The EUT Method for Stochastic Problems in Power System Analysis Dave Thomas (Oluwabukola A. Oke and Preye .M. Ivry The George Green Institute for Electromagnetics Research Department of Electrical and Electronic Engineering The University of Nottingham, UK Email: Dave.thomas@nottingham.ac.uk 1
Nottingham, UK Nottingham Population is Approx. 300,000 The University of Nottingham has approx. 30,000 students Manchester Liverpool Nottingham 180 km London Bristol 2
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The University of Nottingham • Nottingham Civic college 1881 • Full University in 1948 • UK Russel Group University Ranked in the top 1% in the world • 3 Campuses – Nottingham, UK (33,000 Students) – Malaysia (5000 Students) – China (6000 Students) 5
Famous for 20/02/2018 GGIEMR 6 6
Also Famous for George Green 20/02/2018 GGIEMR 7 7
The George Green Institute for Electromagnetics Research Established in 2004 and led by 8 academics: Dave Thomas (Director) Trevor Benson Slaweck Sujecki Ana Vukovic Angela Nothofer 8 Steve Greedy Kristof Cools Phillip Sewell
Background World Total Installed Capacity [MW] 250000 203500 Wind energy Over 15% of accounted for total world 159213 200000 over 20% of electricity world’s total generation to 120903 Prediction electricity be from wind 150000 installations in 93930 by 2020 (DOE, 2009 74122 2010) 100000 59024 ( Wind energy report 47693 39295 2010) 31181 24322 50000 0 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 Probabilistic load flow was introduced in 1974 to properly account for uncertainties in the network during load flow studies. 9
Probabilistic Modeling of Uncertainties Some sources of uncertainties within the network. • Load • Generator • Wind generator • Wind data fitted to the Weibull Distribution • Wind power obtained wind speed and turbine output curve. • Output wind power obtained as Truncated Weibull-Degenerate Distribution. 10 10
Wind Power Distribution Wind Speed follows the v v v v 1 o o f ( v ) ( ) exp[ ( ) ] Weibull Distribution with Active Points PDF PDF of A 1MW Wind Turbine 0.06 600 0.05 Wind Turbine Curve 500 0.04 400 Wind Power (kW) Probability 300 0.03 200 0.02 100 0 0.01 0 5 10 15 20 25 -100 Wind Speed (m/s) 0 0 0.2 0.4 0.6 0.8 1 Output Wind Power (MW) ( ) ( ) P C vo v P C vo v w 1 ci 1 w 1 ci ( ) exp[ ( ) ] 0 P P w r C C C 1 1 1 f ( P ) w v v v v r o co o exp exp P P (4) w r 11 11 International Conference on Environment and Electrical Engineering, 2012
Existing PLF Methods Monte Carlo Simulation (MCS) • Convolution method Analytical • FFT technique methods • Cumulant method PLF Methods • Point Estimate method (PEM) Approximate Methods • Unscented Transforms (UT) method Heuristic • Fuzzy load flow techniques 12 12
Probabilistic Modeling of Uncertainties • Monte Carlo computationally very expensive • UT greatly reduces number of simulations • EUT extends the capability of the UT method 13 13
The Unscented Transforms (UT) Method 0.7 Continous 2nd order UT Works by approximating a 0.6 8th order UT continuous distribution function 0.5 as a discrete distribution using 0.4 f(x) deterministically chosen points 0.3 such that both distributions have 0.2 the same moments 0.1 k k k ˆ ˆ ˆ ˆ E( u ) u w( u )d u w S 0 0 5 10 15 20 25 i i i X Illustration of continuous and discrete PDF Continuous moment Discrete moment Conventional UT • Based on Taylor’s series expansion or some moment related method × Limited to a few orders of approximation × Inaccurate and results in complex points for arbitrary measures like the Rayleigh distribution. 14 14
UT as a Gaussian Quadrature Problem • Integration of the UT equation is carried out using a quadrature technique. • Desired sigma points (Si) correspond to the root of a polynomial orthogonal to weighting function. • No classical orthogonal polynomial associated with the Rayleigh distribution Polynomial orthogonal to Rayleigh distribution is built • from scratch. 15 15
Orthogonal Polynomials • f n and f m are orthogonal if b f , f f ( u ) f ( u ) dW ( u ) 0 n m n m a where (dW(u)=w(u)du) • To derive orthogonal polynomials Moment method is extremely ill-conditioned for arbitrary measures. • Discretization schemes such as the STIELJES PROCEDURE are better alternatives. • 16 16
Implementation Procedure Obtain sigma points Get final sigma points Get PDF for wind and weights using and weights if more power, load and other Gaussian quadrature than 1 random random variables and Stieltjes variable procedure Compute statistical Input basic load flow Run load flow using data (moments and data and sigma points Newton Raphson cumulants) Plot CDF for desired variables using statistical data. 17 17
Simple 3 Bus Test System Empirical CDF MCS 0.411 2nd Order UT 0.41 3rd Order UT 4th Order UT 0.409 5PEM F(V3) 0.408 0.407 0.406 0.405 0.404 1.0139 1.014 1.014 1.0141 V3 5 2nd order UT 4.5 3rd order UT 4 3.5 4th order UT (%) error 3 2.5 5PEM 2 1.5 1 0.5 0 Moments ε V3(mean) ε σ( Std) εμ3( Skewness) εμ4( Kurtosis) 18 18
IEEE 14 Bus Test System CDF of P6-12 1 • A 50MW rated wind 0.8 farm on Bus 6 • Varying active and 0.6 F(P6-12) reactive load on Bus 9 0.4 MCS 2nd Order UT CDF of P6-12 0.2 3rd Order UT 0.33 4th Order UT 5PEM 0 0.32 -0.095 -0.09 -0.085 -0.08 -0.075 -0.07 P6-12 Method Computation F(P6-12) 0.31 Time (S) MCS 0.3 2 nd Order UT 2nd Order UT 0.22054 3rd Order UT 0.29 3 rd Order UT 4th Order UT 0.50433 5PEM 4 th Order UT 0.97112 -0.0884 -0.0882 -0.088 -0.0878 -0.0876 P6-12 5PEM 0.9478 19 19 MCS 835.574
Summary The UT method has been introduced as a method for carrying out PLF. The performance of the UT method has been evaluated by comparing results obtained with those from MCS and 5PEM, using a simple 3 bus test system and the IEEE 14 bus test system. Correlation between the random variables to be considered. Method to be extended for conducted emissions 20 20
Wind Farm Sites and Dependence Source: http://www.renewables-map.co.uk 21 21 Probabilistic Methods Applied to Power Systems 2012
Dependence Versus Correlation Dependence measures the statistical relationship between two random variables Correlation is the strength of relationship between two or more random variables Linear Correlation measures how two random variables are proportional to each other. Zero correlation does NOT imply zero dependence 1000 random variates from 2 identical Gamma marginal distributions and identical correlation but different dependence structure 22 22 Probabilistic Methods Applied to Power Systems 2012
Measures of Dependence Pearson Product Moment Correlation coefficient Measures linear dependence between variables Simple and commonly used Spearman Rank-Order Correlation Coefficient Nonparametric Robust and resistant to data defeats Kendall’s Tau Nonparametric Natural Blomqvist Beta Nonparametric Fast with low computational complexities 23 23 Probabilistic Methods Applied to Power Systems 2012
Techniques for Dependent Variable Generation Rosenblatt Transformation Only applicable to Gaussian distribution Requires prior full knowledge of random variables joint distribution Nataf Transformation Based on linear dependence Transforms correlated variables into standard normal variables Copulas They join marginal distributions of a set of variables to their joint distribution function Captures dependence structure of any set of random variables 24 24 Probabilistic Methods Applied to Power Systems 2012
Copulas Mathematical Basis Fx , , x ( x , , x ) C ( F ( x ), F ( x ), , F ( x )) 1 n 1 n X 1 X 2 X n 1 2 n Joint Distribution Function Marginal distribution Functions Families Elliptical Copulas • Easily determined from covariance matrix of the marginal distributions • Applicable to symmetrical and radial distributions • Poor for distributions with strong tail dependence • Examples: Gaussian Copula and Student T copula Archimedean Copulas • Flexible and uniquely constructed through a generator • Dependence parameters other than Pearson rho needed to generate copula function • Examples: Gumbel copula, Frank copula, Clayton copula 25 25 Probabilistic Methods Applied to Power Systems 2012
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