Transfer Function Based Performance Assessment of Power ...

Accepted Manuscript Title: Transfer Function Based Performance Assessment of Power Distribution Facilities: A Case Study of Distribution Transformers Author: Chidozie Chukwuemeka Nwobi-Okoye PII: DOI: Reference:

S2314-7172(16)30105-2 http://dx.doi.org/doi:10.1016/j.jesit.2016.12.001 JESIT 135

To appear in: Received date: Revised date: Accepted date:

24-9-2016 6-11-2016 6-12-2016

Please cite this article as: Chidozie Chukwuemeka Nwobi-Okoye, Transfer Function Based Performance Assessment of Power Distribution Facilities: A Case Study of Distribution Transformers, http://dx.doi.org/10.1016/j.jesit.2016.12.001 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Transfer Function Based Performance Assessment of Power Distribution Facilities: A Case Study of Distribution Transformers

Chidozie Chukwuemeka Nwobi-Okoye 1*

1

Chukwumeka Odumegwu Ojukwu University (formerly Anambra State University), Anambra State, Nigeria

* Corresponding author’s email: [email protected]; [email protected] Tel.: 234-080-60013240

1

Abstract This research reports on the application of transfer function in performance assessment of power distribution facilities of utility companies: Shell and PHCN. It involves taking input-output data from transformers over a 1-year period and developing transfer functions of the bimonthly transformation processes for the period. The results indicate that the distribution facility of Shell had higher coefficients of performance (COP) than PHCN facilities. The average efficiencies of Shell's distribution facility were well within 90-98 percent recommended by the IEC while that of PHCN's facility were below. Generally, there were correlation between coefficient of performance and efficiency, but in some cases a high efficiency corresponded to low COP, a paradox confirming the superiority of COP as a metric for performance assessment. Keywords: Transfer function; Transformer; Electricity Power distribution; Performance indicators; Modelling Nomenclature, Symbols and Notations k = lag variable βt= pretreated output series αt= prewhitened input series v(B) = Transfer function B = backshift operator Yt = process output at time t Xt = process input at time t yt = differenced output series xt = differenced input series 𝑌̂𝑡 = output forecast 𝑋̂𝑡 = input forecast at = error term/white noise υk = impulse response weight at lag k h=ACF/PACF lag q=order of moving average operator p=order of autoregressive operator d= number of differencing θ = autoregressive operator φ = autoregressive operator Ξ = coefficient of output variable of differential equation H = coefficient of input variable of differential equation γ = cross correlation function χ = covariance function b = transfer function lag ω = coefficient of difference equation input variable δ = coefficient of difference equation output variable r = order of the output series s = order of the input series S = sample standard deviation σ = population standard deviation ρ = auto correlation function 2

μ=mean V=voltage I=current t=time E=Energy P=Power MW=Megawatts η=efficiency ACF=Auto Correlation Function PACF= Partial Auto Correlation Function  = difference operator

 = difference equation variable for output CCF= Cross Correlation Function TF= Transfer Function DF= Degree of Freedom RMSE= Root Mean Square Error MAPE = Mean Absolute Percentage Error MAE = Mean Absolute Error MaxAPE= Maximum Absolute Percentage Error MaxAE= Maximum Absolute Error BIC= Bayesian Information Criterion g= the steady state gain 1. Introduction A cardinal objective of the relentless pursuit for improved environment is the efficient utilization of natural resources for power generation, as well as development of efficient power conversion and distribution systems. Thus, the effective measurement of efficiency and performance of power generation, conversion or distribution systems is very necessary. Effective and accurate determination and modelling of the efficiency of power generation and distribution facilities helps to evaluate the performance of the facilities, diagnos fault and aids preventive maintenance (Nwobi-Okoye and Igboanugo, 2012; Nwobi-Okoye and Igboanugo, 2015). According to Nwobi-Okoye and Igboanugo (2012), the five (5) Ms of production namely: men, machines, material, method and money, could all be used at the same time in performance assessment of electrical power facilities . If all the five (5) Ms are involved performance assessment, they termed it the Macro level analysis and evaluations. Alternatively, when the performance assessment took into consideration one of the five (5) Ms they termed it micro level performance assessment. Macro level performance assessment methods include, Data Envelopment Analysis (DEA), Stochastic Frontier Analysis (SFA) and Analytic Network Process (ANP) (Jha and Shrestha, 2006; Atmaca and Basar, 2012). On the other hand, techniques that belong to micro level performance assessment methods include Reliability analysis, Energy/Power Input-Output Methods etc (El Khashab, 2015; NwobiOkoye and Igboanugo, 2012; Nwobi-Okoye and Igboanugo, 2015; Ray, 2007; Fink and Beaty, 2006; Petkov, 1996; Tang and Hui, 2001). Modelling performance through reliability analysis is very important in preventive maintenance of power distribution facilities. Adoghe et al. (2013) and Yssaad et al. (2014) modelled the performance of some electrical power distribution zones in Nigeria and Algeria respectively using Reliability Centred 3

Maintenance (RCM) method. The presented method is able to deal with uncertain outage parameters and maximized the possibility of reliability improvement and loss reduction. The performance assessment of hydropower plants in Nepal was carried out with data envelopment analysis (DEA) by Jha and Shrestha (2006) and Jha et al. (2007). According to Berg (2010), DEA involves a holistic measure of the efficiency of Utilities where the input components would include man-hours, losses, capital (lines and transformers only), and goods and services. The output variables would include number of customers, energy delivered, length of lines, and degree of coastal exposure. DEA was used by Liu et al. (2010) for efficiency assessment of major thermal power plants in Taiwan. The results they obtained showed that all power plants they studied achieved acceptable overall operational efficiencies between 2004–2006 which was the study period. From their findings also, the combined cycle power plants were the most efficient among all plants. DEA was used by Sözen et al. (2010) to conduct efficiency analyses of some government owned thermal power plants used for electricity generation in Turkey. Two efficiency indexes were used in the research and they were: operational and environmental performance. According to Zhou et al. (2008), prior to 1990, the use of DEA in electricity industry mainly focused on electricity generation plants. Since the earlier 1990s, DEA has gradually become a popular benchmarking tool for studying the efficiency of electricity distribution utilities. Zhou et al. (2008) pointed out that the study by Weyman-Jones (1991), in which the technical efficiency of the UK electricity distribution industry was studied, is probably the first publication in this line. Giannakis et al. (2005) calculated the technical efficiency of the electricity distribution utilities in the UK between 1991/92 and 1998/99 using Data Envelopment Analysis technique in their findings they showed that cost efficiency does not necessarily correlate with high quality service. Growitsch et al. (2009) used stochastic frontier analysis (SFA) method for efficiency analysis of electricity distribution networks from seven European countries the result they obtained showed that that introducing the quality dimension into the analysis affects estimated efficiency significantly, especially that smaller utilities’ efficiency seems to decrease. Atmaca and Basar (2012) used the multi-criteria decision making technique of Analytic Network Process (ANP), a multi-criteria evaluations of six different energy plants were performed with respect to the major criteria such as technology and sustainability, economical suitability, life quality and socio-economic impacts. The macro level performance assessment largely depends on the performance at the micro level. Micro level performance relies heavily on the efficiency and reliability of the transformers, conductors etc. Efficiency has been an important benchmark used by regulators in assessing the performance of power generation and distribution companies (electricity utilities), and highly efficient electricity utilities are often rewarded with incentives to further boost their performance (Nwobi-Okoye and Igboanugo, 2012; Nwobi-Okoye and Igboanugo, 2015; Growitsch et al., 2009). Normally, the electrical engineer determines the efficiency of a distribution facility like the transformer through the following equation (Ray, 2007; Fink and Beaty, 2006): 𝜂 = 𝑉𝑜𝑢𝑡 𝐼𝑜𝑢𝑡 𝑡⁄𝑉𝑖𝑛 𝐼𝑖𝑛 𝑡

(1)

𝜂 = 𝐸𝑜𝑢𝑡 ⁄𝐸𝑖𝑛 = 𝑃𝑜𝑢𝑡 ⁄𝑃𝑖𝑛

(2) 4

It has been established that this method of measuring the efficiency of systems is statistically defective (Nwobi-Okoye and Igboanugo, 2012; Nwobi-Okoye and Igboanugo, 2015; Nwobi-Okoye et al., 2016; Nwobi-Okoye and Okiy, 2016). This is because energy input to the power distribution facility is stochastic and the corresponding energy output of the process is equally stochastic as shown in Figure 1. This makes the determination of the relationship between input and output quite complex. In view of the above fact, Nwobi-Okoye and Igboanugo (2012, 1015) developed a superior, statistically robust and highly innovative technique based on transfer function in the performance assessment of power generation systems.

Transfer function modelling is often used to measure transient input-output relationship of non equilibrium systems (Nwobi-Okoye and Igboanugo, 2012; Nwobi-Okoye and Igboanugo, 2015). Box and Jenkins carried out the seminal work on transfer function modelling and analysis (Nwobi-Okoye and Igboanugo, 2012; Nwobi-Okoye and Igboanugo, 2015; Box et al., 2008; Lai, 1979). Lai (1979) stated that transfer function is frequently used to determine the causal relationship between two variables. Generally, a transfer function relates two variables in a process; one of these is the cause (forcing function or input variable) and the other is the effect (response or output variable) (Igboanugo and Nwobi-Okoye, 2011b). The literature is replete with the application of Box-Jenkins transfer function model in various fields such as: physical science, economics, management, engineering, education, computer science, sociology, biology etc (Okiy et al. 2015; Box et al, 2008; DeLurgio, 1998). Some application of transfer function in production include the work of Web and Hardt (1991) who developed a transfer function modelling for quality control of sheet metal forming process. Also, Fearn and Maris (1991) developed a model based on Box-Jenkins methodology for the feedback control of the addition of gluten during the milling of flour. Gluten, an animal protein, is a very essential constituent of flour and determines to a large extent its quality. Box et al. (2008) applied transfer function models to the monitoring and control of the production of carbon IV oxide in a gas furnace. Similar applications are used in industry in the monitoring of the production of chemicals. Schwartz and Rivera (2010) applied Box-Jenkins transfer function modelling to solving problems of production inventory management. Disney and Towill (2002), Schwartz et al. (2005) applied transfer function modelling to supply chain management problems. In economic and social science applications, transfer functions could be used to predict unemployment as demonstrated by Edlund and Karlsson (1993). For applications in prediction of prices, the works of Adebiyi et al. (2014), Nogales and Conejo (2006), Zareipour et al. (2006), García-Martos et al. (2007) are very good examples. Preciado et al. (2006) applied transfer function modeling in agricultural economics, specifically they used the model to obtain a census of fish catches and efforts of their fleets which are considered suitable for the elaboration of representative indices of catch rate (catch per unit effort). As a matter of fact, most applications of transfer functions in economic sciences involve the determination of the causal relationship between one economic variable and another. In another economic science application, the relationship between United States Tax Reform Act and its magnitude on business failures was extensively studied by Choudhury (2007) using Box-Jenkins 5

intervention analysis model. Further applications of Box-Jenkins transfer function model in economics, finance and business could be found in Cooray (2006), Berument et al. (2006), Enders (1995), DeLurgio (1998), Limanond et al. (2011), Forst (2011) etc. Furthermore, transfer functions could be used to model response of citizen to the enactment of certain laws. The effects of such laws normally called interventions have been widely studied using time series analysis by employing Box-Jenkins transfer function model. In a typical such case, Wilson et al (2007) investigated whether a law in New Zealand making all indoor workplaces including bars and restaurants smoke free which became operational in New Zealand in December 2004 increased calls to the Quitline Service. The result they obtained show that the new national smokefree law increased quitting-related behaviour. In environmental management, Box-Jenkins transfer function modelling was used by Issarayangyun and Greaves (2006) as a tool to assess the relative importance of travel speed (which is known as a proxy for traffic conditions) and meteorological conditions on Fine airborne particulate matter (PM) exposure level. This is a typical application of transfer function modeling to pollution control and environmental management. Bruun et al. (2012) showed examples of how the Box-Jenkins transfer function models can be utilised in environmental and marine systems with simulated case studies. Valipour (2012) used Box-Jenkins model to estimate reference potential evapotranspiration at a climate station in Iran. In the life sciences, Box-Jenkins transfer function models are sometimes used to predict the effects of drugs on microbes and the body, as well as biochemical responses of the body. For some applications of Box-Jenkins model to life sciences, Aldeyab (2012) used a multivariate autoregressive integrated moving average (MARIMA) model was built to relate antibiotic use to ESB-producing bacteria incidence rates and resistance patterns over a 5 year period (January 2005–December 2009). The study according to the authors highlights the value of time-series analysis in designing efficient antibiotic stewardship. Chance et al (1985) developed the concept of transfer function for organ performance prediction which relates work output to biochemical input for skeletal and cardiac muscle under steadystate exercise conditions. They used the model to predict the degree to which metabolic homeostasis is effective, and also showed that poorly controlled metabolic states can readily be identified and used in the diagnosis and therapy of metabolic disease in the organs of neonates and adults. Zhang et al (1998) used Box-Jenkins transfer function to model dynamic cerebral autoregulation in humans. In applications in Engineering, Box-Jenkins transfer function model has been used to forecast wind power generation (Foley et al., 2012). According to Su et al. (2014), forecasting the wind speed is indispensible in wind-related engineering studies and is it is absolutely necessary in the management of wind farms, hence they successfully developed a hybrid technique based on autoregressive integrated moving average (ARIMA) model and Kalman filter for forecasting the daily mean wind speed in western China. Li et al. (2014) used transfer function model to forecast the power output of a grid connected to a photovoltaic system. Pardo et al (2002) in their paper developed a transfer function intervention model for forecasting daily electricity load from cooling and heating degree–days in Spain. The result they obtained showed the influence of weather and seasonality and demonstrated that it is significant even when the autoregressive effects and the dynamic specification of the temperature are taken into account. These examples are some of the numerous applications of Box-Jenkins method in Engineering. As the literature shows above, the basic application of transfer function is in forecasting and intervention analysis. Box-Jenkins transfer function model has widespread use because of its superiority to regression analysis in forecasting and modeling of dynamic systems (Kinney, 1978; Lai 6

(1979); Box et al. 2008; Nwobi-Okoye and Igboanugo, 2012; Nwobi-Okoye and Igboanugo, 2015). Nwobi-Okoye and Igboanugo (2012, 2015) and Nwobi-Okoye et al. (2016) applied transfer function to performance evaluation, a marked departure from forecasting and intervention analysis, which are the prevalent applications. This research differs from others in that it marks the application of Box-Jenkins transfer function modeling to performance modeling of power systems as exemplified by distribution transformers; A Follow-up research to performance assessment applications stated previously. The aim of this research therefore is to develop a statistically sound metric analogous to the six sigma concept in measuring the operations efficiency of power distribution systems using transfer function modelling. The hubs of the investigation are some power distribution facilities of Power Holding Company of Nigeria (PHCN) PLC, Anambra State, Nigeria and Independent Power Project (IPP) Distribution Facility of Shell Petroleum Development Company of Nigeria (Shell’s IPP). Transfer function approach was used to appraise the efficiency and performance of some distribution facilities PHCN and Shell’s IPP.

2. Theoretical Background In its simplest form, the mathematical representation of a linear transfer function is (Nwobi-Okoye and Igboanugo, 2012; Nwobi-Okoye and Igboanugo, 2015; Box et al., 2008): Y∞ = gX Where Y∞ is the steady state output g is the steady state gain X is the steady state input

(3)

Discrete models, the subject of the case study, are often represented by difference equations such as equation 4 (Box et al., 2008). 𝑌𝑡 = 𝛿 −1 (𝐵)𝜔(𝐵)𝑋𝑡−𝑏

(4)

The ratio 𝛿 −1 (𝐵)𝜔(𝐵) in Eq (4) is called the transfer function of the system. The transfer function could be related to the impulse response weights, υ, as shown in Eq (5). 𝑌𝑡 = 𝑣0 𝑋𝑡 + 𝑣1 𝑋𝑡−1 + 𝑣2 𝑋𝑡−2 + ⋯

(5)

Writing Eq (5) in operator B format, Eq (6) is obtained. 𝑌𝑡 = (𝑣0 + 𝑣1 𝐵 + 𝑣2 𝐵 2 + ⋯ )𝑋𝑡

(6)

𝑌𝑡 = 𝑣(𝐵)𝑋𝑡

(7)

7

The series Yt and Xt are modelled as an autoregressive integrated moving average (ARIMA) process (Nwobi-Okoye and Igboanugo, 2015).Hence on further analysis the details of which could be obtained in (Nwobi-Okoye and Igboanugo, 2015), Eq (8) is obtained.

𝛾𝛼𝛽 (𝑘) =

𝑣𝑘 𝜎𝛼 𝜎𝛽

(8)

Eq (8) is an expression showing that the impulse response weight, υj is related to the cross-correlation between the pre-whitened series αt and βt. Furthermore, transfer function modelling comprises of three stages namely: identification, estimation and diagnosis (Nwobi-Okoye and Igboanugo, 2012; Nwobi-Okoye and Igboanugo, 2015; Box et al., 2008; Lai, 1979; Igboanugo and Nwobi-Okoye, 2011a; Igboanugo and Nwobi-Okoye, 2011b).

3. Methodology The data used were a 1-year daily input-output data which was got from some power distribution facilities of PHCN and Shell’s IPP. The transfer function modelling of the electric power distribution process was done with the obtained data. The model shown in Eq (4) is a discrete transfer function model which is the most suitable for the current case study according to the works of Nwobi-Okoye and Igboanugo (2012, 2015). Incorporating the noise term to the model shown in Eq (4), Eq (9) is obtained: 𝑌𝑡 = 𝛿 −1 (𝐵)𝜔(𝐵)𝑋𝑡−𝑏 + 𝑁𝑡

(9)

The transfer function modelling procedure consists of the following steps: 1. Plot the gathered input/output data. 2. Achieve level and variance stationarity of Yt and Xt. 3. Fit a univariate model to xt to estimate αt. 4. Fit a univariate model to yt as a benchmark and possible Nt . 5. Use pre-whitened model of αt and pre-treat yt to get βt. 6. Calculate CCF(k) of βtαt-k to identify r,s and b. 7. Examine CCFs for r,s and b. 8. Estimate the Transfer Function (TF) Using Yt and Xt. 8

9. Use the residual of the TF to identify Nt.

4. Results Table 1 shows part of the 1-year data obtained from Power Holding Company of Nigeria (PHCN) PLC, the graphs of the input and output series for the months 1-2 in the year 2012 are shown in Figures 2 and 3.

4.1 Analysis of input series

The input series showed evidence of stationarity which meant that it did not require differencing. Further evidence from the ACF and PACF in Figures 4 and 5 shows that auto regression one (AR (1)) model is the appropriate model to use. The formula for AR (1) models (Box et al., 2008; DeLurgio, 1998; Shumway and Stoffer, 2006) is given by Eq (10): 𝑋𝑡 = 𝜃0 + 𝜑1 𝑋𝑡−1 +𝑒𝑡

(10)

But for AR (1) models, the following applies: 𝐴𝐶𝐹(1) = 𝜑1 = 0.888

(11)

𝜃0 = (1 − 𝜑1 )𝜇

(12)

𝜃0 = (1 − 0.888)9.06

(13)

𝜃0 = 1.015

(14)

Fitting the coefficients 𝜃0 and 𝜑1 into the formula for AR (1) models, equation (15) is obtained. 𝑋𝑡 = 1.015 + 0.888𝑋𝑡−1 +𝑒𝑡

(15)

But 𝑒𝑡 = 𝛼𝑡

(16)

In forecasting form equation (15) is transformed to equation (17): 9

𝑋̂𝑡 = 1.015 + 0.888𝑋𝑡−1

(17)

4.2 Analysis of output series The input series showed evidence of stationarity which meant that it did not require differencing. Further evidence from the ACF and PACF in Figures 6 and 7 shows that auto regression one (AR (1)) model is the appropriate model to use. The formula for AR (1) models (Box et al., 2008; DeLurgio, 1998; Shumway and Stoffer, 2006) is given by equation (18): 𝑌𝑡 = 𝜃0 + 𝜑1 𝑌𝑡−1 +𝑒𝑡

(18)

But for AR (1) models, the following applies: 𝐴𝐶𝐹(1) = 𝜑1 = 0.884

(19)

𝜃0 = (1 − 𝜑1 )𝜇

(20)

𝜃0 = (1 − 0.884)7.76

(21)

𝜃0 = 0.900

(22)

Fitting the coefficients 𝜃0 and 𝜑1 into the formula for AR (1) models, equation (23) is obtained. 𝑌𝑡 = 0.900 + 0.884𝑌𝑡−1 +𝑒𝑡 But

(23)

𝑒𝑡 = 𝛽𝑡

(24)

Eq (23) is shown in forecasting form as Eq (25): 𝑌̂𝑡 = 0.900 + 0.884𝑌𝑡−1

(25)

The CCF between βt and αt is shown in Figure 8. It has one significant CCF at lag zero (0). Evidence from the CCF supports the following transfer function model:

10

𝑦𝑡 = 𝜔0 𝑥𝑡 +𝑁𝑡

(26)

As evidenced by the Ljung-Box statistics shown in Table 2 and analysis of the residuals, the noise term 𝑁𝑡 was disregarded to obtain equation (27). 𝑦𝑡 = 𝜔0 𝑥𝑡

(27)

As shown by Box et al. (2008) and DeLurgio (1998), 𝑣0 = 𝜔0

(28)

But 𝑣0 =

𝛾𝛼𝛽 (0)𝑆𝛽 𝑆𝛼

(29)

𝛾𝛼𝛽 (0) is the cross correlation between α and β at lag zero (0). But 𝑋𝑡 − 𝜇𝑥 = 𝑥𝑡

(30)

And 𝑌𝑡 − 𝜇𝑦 = 𝑦𝑡

(31)

Substituting equation (31) into equation (27), equation (32) is obtained. 𝑌𝑡 = 𝜇𝑦 + 𝜔0 𝑥𝑡

(32)

In forecasting form equation (32) is transformed to equation (33). 𝑌̂𝑡 = 𝜇𝑦 + 𝜔0 𝑥𝑡

(33)

The model fit and statistics are good as shown for months 1-2 in Tables 1 and 2 respectively.

11

For months 1-2 operations of the Power Station we obtained: 𝛾𝛼𝛽 (0) = 0.976 𝑆𝛽 = 0.96 𝑆𝛼 = 1.09 Hence, 𝑣0 =

0.976 × 0.96 1.09

𝑣0 = 0.857 𝜔0 = 0.857 Hence from equation (27) 𝑦𝑡 = 0.857𝑥𝑡 Since 𝜔0 = 0.857 for the months 1-2 operation of the transformer, the transfer function obtained is : 𝑌̂𝑡 = 𝜇𝑦 + 0.857𝑥𝑡

(34)

For further confirmation of the strength of the result of the transfer function model other ARIMA model where used to build the transfer function model and compare the result. Hence, a thorough and very detailed analysis and Examination of the results shown in Tables 3 and 4 shows that Model 1,0,0 has the least MAPE and the highest value of R-squared. This is an indication that Model 1,0,0 is the 12

best. Furthermore as shown in Table 4, the relatively low value of Ljung Box Statistics (14.483) with high significance value is an indication of the goodness of the model. It is noteworthy that as corroborated by literature (Valipour, 2015) higher values of p and q in the ARIMA model used in this study gave generally slightly better results without differencing than the model prescribed by the ACF and PACF in Figures 4, 5, 6 and 7. The best result obtained, having tested different higher models up to p and q values of 5, occurred at ARIMA (4, 0, 4). When ARIMA (4, 0, 4) was used, the MAPE obtained was 1.943 and the R2 value was 0.993. The difference between the MAPE and R2 values for ARIMA (1, 0, 0) and ARIMA (4, 0, 4) was very small. Since the present application was not for forecasting, and the R 2 values of the models were almost the same, for parsimonious reasons ARIMA (1, 0, 0) prescribed by the graphs of the ACF and PACF was preferred for the analysis.

The residual plot of the ACF and PACF shown in Figure 9 shows there is no significant ACF/PACF and the plots have no pattern. These are confirmations that the residuals are white noise.

Tables 5 and 6 show the transfer function models for the twelve-month operation of the transformers.

Table 7 depicts the total monthly energy distributed together with the corresponding coefficient of performance of the Power Station computed on monthly basis.

13

Tables 7 and 8 shows the coefficients of performance for the distribution transformers studied. From Table 7, the value of  0 in the months 1-2 was 0.670 while the value was 1.065 in the months 3-4. This indicates that the performance of the system was more effective in the months 3-4 than in the months 1-2. The lower energy output in the months was partly because operations efficiency was poorer in those months when compared to the months 3-4.

As shown in Figure 9, most of the efficiency values of the low voltage transformer from Shell measured over a 122-days period is well within the range of 90-98percent value specified by the IEC for typical low voltage transformers (IEC, 2007). Similarly, most of the efficiency values of the low voltage transformer from PHCN measured over a 122-days period as shown in Figure 10 is well below the range of 90-98percent value specified by the IEC for typical low voltage transformers (IEC, 2007).

5. Discussion As Figures 2 and 3 shows, in a power distribution system the input and output are stochastic in nature as earlier clarified. The power distribution system which is the subject of this study is a single-inputsingle-output-system (SISO). As stated by Nwobi-Okoye and Igboanugo (2015), the practical implication of Eq (32) is that given some autonomous value 𝜇𝑦 for every unit increase in xt, the output Yt changes by 𝜔0 which is the coefficient of performance. Furthermore, a very crucial consequence of the findings is that the average efficiency over a given time interval, i.e. interval 1, could be higher in value than in another interval, i.e. interval 2, and yet the COP in the first interval could be lower in value than in the second interval. This is in complete agreement with the findings of Nwobi-Okoye and Igboanugo (2015).

Consider Figures 11 and 12 which show the variations of efficiency over two periods for two transformers. Figure 11 is for Shell’s transformer in months 1-2, while Figure 12 is for PHCN’s transformer in months 1-2. Table 9 shows the mean efficiency at each time interval and their associated coefficients of performance. The implication of this is that a unit input power supplied to the transformer will change the output power by 0.67MW in period 1 and by 0.87MW in period 2.

Primarily using the efficiency as the metric, the system managers would assume that period 1 was better than period 2, but in actual fact the reverse is the case. As stated by Nwobi-Okoye and Igboanugo (2015), the primary reason for performance assessment is to improve performance. Thus, better metric for performance assessment/appraisal would help to improve system performance. Suffice it to say that to achieve higher COP, it is very necessary that the managers need to appraise or adjust one or all of manpower, machine, money, method and material as the case may be (2015). Generally, the major benefits of this method as stated by Nwobi-Okoye and Igboanugo (2015) are:

14

(a) (b) (c)

Greater accuracy in efficiency measurement over a given period. Statistically robust efficiency measurements. Better plant fault diagnosis and superior aid to predictive and preventive maintenance.

Generally, Shell’s distribution transformer had higher coefficient of performance and average efficiency than PHCN’s distribution transformers. This case study is another practical application of theoretical proposal by Nwobi-Okoye and Igboanugo (2011c). In two former applications a hydro power generation and gas power generation systems were analysed (Nwobi-Okoye and Igboanugo, 2012; Nwobi-Okoye and Igboanugo, 2015). In the present case a power distribution system component was analysed. The coefficient of performance is a superior metric for measuring the maintenance effectiveness and operations efficiency of systems (NwobiOkoye and Igboanugo, 2012; Nwobi-Okoye and Igboanugo, 2015; Nwobi-Okoye et al., 2016; ; NwobiOkoye and Okiy, 2016). Operating a facility at a very high coefficient of performance ensures long life span for the components of the distribution system, as well as aids in conservation of energy and protection of the environment.

6. Conclusion In conclusion, this research has further elucidated the superiority of coefficient of performance (COP) as a superior metric for evaluating performance over and above efficiency. Also, values of COP obtained from transfer function modelling are more accurate and statistically robust than equivalent model obtained from regression analysis. In continuation, COP is found to be a more accurate performance measure for power distribution systems as exemplified by distribution transformers. Furthermore, Coefficient of performance (COP) could be a more accurate measure of how effective electrical power distribution systems satisfy consumers. Thus with high coefficient of performance it is assumed that electrical power is delivered very satisfactorily to consumers, on the other hand lower coefficient of performance presupposes that the quality of electrical power delivered to consumers is low.

References Adebiyi, A. A., Adewumi, A. O., & Ayo, C. K. (2014). Comparison of ARIMA and artificial neural networks models for stock price prediction. Journal of Applied Mathematics, 2014. Adoghe, A.U., Awosope, C.O.A. and Ekeh, J.C. (2013). Asset maintenance planning in electric power distribution network using statistical analysis of outage data. International Journal of Electrical Power and Energy Systems, 47 (2013): 424–435. Aldeyab, M. A., Harbarth, S., Vernaz, N., Kearney, M. P., Scott, M. G., Darwish Elhajji, F. W., ... & McElnay, J. C. (2012). The impact of antibiotic use on the incidence and resistance pattern of extended‐ spectrum beta‐ lactamase‐ producing bacteria in primary and secondary healthcare settings.British journal of clinical pharmacology, 74(1), 171-179.

15

Atmaca, E. and Basar, H.B. (2012). Evaluation of power plants in Turkey using Analytic Network Process (ANP). Energy, 44 (1): 555–563. Berg, S. (2010). Water Utility Benchmarking: Measurement, Methodology, and Performance Incentives. International Water Association. Berument, H., Ceylan, N.B. and Gozpinar, E. (2006). Performance of soccer on the stock market: Evidence from Turkey. The Social Science Journal 43 (2006): pp. 695–699. Box, G.E.P., Jenkins, G.M. and Reinsel, G.C. (2008).Time Series Analysis Forecasting and Control. McGraw-Hill Inc.,New York, USA. Bruun, J., Somerfield, P., & Allen, J. I. (2012). Applications of Box-Jenkins Transfer Function methods to identify seasonal and longer term trends in Marine time series. ICES CM, 22. Chance, B. Leigh, J.S. Jr, Clark, B. J., Maris, J, Kent, J, Nioka, S and Smith, D (1985). Control of oxidative metabolism and oxygen delivery in human skeletal muscle: a steady-state analysis of the work/energy cost transfer function. PNAS 82(24): 8384-8388. Choudhury, A.H. (2007). Intervention Impact of Tax Reform Act on the Business Failure Process. Academy of Accounting and Financial Studies Journal, 11(3):pp. 57-68. Cooray, T. (2006). Analysis And Evaluation of Econometric Time Series Models: Dynamic Transfer Function Approach. Proceeding of the 62th Annual Session of the SLAAS (2006) pp 96. DeLurgio, S.A. (1998). Forecasting Principles and Applications, 3rd Edn, McGraw-Hill, New York, USA, (1998). Disney, S.M. and Towill D.R. (2002). A discrete transfer function model to determine the dynamic stability of a vendor managed inventory supply chain. International Journal of Production Research, 40(1):179-204. Edlund, P. and Karlsson S. (1993). International Journal of Forecasting 9(1):61-76. Enders, W. (1995). Applied Econometric Time Series. John Wiley and Sons, New York, USA. El Khashab, H., & Al Ghamedi, M. (2015). Comparison between hybrid renewable energy systems in Saudi Arabia. Journal of Electrical Systems and Information Technology, 2(1), 111-119. Fearn, T. and Maris, P.I. (1991). An Application of Box-Jenkins Methodology to the Control of Gluten Addition in a Flour Mill. Applied Statistics, 40:477-484. Fink, D.G. and Beaty, H.W. (2006). Standard Handbook for Electrical Engineers, 15th Ed, McGrawHill, New York, USA. Foley, A. M., Leahy, P. G., Marvuglia, A., & McKeogh, E. J. (2012). Current methods and advances in forecasting of wind power generation. Renewable Energy, 37(1), 1-8. 16

Forst, F.G. (2011). Forecasting Restaurant Sales Using Multiple Regression and Box-Jenkins Analysis. Journal of Applied Business Research, 8(2):15-19. García-Martos, C., Rodriguez, J. and Sanchez, M.J. (2007). Mixed Models for Short-Run Forecasting of Electricity Prices: Application for the Spanish Market. IEEE Transactions on Power Systems, 22(2). Giannakis, D., Jamasb, T. and Pollitt, M. (2005). Benchmarking and incentive regulation of quality of service: An application to the UK electricity distribution networks. Energy Policy 33 (2005): 2256– 2271. Growitsch, C., Jamasb, T. and Pollitt, M.G. (2009). Quality of Service, Effciency, and Scale in Network Industries: An Analysis of European Electricity Distribution, Applied Economics, 44(20): 2555-2570. International Electrotechnical Commission (2007). Efficient Electrical Energy Transmission and Distribution.IEC Newsletter. Igboanugo, A.C. and Nwobi-Okoye, C.C. (2011a). Production Process Capability Measurement and Quality Control Using Transfer Functions. Journal of the Nigerian Association of Mathematical Physics, 19(1): 453-464. Igboanugo, A.C. and Nwobi-Okoye, C.C. (2011b). Optimisation of Transfer Function Models using Genetic Algorithms. Journal of the Nigerian Association of Mathematical Physics, 19(1): 439-452. Igboanugo, A.C. and Nwobi-Okoye, C.C. (2011c).Transfer Function Modelling as a Tool for Solving Manufacturing System Dysfunction. Proceedings of NIIE 2011 Conference, August 4-6, 2011, pp. 2232. Issarayangyun, T. and Greaves, S (2006). Analysis of Minute-by-Minute Pollution Exposure inside a Car – a Time-Series Modelling Approach. CAITR 2006. Jha, D.K. and Shrestha, R. (2006). Measuring efficiency of hydropower plants in Nepal using data envelopment analysis. IEEE Transactions on Power Systems, 21(4): 1502-1511. Jha, D.K., Yorino, N. and Zoka, Y. (2007). A Modified DEA Model for Benchmarking of Hydropower Plants. Power Tech, 2007 IEEE Lausanne, July 1-5, 2007, pp. 1374-1379.

Kinney, W.R. (1978). ARIMA and Regression in Analytical Review: An Empirical Test, The Accounting Review, 53 (1) : 48-60. Lai, P. (1979). Transfer Function Modelling Relationship between Time Series Variables. Concepts and Techniques in Modern Geography (CATMOG), London School of Economics, No. 22 (1979). Li, Y., Su, Y., & Shu, L. (2014). An ARMAX model for forecasting the power output of a grid connected photovoltaic system. Renewable Energy, 66, 78-89.

17

Limanond, T., Jomnonkwao, S., Srikaew, A. (2011). Projection of future transport energy demand of Thailand. Energy Policy, 39(5):2754–2763. Liu, C.H., Lin, S.J. and Lewis, C. (2010). Evaluation of thermal power plant operational performance in Taiwan by data envelopment analysis. Energy Policy, 38 (2): 1049–1058. Nogales F.J. and Conejo A.J. (2006). The Journal of the Operational Research Society, 57(4):350-356. Nwobi-Okoye, C.C. and Igboanugo, A.C. (2012). Performance Evaluation of Hydropower Generation System Using Transfer Function Modelling. International Journal of Electrical Power and Energy Systems, 43(1): 245-254. Nwobi-Okoye, C.C. and Igboanugo, A.C. (2015). Performance Appraisal of Gas Based Electric Power Generation System Using Transfer Function Modelling. Ain Shams Engineering Journal, 6 (2015): 541-551. Nwobi-Okoye, C.C., Okiy, S. and Igboanugo, A.C. (2016). Performance Evaluation of Multi-InputSingle-Output (MISO) Production Process Using Transfer Function and Fuzzy Logic: Case Study of a Brewery. Ain Shams Engineering Journal, (2016) 7: 1001–1010. http://dx.doi.org/10.1016/j.asej.2015.07.008. Nwobi-Okoye, C.C., and Okiy, S. (2016). Performance Assessment of Multi-Input-Single-Output (MISO) Production Process Using Transfer Function and Fuzzy Logic: Case Study of Soap Production. Cogent Engineering, (2016). http://dx.doi.org/10.1080/23311916.2016.1257082. Okiy, S., Nwobi-Okoye, C.C. and Igboanugo, A.C. (2015). Transfer Function Modelling: A Literature Survey. Research Journal of Applied Sciences, Engineering and Technology, 11(11): 1265-1279. Pardo, A., Meneu, V. and Valor, E. (2002). Temperature and seasonality influences on Spanish electricity load. Energy Economics, 24(1):55-70. Petkov, R. (1996). Optimum Design of a High-Power, High-Frequency Transformer. IEEE Transactions on Power Electronics, 11(1): 33-42. Preciado, I., Punzon, A., Gallego, J.L. and Vila, Y. (2006). Using time series methods for completing fisheries historical series. Bol. Inst. Esp. Oceanogr. 22 (1-4):pp. 83-90. Ray, S. (2007). Electrical Power Systems: Concepts Theory and Practice. PHI Learning Private Limited, New Delhi, India. Schwartz, J.D., Rivera, D.E., and Kempf, K.G. (2005). Towards Control-Relevant Forecasting in Supply Chain Management. American Control Conference June 8-10, 2005. Portland, OR, USA. Shumway, R.H. and Stoffer, D.S. (2006). Time Series Analysis and Its Applications in R. Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA.

18

Sözen, A., Alp, İ. and Özdemir, A. (2010). Assessment of operational and environmental performance of the thermal power plants in Turkey by using data envelopment analysis. Energy Policy, 38 (10): 6194–6203. Su, Z., Wang, J., Lu, H., & Zhao, G. (2014). A new hybrid model optimized by an intelligent optimization algorithm for wind speed forecasting. Energy Conversion and Management, 85, 443-452. Tang, S.C. and Hui, S.Y. (2001). A Low-Profile Low-Power Converter with Coreless PCB Isolation Transformer. IEEE Transactions on Power Electronics, 16 (3): 311-315. Valipour, M. (2015). Long term runoff study using SARIMA and ARIMA models in the United States. Meteorological Applications, 22(3), 592-598. Valipour, M. (2012). Ability of Box-Jenkins Models to Estimate of Reference Potential Evapotranspiration (A Case Study: Mehrabad Synoptic Station, Tehran, Iran). IOSR Journal of Agriculture and Veterinary Science, 1(2012), 1-11. Webb, R.D. and Hardt, D.E. (1991). Transfer Function Description of Sheet Metal Forming for Process Control. Transactions of ASME, 113:pp 44-52. Weyman-Jones, T.G. (1991). Productive efficiency in a regulated industry: The area electricity boards of England and Wales. Energy Economics 13(1991)116–122. Wilson, N., Sertsou, G., Edwards, R., Thomson, Grigg, M. and Li, J. (2007).A new national smokefree law increased calls to a national quitline. BMC Public Health, 7:75. Yssaada, B., Khiatb, M. and Chaker, A. (2014). Reliability centered maintenance optimization for power distribution systems. International Journal of Electrical Power and Energy Systems, 55 (2014): 108–115. Zhang, R., Zuckerman, J.H., Giller, C.A. and Levine, B.D. (1998).Transfer function analysis of dynamic cerebral autoregulation in humans. Am J Physiol Heart Circ Physiol 274(1): H233-H241. Zhou, P., Ang, B.W. and Poh, K.L. (2008). A survey of data envelopment analysis in energy and environmental studies. European Journal of Operational Research 189 (2008): 1–18.

19

Variability in Power Input (MWH)

Input Xt-b

Variability in Power Output (MWH)

Process (Transformers, Relays & Switches)

Output Yt

Figure 1: Schematic of the input-output relationship of a power distribution system

Average Daily Power Input (MW)

14.00 12.00 10.00 8.00 6.00 4.00 2.00 0.00 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 Day

Figure 2: Input Series for months 1-2 (2012)

20

Average Daily Power Output (MW)

12.00 10.00 8.00 6.00 4.00 2.00 0.00 1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49 51 53 55 57 59 Day

Figure 3: Output Series for months 1-2 (2012)

Figure 4: Input series ACF

21

Figure 5: Input series PACF

Figure 6: Output series ACF

22

Figure 7: Output series PACF

Figure 8: CCF of the pre-whitened series

23

Figure 9: Residual plot of ACF and PACF vs lag for months 1-2 (ARIMA 1,0,0)

Efficiency (%)

100 95 90 85 80 75

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121

70

Day

Figure 10: Efficiencies of the low voltage transformer from Shell

24

100.00

Efficiency (%)

95.00 90.00 85.00 80.00 75.00 70.00 65.00

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116

60.00

Day

Figure 11: Efficiencies of the low voltage transformer from PHCN 105

Efficiency (%)

100 95 90 85 80 75 0

10

20

30

40

50

60

Day

Figure 12: Daily efficiency measurements in period 1(Months 1-2 for Shell’s transformer)

25

70

95.00

Efficiency (%)

90.00 85.00 80.00 75.00 70.00 65.00 60.00 0

10

20

30

40

50

60

70

Day

Figure 13: Daily efficiency measurements in period 2 (Months 1-2 for 132KVA/11KVA PHCN’s transformer)

26

Table 1: Model fit 1-2

Fit Statistic Stationary R-squared R-squared RMSE MAPE MaxAPE MAE MaxAE Normalized BIC

Value 0.992 0.992 0.191 2.052 25.153 0.114 0.953 -3.103

Table 2: Model Statistics 1-2 Model statistics

Model Transfer Model

Function

Fit

Ljung-Box Q(18) Number of Stationary RPredictors squared Statistics DF

Sig.

Number Outliers

1

0.561

0

0.992

14.483

27

17

of

Table 3: Summary of extensive analysis of the results of the transfer function modelling for months 1-2 STATISTICS ARIMA ARIMA ARIMA

R-squared MAPE

ARIMA ARIMA

ARIMA ARIMA ARIMA

ARIMA

ARIMA

(1,0,0)

(0,1,0)

(1,1,0)

(1,1,1)

(0,1,1)

(0,0,1)

(1,0,1)

(1,2,0)

(1,2,1)

(0,2,1)

0.992 2.052

.221 5.119

.276 4.793

.319 4.942

.284 4.813

.711 3.408

.716 3.280

-.517 7.083

.130 5.095

.030 5.211

28

Table 4: Summary of Ljung Box diagnostics test of the transfer function modelling for months 1-2 Ljung-Box Q(18) Model

Statistics

DF

Sig.

14.483

17

0.561

ARIMA (0,1,0) model

17.215

18

.508

ARIMA (1,1,0)Model

18.210

17

.376

ARIMA(1,1,1) Model

12.450

16

.712

ARIMA(0,1,1)Model

16.884

17

.462

ARIMA(0,0,1)Model

13.190

17

.934

ARIMA(1,0,1)Model

14.398

16

.845

ARIMA(1,2,0) Model

31.539

17

.017

ARIMA (1,2,1) model

20.109

16

.215

ARIMA (0,2,1)Model

21.184

17

.218

ARIMA(1,0,0) Model

29

Table 5: Transfer Function Models of the PHCN’s Power Distribution Transformers Months 1-2

Transfer Function Model Transfer Function Model (v(B)) (v(B)) (132KVA-33KVA Transformer) (132KVA-11KVA 𝑌̂𝑡 = 𝜇𝑦 + 0.862𝑥𝑡 Transformer) 𝑌̂𝑡 = 𝜇𝑦 + 0.857𝑥𝑡

3-4

𝑌̂𝑡 = 𝑌𝑡−1 + 0.859𝑥𝑡

5-6

𝑌̂𝑡 = 𝑌𝑡−1 + 0.851𝑥𝑡

𝑌̂𝑡 = 𝜇𝑦 + 0.881𝑥𝑡

7-8

𝑌̂𝑡 = 𝑌𝑡−1 + 0.864𝑥𝑡

𝑌̂𝑡 = 𝑌𝑡−1 + 0.869𝑥𝑡

9-10

𝑌̂𝑡 = 𝜇𝑦 + 0.871𝑥𝑡

𝑌̂𝑡 = 𝜇𝑦 + 0.859𝑥𝑡

11-12

𝑌̂𝑡 = 𝑌𝑡−1 + 0.869𝑥𝑡

𝑌̂𝑡 = 𝑌𝑡−1 + 0.891𝑥𝑡

30

𝑌̂𝑡 = 𝜇𝑦 + 0.876𝑥𝑡

Table 6: Transfer Function Models of the Shell’s Power Distribution Transformer Transfer Function Model (v(B)) (132KVA-11KVA Transformer)

Months 1-2

𝑌̂𝑡 = 𝜇𝑦 + 0.67𝑥𝑡

3-4

𝑌̂𝑡 = 𝑌𝑡−1 + 1.065𝑥𝑡

5-6

𝑌̂𝑡 = 𝑌𝑡−1 + 1.041𝑥𝑡

7-8

𝑌̂𝑡 = 𝑌𝑡−1 + 1.081𝑥𝑡

9-10

𝑌̂𝑡 = 𝑌𝑡−1 + 0.978𝑥𝑡

11-12

𝑌̂𝑡 = 𝑌𝑡−1 + 1.022𝑥𝑡

Table 7: Energy Transformed Vs Coefficient of Performance of the Shell’s Low Voltage Transformer Years

Coefficient of Performance 0

1-2 3-4 5-6 7-8 9-10 11-12

0.670 1.065 1.041 1.081 0.978 1.022

Table 8: Energy Transformed Vs Coefficient of Performance of the PHCN’s Low Voltage Transformer Years

Coefficient of Performance Coefficient of Performance 0 0 (132KVA-11KVA Transformer)

(132KVA-33KVA Transformer)

31

1-2 3-4 5-6 7-8 9-10 11-12

0.857 0.859 0.851 0.864 0.871 0.869

0.862 0.876 0.881 0.869 0.859 0.891

Table 9: Efficiency Vs Coefficient of Performance for two Transformers Period 1 2

Average Efficiency (%) 93.79 85.61

32

COP 0.670 0.857