which are significantly bad observations, can skew the results because they have more impact. This impact is because the square of a number grows large, faster than the number. It is better to reject the outliers using some other method prior to using least squares on the remaining data. Of course, this must be substantiated because rejecting data otherwise is bad practice. It is also attributed to the choice of starting values. Fig. 2 shows the predicted and actual values of Electricity Demand for Bayelsa State using Least square method.

LAGRANGE INTERPOLATION
Assuming the existence of a set of data points The function Evaluating ) (x I at the four data points For a four-data point ( can be expressed as follows: However, this research is limited to two data points' model as shown below: The results using Lagrange Interpolation shows that power demand by Bayelsa State in 2025 is 76.08 MW. The method gives an error of 1.48, indicating the deviation of forecast values from the real values using the method. Table 3 shows error values using Lagrange method.
In comparison with other methods, the Lagrange Interpolation models is a simplicity, that is, it can be determined without solving a system of simultaneous equations or performing repetitive calculations as in the case of some of the methods hence the results are acceptable. However, the error between the real and interpolated values might be as a result of the few data points used in the analysis, the more the data points the less the error. Fig. 3 shows the predicted and actual values of Electricity Demand for Bayelsa State using Lagrange model.

SECOND-ORDER POLYNOMIAL
Assuming a polynomial of second-order degree, say The minimum error is at the partial derivative of the error function with respect to coefficients are all zero. Hence the equation resulting from evaluating the partial derivatives with respect to     The results using Second Order Polynomial show that power demand by Bayelsa State in 2025 is 272.25 MW. The method gives an error of 0.66, indicating the deviation of forecast values from the real values using the method and the least error amongst the forecast methods. It was noticed from daily load demand log sheets that peak loads occur between 6am-9am and 7pm-9pm. This is because it is a newly created state where most of the work forces are civil servants. In the morning, they prepare for work. At close of work by 4pm, they get home between 4.30pm-6pm and carry out their domestic functions between 6pm-9pm. Table 4 shows the Electricity Power Demand forecast values for Bayelsa State using Second Order polynomial method while Table 5 shows the error values for this method.
Although the Second Order Polynomial gives the highest load demand for Bayelsa State by 2025, it gives the least error recorded so far from all the models employed in the forecast because a second-order polynomial often provides a good approximation to the shape of ) (x f near and optimum. Fig. 4 shows the predicted and actual values of Electricity demand for Bayelsa State using Second-Order polynomial method.

EXPONENTIAL FUNCTION
Bayelsa State is a fast developing state and it is good to put industrial development or expansion of existing industries into account in this research. To account for these anomalies, the exponential function will be employed to obtain a best fit for the period this research was carried out. Exponential functions are expressed as; To solve the exponential function, equation 29 can be linearized by multiplying through by the log function, hence we have; Substituting in equation 30 we get;

Fig. 4. Second-order polynomial method results
Equation (31) is the linearized exponential function. Obtaining a matrix of the linearized exponential function, we have; And applying Crammer's rule we get; The results using the Exponential Function method show that power demand by Bayelsa State in 2025 is 114.44 MW. The method gives an error of 1.19 indicating the deviation of forecast values from the real values using the above method. Table 6 shows the Electrical Power Demand forecast values for Bayelsa State using Exponential function method while Table 7 shows the error values for this method.
The small error obtained using exponential function is attributed to some reasons. Exponential smoothing is easier to implement and more efficient to compute, as it does not require maintaining a history of previous input data values. Furthermore, there are no sudden effects in the output as occurs with some forecasting methods. With exponential smoothing, the effect of the unusual data fades uniformly as it has a big impact when it first appears. Fig. 5 shows the actual and predicted values of Electrical power demand for Bayelsa State using Exponential function method.     Table 8 shows the Electrical Power Demand forecast values for Bayelsa State using Power function method while Table 9 shows the error values for this method.
In considering the power function model, the data may not actually be linear but the use of power function transform the equations into a linear form through mathematical manipulations before applying the simple linear regression to fit the equations to the data, hence the least error is obtained.

Error Estimation
The results using Time Series shows that power demand by Bayelsa State in 2025 is 85.19MW. The method gives an error of 1.27, indicating the deviation of forecast values from the real values using Time Series. Table 10 shows the Electrical  Power Demand forecast values for Bayelsa State  using Time series method while Table 11 shows the error values for this method.
In using the Time Series model, there is a positive relationship between load demand and year; as the year increases, the load demand increases. Although Time Series gives an error more than some of the models, it gives a more reliable forecast because it considers general mathematical description of movement present in the series. The increased error may be due to the decomposition of series into its basic components (T+C+S+I) which are not presented in the other models. The method gives a better fit because time is a physical concept, hence parameters and other characters of mathematical models can be real world interpretation.

CONCLUSION
In summary, there exist a very strong positive relationship between Power Demand and Year; that is, as the years increases, demand in electricity increases. This positive relationship between the two variables (Power Demand and Time) is shown in all the methods adopted in the analysis. This positive relationship is attributed to the fast rate of development of Bayelsa State; Bayelsa State is a "virgin" state and it is a good site for investors, hence the sudden increase in population and power demand.

Fig. 7. Time series method results
In all the models employed in the analysis, the Second-Order Polynomial Model gives the best fit; it gives the least error of 0.66 and 272.25MW providing a good approximation to the shape of ) (x f function which describes the trend of the power demand distribution in Bayelsa State. Also, the daily peak load periods in Bayelsa State occur between 6am -9am and 7pm -9pm respectively.

RECOMMENDATIONS
Based on the data provided and the analysis carried out, the following recommendations are made: • Load demand forecasting should be carried out periodically to enable power system planning and development. • Researchers who are interested in load forecasting should adopt Second-Order Polynomial Model because it is a holistic approach and is more adequate compared to other methods used in this research work. • Bayelsa State is fast developing; hence there is the need to step-up power generation sector (building power plants) to meet the teeming population and industrial development of the state.

• The distribution network infrastructures in
Bayelsa State is of the rural setting and should be improved to reduce the power interruptions usually experienced in the state. • Injection substations should be installed in Bayelsa State to improve the power situation and flexibility in operation of the system.