Power Systems Reliability Applications


CHAPTER 10


Power Systems Reliability Applications


10.1  INTRODUCTION


Electrical energy is an essential ingredient for the development of modern society. Almost all aspects of daily life depend on the use of electrical energy and the performance of a power utility. The power utility is measured in terms of the quality and reliability of the supply. Electric power utilities have invested a substantial amount of capital in generating stations, transmission lines, and distribution facilities to supply electrical energy to their customers as economically as possible and with a reasonable assurance of continuity, safety, and quality. This creates the difficult problem of balancing the need for continuity of power supply with the costs involved.


10.2  EQUIVALENT TRANSITION RATE MATRIX


There are a number of methods for constructing the transition rate matrix. For general systems using the Kronecker algebra helps to form a transition rate matrix. This is one method out of a number of methods that help to form a transition rate matrix. Also, the rules and properties can be introduced. The Kronecker algebra is used in this textbook for constructing the transition rate matrix of a system contain a number of power stations.


10.3  TWO-STATE MODEL


The two-state model refers to a generator passing through two-states. In this case, the power generator is passing through Up-State and Down-State. This means the generator is operating and is failing, respectively. The three methods to calculate the transient probabilities for the considered system are discussed earlier in Chapter 8. The results are obtained using the SMM illustrated in Table 10.1 and Figure 10.1. Table 10.2 and Figure 10.2 illustrate the results obtained by the Rung-Kutta method. Finally, Table 10.3 and Figure 10.3 show the results obtained by the Adams method. These three methods are recommended to calculate the transient and steady-state probabilities for any model.


























































































































TABLE 10.1    Two-State Model Results Using SMM

Time (year)


P1(t)


P2(t)


0


1


0


0.5


0.9


0.1


1


0.85


0.15


1.5


0.825


0.175


2


0.8125


0.1875


2.5


0.8063


0.1938


3


0.8031


0.1969


3.5


0.8016


0.1984


4


0.8008


0.1992


4.5


0.8004


0.1996


5


0.8002


0.1998


5.5


0.8001


0.1999


6


0.8


0.2


6.5


0.8


0.2


7


0.8


0.2


7.5


0.8


0.2


8


0.8


0.2


15


0.8


0.2


15.5


0.8


0.2


16


0.8


0.2


16.5


0.8


0.2


17


0.8


0.2


17.5


0.8


0.2


18


0.8


0.2


18.5


0.8


0.2


19


0.8


0.2


19.5


0.8


0.2


20


0.8


0.2



Image
FIGURE 10.1    Two-state model results using SMM.













































































































































































TABLE 10.2    Two-State Model Results Using Runge-Kutta Method

Time (year)


P1(t)


P2(t)


0


1


0


0.5


0.9245


0.0755


1


0.8775


0.1225


1.5


0.8482


0.1518


2


0.83


0.17


2.5


0.8187


0.1813


3


0.8116


0.1884


3.5


0.8072


0.1928


4


0.8045


0.1955


4.5


0.8028


0.1972


5


0.8017


0.1983


5.5


0.8011


0.1989


6


0.8007


0.1993


6.5


0.8004


0.1996


7


0.8003


0.1997


7.5


0.8002


0.1998


8


0.8001


0.1999


8.5


0.8001


0.1999


9


0.8


0.2


9.5


0.8


0.2


10


0.8


0.2


10.5


0.8


0.2


11


0.8


0.2


11.5


0.8


0.2


12


0.8


0.2


12.5


0.8


0.2


13


0.8


0.2


13.5


0.8


0.2


14


0.8


0.2


14.5


0.8


0.2


15


0.8


0.2


15.5


0.8


0.2


16


0.8


0.2


16.5


0.8


0.2


17


0.8


0.2


17.5


0.8


0.2


18


0.8


0.2


18.5


0.8


0.2


19


0.8


0.2


19.5


0.8


0.2


20


0.8


0.2



Image
FIGURE 10.2    Two-state model results using the Runge-Kutta method.









































































































































TABLE 10.3    Two-State Model Results Using Adams Method

Time (hr)


P1(t)


P2(t)


0


1


0


0.5


0.91


0.09


1


0.86


0.14


1.5


0.83


0.17


2


0.815


0.185


2.5


0.805


0.195


3


0.8


0.2


3.5


0.8


0.2


4


0.8


0.2


4.5


0.8


0.2


5


0.8


0.2


5.5


0.8


0.2


6


0.8


0.2


6.5


0.8


0.2


7


0.8


0.2


7.5


0.8


0.2


8


0.8


0.2


8.5


0.8


0.2


9


0.8


0.2


9.5


0.8


0.2


10


0.8


0.2


11


0.8


0.2


11.5


0.8


0.2


12


0.8


0.2


13


0.8


0.2


14


0.8


0.2


15


0.8


0.2


16


0.8


0.2


17


0.8


0.2


18


0.8


0.2


19


0.8


0.2


20


0.8


0.2



Image
FIGURE 10.3    Two-state model results using the Adams method.

The curve fitting technique, applied to the three methods, is used to find out the results for the two-state model. The results are illustrated in Figures 10.4a, b using SMM, Figures 10.5a, b using the Runge-Kutta method, and Figures 10.6a, b using Adams method. The suitable equations for the three methods are summarized in Table 10.4.



Image
FIGURE 10.4a    First probability using SMM.


Image
FIGURE 10.4b    Second probability using SMM.


Image
FIGURE 10.5a    First probability using the Runge-Kutta method.


Image
FIGURE 10.5b    Second probability using the Runge-Kutta method.


Image
FIGURE 10.6a    First probability using the Adams method.


Image
FIGURE 10.6b    Second probability using the Adams method.

10.4  THREE-STATE MODELS


The three-state model (Figure 10.7) is considered for a case of two generators under operation in a power station. The first state considered that both generators are working, where the second state illustrates that one is working and the others fail. In the third state, both generators are in a failstate. Using the Laplace transforms will result:


P1(t)λ2(λ+μ)2e2(μ+λ)t+2λμ(λ+μ)2e(μ+λ)t+μ2(λ+μ)2P2(t)2μλ(λ+μ)2+2λ(λμ)(λ+μ)2e(μ+λ)t2λ2(λ+μ)2e2(μ+λ)tP3(t)λ2(λ+μ)2e2(μ+λ)t2λ2(λ+μ)2e(μ+λ)t+λ2(λ+μ)2













































TABLE 10.4    Two-State Results Curve Fitting, y = a + b e-cx

Method


Probability


a


b


c


SMM


1


0.7999963±0.000003416


0.2000039±0.00001428


1.386269±0.0002655


2


0.2000037±0.000003416


–0.2000039±0.00001428


1.386269±0.0002655


Runge-Kutta


1


0.7999955±0.000005142


0.2000096±0.00002047


0.948314±0.000219


2


0.2000045±0.000005142


–0.2000096±0.00002047


0.948314±0.000219


Adams


1


0.7996066±0.0003309


0.2007632±0.001368


1.236986±0.02122


2


0.2003934±0.0003309


–0.2007631±0.001368


1.236985±0.02122



Image
FIGURE 10.7    Three-state of the two-generation model.
Source: Modified from Billinton and Allan, 1992.

The results are illustrated in three cases. These cases are summarized in Tables 10.5, 10.6, 10.7, 10.8 and Figures 10.8, 10.9, 10.10.






























TABLE 10.5    Results Summary for Three Cases Using Laplace Transforms

Case #


μ (per year)


λ (per year)


Table


Figure


1


0.9


0.1


10.5


10.5


2


0.85


0.15


10.6


10.6


3


0.75


0.25


10.7


10.7






































































































































































TABLE 10.6    3-State Model Using Laplace Transforms, μ = 0.9 per year, λ = 0.1 per year

TIME (YEAR)


P1(T)


P2(T)


P3(T)


0


1


0


0


1


0.877571652


0.118432584


0.003995764


2


0.834543507


0.157980042


0.007476451


3


0.81898646


0.171984494


0.009029046


4


0.81330017


0.177062789


0.009637042


5


0.811213284


0.17892102


0.009865695


6


0.810446237


0.179603277


0.009950486


7


0.810164147


0.179854082


0.009981771


8


0.810060384


0.179946324


0.009993292


9


0.810022214


0.179980254


0.009997532


10


0.810008172


0.179992736


0.009999092


11


0.810003006


0.179997328


0.009999666


12


0.810001106


0.179999017


0.009999877


13


0.810000407


0.179999638


0.009999955


14


0.81000015


0.179999867


0.009999983


15


0.810000055


0.179999951


0.009999994


16


0.81000002


0.179999982


0.009999998


17


0.810000007


0.179999993


0.009999999


18


0.810000003


0.179999998


0.01


19


0.810000001


0.179999999


0.01


20


0.81


0.18


0.01


21


0.81


0.18


0.01


22


0.81


0.18


0.01


23


0.81


0.18


0.01


24


0.81


0.18


0.01


25


0.81


0.18


0.01


26


0.81


0.18


0.01


27


0.81


0.18


0.01


28


0.81


0.18


0.01


29


0.81


0.18


0.01


30


0.81


0.18


0.01






































































































































































TABLE 10.7    Three-State Model, μ = 0.85 per year, λ = 0.15 per year

Time (year)


P1(t)


P2(t)


P3(t)


0


1


0


0


1


0.819354301


0.17165523


0.008990469


2


0.757422599


0.225755387


0.016822014


3


0.735251474


0.244433172


0.020315354


4


0.727178036


0.25113862


0.021683344


5


0.724219198


0.253582988


0.022197814


6


0.72313222


0.254479186


0.022388594


7


0.722732549


0.254808467


0.022458984


8


0.722585546


0.254929548


0.022484907


9


0.72253147


0.254974083


0.022494447


10


0.722511577


0.254990466


0.022497957


11


0.722504259


0.254996493


0.022499248


12


0.722501567


0.25499871


0.022499724


13


0.722500576


0.254999525


0.022499898


14


0.722500212


0.254999825


0.022499963


15


0.722500078


0.254999936


0.022499986


16


0.722500029


0.254999976


0.022499995


17


0.722500011


0.254999991


0.022499998


18


0.722500004


0.254999997


0.022499999


19


0.722500001


0.254999999


0.0225


20


0.722500001


0.255


0.0225


21


0.7225


0.255


0.0225


22


0.7225


0.255


0.0225


23


0.7225


0.255


0.0225


24


0.7225


0.255


0.0225


25


0.7225


0.255


0.0225


26


0.7225


0.255


0.0225


27


0.7225


0.255


0.0225


28


0.7225


0.255


0.0225


29


0.7225


0.255


0.0225


30


0.7225


0.255


0.0225






































































































































































TABLE 10.8    Three-State Model, μ = 0.75 per year, λ = 0.25 per year

Time (year)


P1(t)


P2(t)


P3(t)


0


1


0


0


1


0.708913246


0.266113229


0.024973525


2


0.614395459


0.338876724


0.046727817


3


0.581325073


0.362243389


0.056431538


4


0.569389331


0.370379157


0.060231512


5


0.565029568


0.373309838


0.061660594


6


0.563429916


0.374379544


0.06219054


7


0.562842008


0.374771926


0.062386067


8


0.562625806


0.37491612


0.062458074


9


0.56254628


0.374969146


0.062484575


10


0.562517025


0.37498865


0.062494325


11


0.562506263


0.374995825


0.062497912


12


0.562502304


0.374998464


0.062499232


13


0.562500848


0.374999435


0.062499717


14


0.562500312


0.374999792


0.062499896


15


0.562500115


0.374999924


0.062499962


16


0.562500042


0.374999972


0.062499986


17


0.562500016


0.37499999


0.062499995


18


0.562500006


0.374999996


0.062499998


19


0.562500002


0.374999999


0.062499999


20


0.562500001


0.374999999


0.0625


21


0.5625


0.375


0.0625


22


0.5625


0.375


0.0625


23


0.5625


0.375


0.0625


24


0.5625


0.375


0.0625


25


0.5625


0.375


0.0625


26


0.5625


0.375


0.0625


27


0.5625


0.375


0.0625


28


0.5625


0.375


0.0625


29


0.5625


0.375


0.0625


30


0.5625


0.375


0.0625



Image
FIGURE 10.8    Three-state model, μ = 0.9 per year, λ = 0.1 per year.


Image
FIGURE 10.9    Three-state model, μ = 0.85 per year, λ = 0.15 per year.


Image
FIGURE 10.10    Three-state model, μ = 0.75 per year, λ = 0.25 per year.

The Laplace Transforms and Adams Methods have been applied to obtain the transient probabilities shown in the present chapter. The curve fitting technique applied to the three cases of the three-state model. The results obtained are summarized in Figures 10.11a,b,c, 10.12a,b,c, and 10.13a,b,c as the values of the repair and failure rates printed. The results of the fitting equations for the three cases are summarized in Table 10.9.



Image
FIGURE 10.11a    The first probability for the three-state model using Laplace transforms (μ = 0.9, λ = 0.1).


Image
FIGURE 10.11b    The second probability for the three-state model using Laplace transforms (μ = 0.9, λ = 0.1).


Image
FIGURE 10.11c    The third probability for the three-state model using Laplace transforms (μ = 0.9, λ = 0.1).


Image
FIGURE 10.12a    The first probability for the three-state model using Laplace transforms (μ = 0.85, λ = 0.15).


Image
FIGURE 10.12b    The second probability for the three-state model using Laplace transforms (μ = 0.85, λ = 0.15).


Image
FIGURE 10.12c    The third probability for the three-state model using Laplace transforms (μ = 0.85, λ = 0.15).


Image
FIGURE 10.13a    The first probability for the three-state model using Laplace transforms (μ = 0.75, λ = 0.25).


Image
FIGURE 10.13b    The second probability for the three-state model using Laplace transforms (μ = 0.75, λ = 0.25).


Image
FIGURE 10.13c    The third probability for the three-state model using Laplace transforms (μ = 0.75, λ = 0.25).

10.5  SYSTEM COMBINATION


Any number of Markov models (representing a number of systems) can form one large system representing an electric power station. This clear when the Kronecker multiplication is used.


































































TABLE 10.9    Three-State Results Curve Fitting Using Laplace Transforms, y = a + b e-cx

Method


Probability


a


b


c


LT,


1


0.8100282±0.00003175


0.1899086±0.0001303


1.029159±0.001632


μ = 0.9


2


0.1799455±0.00005997


–0.1798336±0.0002469


1.063473±0.003417


λ = 0.1


3


0.01003722±0.00005733


–0.01026308±0.00005997


0.6513805±0.02942


LT,


1


0.7225763±0.00007297


0.277281±0.0002973


1.045382±0.002606


μ = 0.85


2


0.254857±0.0001317


–0.2546274±0.00054


1.104067±0.00557


λ = 0.15


3


0.0225983±0.0001371


–0.02314623±0.0005211


0.6590729±0.03037


LT,


1


0.5627287±0.0001916


0.4369243±0.0007806


1.082793±0.004574


μ = 0.75


2


0.3746189±0.0003001


–0.3742009±0.001239


1.215129±0.01003


λ = 0.25


3


0.06282769±0.0003926


–0.06439482±0.001466


0.6609917±0.03079


10.6  PROPOSED MODEL ADVANTAGE


The Kronecker product is very efficient to form a large model of the electric power system, where this an advantage recorded for the Kronecker product.


10.7  MEMBERSHIP FUNCTION


Figure 10.14 illustrates the four layers connection for three inputs and one output. These four layers represent the Neuro-Fuzzy model, which is used to represent the developed estimated peak load model in this study. It is clear that the results obtained automatically through the training of data performed in the Adaptive Neuro-Fuzzy Inference System (ANFIS).



Image
FIGURE 10.14    (See color insert.) ANFIS structure with three inputs and one output.
Source: Reprinted from Qamber and Al-Hamad, 2016.

Figure 10.15 shows the proposed model using the Neuro-Fuzzy, where the proposed model is three inputs and one output model. The three inputs are the year, population, and GDP. The output is the peak load for the country. Therefore, the three inputs used as follows:



1.  Input 1: Year.


2.  Input 2: Population.


3.  Input 3: GDP.



Image
FIGURE 10.15    (See color insert.) Artificial neuro-fuzzy logic model using long term load forecasting.
Source: Reprinted from Qamber and Al-Hamad, 2016.

Figure 10.16 illustrates the actual data collected and presented for member state’s peak demand, where the estimated peak load demand for the coming years generated by Neuro-Fuzzy model. Figure 10.17 membership function used in the neuro-fuzzy logic model for long term load forecasting.


10.7.1  VALIDATION OF MEMBERSHIP FUNCTIONS


The ANFIS or adaptive network-based fuzzy inference system can be a shortage as “ANFIS.” ANFIS is a kind of artificial neural networkmodeling standard. It was adopted to effectively tune the membership function to decrease the output error and maximize performance indicators. At the same time, ANFIS Editor Display made up four types. These types are Load data, General is, Train FIS, and Test FIS. The load data is used for training, testing, and checking. In the research, the ANFIS approach is recommended, where it is used by adding the Artificial Neural Network to the Fuzzy System, which has successfully simulated for the load forecasting.



Image
FIGURE 10.16    (See color insert.) Actual data used versus the output of the neurofuzzy model.
Source: Reprinted from Qamber and Al-Hamad, 2016.


Image
FIGURE 10.17    (See color insert.) Membership function used in the neuro-fuzzy logic model for long term load forecasting.
Source: Reprinted from Qamber and Al-Hamad, 2016.

10.7.2  LOAD FORECASTED SURFACE


The electricity load forecasting is an active research topic with important practical effects for almost any industry. It is well known that the load forecasting is a technique used by electric power or electric energyproviding companies to estimate the power or energy needed to satisfy the demand and supply stability. The accuracy of forecasting is of great worth for the operational and managerial loading of a utility company. The most accurate estimation of energy consumption and supplies has a helpful impact on active budgets. In the instance of the electrical industry, accurate electrical load estimating is a suitable means to provide well-grounded information and capable energy management for arrangement in both the mid and long term and for the network operation in the short term. Back to the term (surface). The term (surface) is viewer represented by a graphical interface that lets the researcher examines the output surface of the fuzzy inference system (FIS) for two inputs. Figure 10.18 illustrates an example of a load forecasting of a country. The inputs are the population and years to find the estimates load for further years, as shown in Figure 10.18.



Image
FIGURE 10.18    (See color insert.) A surface graph for a country, estimating the load.
Source: Reprinted from Qamber and Al-Hamad, 2016.

10.8  THE EFFECT OF TEMPERATURE


To design high-precision estimators for a load-cell-based weighting system with temperature compensation, ANFIS is used to model the relationship with actual weights of samples. In addition, ANFIS can improve the precision of load cell. ANFIS is used to model the relationship between the reading of load cells and the actual weight of samples considering the temperature-varying effect and nonlinearity of the load cells. The model of the load-cell-based weighting system can accurately estimate the weight of test samples from the load cell reading. The proposed ANFIS-based method is convenient to use and can improve the 117 precision of digital load cell measurement systems. The temperature and humidity are two parameters considered. The humidity and the temperature data are fed to the fuzzy logic (FL) system, where the output is the load. The load is proportional to these two parameters. The neural networks used for training and comparing the set of the past load to predict the future. In this case, the load does not just depend on the temperature and humidity but also depends on the other data like a number of customers for Load Forecasting of the Power System Planning using Fuzzy-Neural.


10.8.1  FIRST MODEL


There are a number of methods used to estimate the electric load for the countries. Kingdom of Bahrain taken as an example in the present chapter to estimate its electric load for the future up to the year 2025 using the polynomial. Table 10.10 shows the estimated peak loads for the Kingdom of Bahrain with calculated results using the Polynomial. The electric load percentage error using the polynomial is calculated and found as 2.471288523. The estimated load versus the years is plotted and illustrated in Figure 10.19. The percentage error of the peak load for the results of the polynomial is shown in Figure 10.20. The results of the percentage error for the peak load converted to absolute values and shown in Figure 10.21. Furthermore, when the square of the peak load error versus the years obtained, which is shown in Figure 10.22, the shape of that becoming almost a sinusoidal.


10.8.2  SECOND MODEL


There are a number of methods used to estimate the electric load for the countries. Kingdom of Bahrain taken as an example in the present chapter to estimate its electric load for the future up to the year 2025 using the exponential. Table 10.11 shows the peak loads for the Kingdom of Bahrain with calculated estimated peak loads results using the Exponential. The electric load percentage error using the exponential is calculated and found as5.100549189. The estimated load versus the years is plotted and illustrated in Figure 10.23. The percentage error of the peak load for the results of the exponential is shown in Figure 10.24. The results of the absolute value percentage error for the peak load is found and shown in Figure 10.25. Furthermore, when the square of the peak load error versus the years obtained, which is shown in Figure 10.26, the shape of that becoming almost a sinusoidal.






















































































































TABLE 10.10    The Estimated Peak Loads for the Kingdom of Bahrain with Calculated Results Using the Polynomial

Year


Peak Load Actual (MW)


Peak Load Calculated (MW)


% Peak Load Error


2003


1535


1449.329226


5.581157948


2004


1632


1633.227512


–0.075215196


2005


1725


1811.186738


–4.996332609


2006


1906


1983.206902


–4.050729381


2007


2136


2149.288006


–0.622097636


2008


2314


2309.430048


0.197491443


2009


2438


2463.63303


–1.051395796


2010


2708


2611.89695


3.54885709


2011


2871


2754.22181


4.067509248


2012


2880


2890.607608


–0.368319722


2013


2917


3021.054346


–3.567169883


2014


3152


3145.562022


0.204250571


2015


3441


3264.130638


5.140057033


2016


3415


3376.760192


1.119760117


2017


3572


3483.450686


2.478984169


2018



3584.202118


2019



3679.01449


2020



3767.8878


2021



3850.82205


2022



3927.817238


2023



3998.873366


2024



4063.990432


2025



4123.168438



Image
FIGURE 10.19    The estimated load versus the years.


Image
FIGURE 10.20    The percentage error for the load versus the years.


Image
FIGURE 10.21    The absolute value of percentage error for the load versus the years.


Image
FIGURE 10.22    The square of the percentage error for the load versus the years.





















































































































TABLE 10.11    The Estimated Peak Loads for the Kingdom of Bahrain with Calculated Results Using the Exponential

Year


Actual Peak Load (MW)


Estimated Peak Load (MW)


% Error


2003


1535


1570.298109


–2.299551106


2004


1632


1681.342525


–3.023439059


2005


1725


1800.239503


–4.361710307


2006


1906


1927.544339


–1.130343093


2007


2136


2063.851601


3.377734032


2008


2314


2209.797899


4.503115861


2009


2438


2366.064862


2.950579889


2010


2708


2533.382322


6.448215581


2011


2871


2712.53172


5.519619656


2012


2880


2904.349756


–0.845477656


2013


2917


3109.732302


–6.60720952


2014


3152


3329.638576


–5.635741633


2015


3441


3565.095633


–3.606382826


2016


3415


3817.203153


–11.77754475


2017


3572


4087.138583


–14.42157287


2018



4376.162631


2019



4685.62516


2020



5016.971486


2021



5371.749133


2022



5751.615059


2023



6158.343393


2024



6593.833726


2025



7060.119975



Image
FIGURE 10.23    The estimated load versus the years.


Image
FIGURE 10.24    The percentage error for the load versus the years.


Image
FIGURE 10.25    The absolute value of percentage error for the load versus the years.


Image
FIGURE 10.26    The square of the percentage error for the load versus the years.

10.8.3  THIRD MODEL


There are a number of methods used to estimate the electric load for the countries. Kingdom of Bahrain taken as an example in the present chapter to estimate its electric load for the future up to the year 2025 using the linear. Table 10.12 shows the peak loads for the Kingdom of Bahrain with calculated results using the Linear. The electric load percentage error using the linear is calculated and found as 2.533884781. The estimated load versus the years is plotted and illustrated in Figure 10.27. The percentage error of the peak load for the results of the linear is shown in Figure 10.28. The results of the percentage error for the absolute value peak load are found and shown in Figure 10.29. Furthermore, when the square of the peak load error versus the years obtained, which is shown in Figure 10.30, the shape of that becoming almost a semi-sinusoidal.


10.9  COMPARISON BETWEEN THE MODELS


Three techniques were used to find the estimated peak load for the Kingdom of Bahrain using the polynomial, exponential, and linear. The average percentage error was determined and found to be 2.471288523 using the polynomial technique. The average percentage errors were determined and found to be 5.100549189using the Exponential technique. The average percentage errors were determined and found to be 2.533884781using the linear technique. Comparing the three techniques results of the average percentage error, where it is found that the polynomial results have the minimum percentage average error, then the linear results and finally the exponential results. Therefore, the exponential technique is recommended for further research work. When the percentage error of the peak load squared, the sinusoidal form is becoming as a result for both the polynomial and the exponential, where the linear will take the shape of semi-sinusoidal form. In this case, the shape of the sinusoidal, which has been taken, will be homogeneous. Out of the results highlighted, it is recommended to use the polynomial, because it has less percentage error (better accuracy) and a sinusoidal form, which is homogeneous.






















































































































TABLE 10.12    The Estimated Peak Loads for the Kingdom of Bahrain with Calculated Results Using the Exponential

Year


Actual Peak Load (MW)


Estimated Peak Load (MW)


% Error (Peak Load)


2003


1535


1503.0584


2.080885993


2004


1632


1657.2612


–1.547867647


2005


1725


1811.464


–5.012405797


2006


1906


1965.6668


–3.130472193


2007


2136


2119.8696


0.755168539


2008


2314


2274.0724


1.725479689


2009


2438


2428.2752


0.398884331


2010


2708


2582.478


4.635228951


2011


2871


2736.6808


4.678481365


2012


2880


2890.8836


–0.377902778


2013


2917


3045.0864


–4.391031882


2014


3152


3199.2892


–1.500291878


2015


3441


3353.492


2.543097937


2016


3415


3507.6948


–2.714342606


2017


3572


3661.8976


–2.516730123


2018



3816.1004


2019



3970.3032


2020



4124.506


2021



4278.7088


2022



4432.9116


2023



4587.1144


2024



4741.3172


2025



4895.52



Image
FIGURE 10.27    The estimated load versus the years.


Image
FIGURE 10.28    The percentage error for the load versus the years.


Image
FIGURE 10.29    The absolute value of percentage error for the load versus the years.


Image
FIGURE 10.30    The square of the percentage error for the load versus the years.

KEYWORDS



•  adaptive network-based fuzzy inference system


•  adaptive neuro-fuzzy inference system


•  equivalent transition rate matrix


•  load forecasted surface


•  membership function


•  three-state models


REFERENCES


Ameen, I., & Novati, P. The solution of fractional order epidemic model by implicit Adams methods. Applied Mathematical Model, 2017, 43, 78–84.


Billinton, R., & Allan, R. N. Reliability Evaluation of Engineering Systems: Concepts and Techniques (2nd edn.). Pitman: New York, 1992.


Qamber, I. S., & Al-Hamad, M. Y. Trading opportunities forecasted benefits at peak load for GCC countries. Electric Power and Water Desalination Conference, 2016, Doha, Qatar.


Yanga, X., & Ralescu, D. Adams method for solving uncertain differential equations. Applied Mathematics and Computation, 2015, 270, 993–1003.

Aug 1, 2021 | Posted by in Electrical Engineer | Comments Off on Power Systems Reliability Applications
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