A Model for Hydro Inow and Wind Power Capacity for the Brazilian Power Sector
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1 A Model for Hydro Inow and Wind Power Capacity for the Brazilian Power Sector Gilson Matos Cristiano Fernandes PUC-Rio Electrical Engineering Department GAS Workshop Tenerife 9-11th January 2014 Gilson Matos Cristiano Fernandes GAS Workshop 1 / 53
2 Outline Introduction 1 Introduction Gilson Matos Cristiano Fernandes GAS Workshop 2 / 53
3 Outline Introduction 1 Introduction Gilson Matos Cristiano Fernandes GAS Workshop 3 / 53
4 Introduction Motivation In Brazil, the peak of wind power production in the Northeast coincides with the dry season of the hydro system in the Southeast (Seasonal complementarity). This complementarity between the hydro and wind power systems can be exploited for optimal power generation planning and energy dispatch. In practice this would entail the need of jointly forecasting/simulating time series of wind and hydro ow. Gilson Matos Cristiano Fernandes GAS Workshop 4 / 53
5 Figure: Brazil geographical regions. Gilson Matos Cristiano Fernandes GAS Workshop 5 / 53
6 Figure: Monthly streamow of Paraibuna river (MG), from 1976 to Figure: Boxplots and averages of the monthly streamow of Paraibuna river. Gilson Matos Cristiano Fernandes GAS Workshop 6 / 53
7 Figure: Monthly capacity factor of a wind farm in Northeast of Brazil, from 1976 to Figure: Boxplots and averages of the monthly capacity factor in Northeast of Brazil. Gilson Matos Cristiano Fernandes GAS Workshop 7 / 53
8 Figure: Streamow and wind power factor series. Figure: Scatter plot and regression line. Gilson Matos Cristiano Fernandes GAS Workshop 8 / 53
9 Objectives To model streamow and wind power processes, we developed univariate GAS models with predictive gamma and beta densities, respectively. GAS structures for the univariate cases: SARIMA and unobserved components. Univariate models were used to construct a bivariate GAS model, with marginal gamma and beta distributions. The Bivariate GAS model implies negative correlated variables which captures the observed seasonal complementarity between the two series. Gilson Matos Cristiano Fernandes GAS Workshop 9 / 53
10 Notation y t : variable of interest. f t : time-varying vector. x t : vector of exogenous variables. θ: vector of static parameters. Y t = (y 1, y 2,..., y t ), F t = (f 0, f 1,..., f t ) and X t = (x 1, x 2,..., x t ). The available information at t is given by {f t, Ϝ t }, where Ϝ t = {Y t 1, F t 1, X t }, t = 1, 2,..., n. Assume that y t is generated by y t p(y t /f t, Ϝ t, θ) t and s t are the score vector and the weighted score, respectively. Gilson Matos Cristiano Fernandes GAS Workshop 10 / 53
11 Outline Introduction 1 Introduction Gilson Matos Cristiano Fernandes GAS Workshop 11 / 53
12 Gamma Model - SARIMA structure for parameter evolution Consider y 1t = λ tu t, u t s Gamma(α, α 1 ) iid. Multiplicative error model (MEM): f t = λ t. An appropriate link function is given by: [ r ] f t = φ k g(x t k+1 ) + ln λ t (1) k=1 or [ ] r λ t = exp f t + φ k g(x t k+1 ) (2) k=1 Gilson Matos Cristiano Fernandes GAS Workshop 12 / 53
![An appropriate link function is given by: [ r ] f t = φ k g(x t k+1 ) + ln λ t (1)](/docs-images/47/20607389/images/page_12.jpg "k=1 or [ ] r λ t = exp f t + φ k g(x t k+1 ) (2) k=1 Gilson Matos Cristiano")
13 Gamma Model - SARIMA structure for parameter evolution (y 1t /f t, Ϝ t, θ) Gamma ( α, α 1 λ t ), E (y1t /f t, Ϝ t, θ) = λ t (3) Var (y 1t /f t, Ϝ t, θ) = λ 2 t /α ( ) s t = α d yt 1, d = 0, 1/2 or 1 (4) λ t p q f t+1 = w + A i s t i+1 + B j f t j+1 (5) i=1 j=1 >> Seasonality is captured by making A i and B j nonzero, for i, j multiples of 12 and in their vicinity. Gilson Matos Cristiano Fernandes GAS Workshop 13 / 53
14 Gamma Model - Unobserved component structure for parameter evolution (y 1t /f t, Ϝ t, θ) Gamma ( α, α 1 λ t ), E (y1t /f t, Ϝ t, θ) = λ t (6) Var (y 1t /f t, Ϝ t, θ) = λ 2 t /α ( ) s t = α d y1t 1, d = 0, 1/2 or 1 (7) λ t r f t+1 = w + f 1,t+1 + f 2,t+1 + f 3,t+1 + φ k g(x t k+1 ) (8) k=1 In this model λ t = exp(f t). The scaled score can be seen as an error component. Gilson Matos Cristiano Fernandes GAS Workshop 14 / 53
15 Gamma Model - Unobserved component structure for parameter evolution random walk f 1,t+1 = f 1,t + a 1 s t (9) seasonality f 2,t+1 = 11 i=1 f 2,t+1 i + a 2 s t (10) autoregressive f 3,t+1 = φf 3,t + a 3 s t (11) Gilson Matos Cristiano Fernandes GAS Workshop 15 / 53
16 Pearson residuals: Univariate Gamma Models r t/t 1 = s t (d = 1/2) (12) Also, from MEM specication (y 1t = λ t u t ), a natural residual is given by: u t = (u t 1) (13) = s t (d = 0) Gilson Matos Cristiano Fernandes GAS Workshop 16 / 53
17 Beta Model - SARIMA structure for parameter evolution Wind power capacity is dened in the range [0, k], for k 100. In the application we adopted k = 70. For this series, we considered a beta distribution, with the timevarying parameter given by: [ ] r β t = exp f t + φ k g(x t k+1 ) (14) k=1 Gilson Matos Cristiano Fernandes GAS Workshop 17 / 53
18 Beta Model - SARIMA structure for parameter evolution β t (y 2t /f t, Ϝ t, θ) Beta (β t, α), E (y 2t /f t, Ϝ t, θ) = k β t + α (15) Var (y 2t /f t, Ϝ t, θ) = k 2 β tα (β t + α) 2 (β t + α + 1) s t = ln y 2t [ln k + ψ 1 (β t) ψ 2 (β t + α)] β 1 2d t [ψ 2 (β t) ψ 2 (β t + α)] 1 d, d = 0, 1/2 or 1 (16) p q f t+1 = w + A i s t i+1 + B j f t j+1 (17) i=1 j=1 Gilson Matos Cristiano Fernandes GAS Workshop 18 / 53
![2t [ln k + ψ 1 (β t) ψ 2 (β t + α)] β 1 2d t [ψ 2 (β t) ψ 2 (β t + α)] 1 d, d = 0, 1/2 or 1 (16) p q f](/docs-images/47/20607389/images/page_18.jpg "t+1 = w + A i s t i+1 + B j f t j+1 (17) i=1 j=1 Gilson Matos Cristiano Fernandes GAS Workshop 18 / 53")
19 Beta Model - Unobserved component structure for parameter evolution β t (y 2t /f t, Ϝ t, θ) Beta (β t, α), E (y 2t /f t, Ϝ t, θ) = k β t + α (18) Var (y 2t /f t, Ϝ t, θ) = k 2 β tα (β t + α) 2 (β t + α + 1) s t = ln y 2t [ln k + ψ 1 (β t) ψ 2 (β t + α)] β 1 2d t [ψ 2 (β t) ψ 2 (β t + α)] 1 d, d = 0, 1/2 or 1 (19) r f t+1 = w + f 1,t+1 + f 2,t+1 + f 3,t+1 + φ k g(x t k+1 ) (20) k=1 Components f 1,t, f 2,t and f 3,t are the same as in the gamma model. Gilson Matos Cristiano Fernandes GAS Workshop 19 / 53
![α)] β 1 2d t [ψ 2 (β t) ψ 2 (β t + α)] 1 d, d = 0, 1/2 or 1 (19) r f t+1 = w + f 1,t+1 + f 2,t+1 + f 3,t+1 + φ k g(x t k+1 )](/docs-images/47/20607389/images/page_19.jpg "(20) k=1 Components f 1,t, f 2,t and f 3,t are the same as in the gamma model.")
20 Outline Introduction 1 Introduction Gilson Matos Cristiano Fernandes GAS Workshop 20 / 53
21 Bivariate Gamma-Beta Distribution We considered a modied version of a bivariate distribution proposed by Nadarajah (2009). Let u and v be gamma variables with: u Gamma(α, λ/α), v Gamma(β, λ/α). (21) Now, dene y 1 and y 2 as follows: y 1 = u, v y 2 = k u + v (22) Thus, y 1 and y 2 have gamma and beta distributions, respectively. The bivariate Gamma Beta(α, β, λ) is easily obtained through a change of variables. p(y 1, y 2 ) = ky α+β 1 1 y β 1 2 (k y 2 ) (β+1) exp{ kαy 1 /[(k y 1 )λ]} (λ/α) α+β Γ(α)Γ(β) (23) Gilson Matos Cristiano Fernandes GAS Workshop 21 / 53
22 Bivariate Gamma-Beta Distribution Moments: E(y 1 ) = λ, V (y 1 ) = λ 2 /α (24) β E(y 2 ) = k α + β, V (y 2) = k 2 αβ (α + β) 2 (α + β + 1) (25) ( Corr(y 1, y 2 ) = 1 + α + 1 ) 1 2 (26) β Gilson Matos Cristiano Fernandes GAS Workshop 22 / 53
23 Bivariate Gamma-Beta GAS Model ((y 1t, y 2t )/f t, Ϝ t, θ) Gamma Beta(α, β t, λ t ) (27) s t = I d 1 t/t 1 t, d = 0, 1/2 or 1 (28) f t+1 = w + p A i s t i+1 + i=1 q B j f t j+1 (29) j=1 Gilson Matos Cristiano Fernandes GAS Workshop 23 / 53
24 Bivariate Gamma-Beta GAS Model The chosen parameterization is given by: r 1 r 2 (β t, λ t ) = exp f 1,t + φ 1k g 1 (x t k+1 ), exp f 2,t + φ 2k g 2 (x t k+1 ) k=1 k=1 (30) Score vector and the Fisher Information matrix to f t = (f 1t, f 2t ) : [ βt {ln [y t = 1t y 2t /(k y 2t )] ln λ t + ln α ψ 1 (β t)} kαy 1t /[(k y 2t )λ t] (α + β t) ] (31) [ β 2 I t/t 1 = t ψ 2 (β t) β t β t (α + β t) ] (32) Gilson Matos Cristiano Fernandes GAS Workshop 24 / 53
25 Bivariate Gamma-Beta GAS Model Since the conditional moments of each variable are the same of the univariate models, the dependence structure will come from eq.(29). To illustrate, a GAS(1,1) model without exogenous variables has updating equation given by: ( ) ( ) ( ) ( ) ( ) ( ) f1,t+1 w1 a11 a = + 12 s1,t b11 b + 12 f1,t f 2,t+1 w 2 a 21 a 22 s 2,t b 21 b 22 f 2,t (33) Gilson Matos Cristiano Fernandes GAS Workshop 25 / 53
26 Outline Introduction 1 Introduction Gilson Matos Cristiano Fernandes GAS Workshop 26 / 53
27 Motivation: Seasonal stabilization of energy supply in Brazil by exploiting seasonal complementarity between wind and hydro regimes. Univariate and Bivariate GAS models. Generation of integrated scenarios including consideration of the dispatch from wind farms in energy planning. Gilson Matos Cristiano Fernandes GAS Workshop 27 / 53
28 Data Monthly averages of streamow and wind power capacity factor from January 1976 to July 2009 (403 observations). Last 24 observations were separated to evaluate out-of-sample prediction accuracy. Covariates: Natural Auent Energy (ENA) series of the four power grid subsystems of Brazil (lag r = 2). Gilson Matos Cristiano Fernandes GAS Workshop 28 / 53
29 Figure: Streamow and wind power factor series. Figure: Scatter plot of the series and regression line. Gilson Matos Cristiano Fernandes GAS Workshop 29 / 53
30 Model selection: AIC/BIC. Methodology Diagnostics: quantile residuals - Ljung-Box (LB) and Jarque-Bera (JB). Goodness of t: RMSE, MAE and ( ) n n 1 MASE = ŷ t=1 t/t 1 y t n n y t=2 t y t 1 smape = 1 n n t=1 ŷ t/t 1 y t (ŷ t/t 1 + y t ) pseudo R 2 = [ corr(ŷ t/t 1, y t ) ] 2 (34) (35) (36) Gilson Matos Cristiano Fernandes GAS Workshop 30 / 53
31 Forecasting k-step ahead forecasts, as given by E (y t+k /f t, Ϝ t, θ), obtained by Monte Carlo simulation. Algorithm: For k 1, independently generate m paths, each one with extension of K, according to the steps below: 1 Update f t+k from {Ϝ t+k 1 } and f t+k 1 ; 2 Simulate y t+k from the predictive distribution, using ˆθ and the information sets {Ϝ t+k } and f t+k ; 3 Return to step (a), until k = K; For each horizon k = 1, 2,..., K mean, condence interval and quantiles can be obtained. Gilson Matos Cristiano Fernandes GAS Workshop 31 / 53
32 Diagnostics using Quantile Residuals Diagnostics using quantile residuals (Kalliovirta, 2009), dened by: R t,θ = Φ 1 (F (y t/f t, Ϝ t, θ)) (37) where Φ 1 (.) is the inverse normal CDF and F (y t/f t, Ϝ t, θ) is the conditional CDF. Observed quantile residuals r t, ˆθ obtained by substituting θ for ˆθ. In the multivariate case, the theoretical quantile residuals are dened from the marginal distributions, ie: R t,θ = [ R t1,θ, R t2,θ,..., R tk,θ ] (38) Gilson Matos Cristiano Fernandes GAS Workshop 32 / 53
33 Gamma GAS Models Introduction Univariate Gamma GAS Models Table: Model selection for univariate Gamma GAS models. Statistic SARIMA UCM Log-Lik ,0-1442,0 AIC 2928,1 2922,0 BIC 3025,7 2996,2 Gilson Matos Cristiano Fernandes GAS Workshop 33 / 53
34 Table: Goodness of t for univariate Gamma GAS models. Period Measures SARIMA UCM In-sample Out-of-sample (1) RMSE 16,7 16,5 MAE 11,6 11,5 MASE 0,60 0,60 smape 7,8 7,7 pseudo R 2 0,76 0,77 RMSE 12,9 16,3 MAE 10,2 12,8 MASE 0,53 0,66 smape 7,9 9,9 pseudo R 2 0,89 0,75 Note: (1) Predictions k-step ahead obtained by simulation. Gilson Matos Cristiano Fernandes GAS Workshop 34 / 53
35 Figure: 1-step ahead for streamow series (SARIMA evolution). Gilson Matos Cristiano Fernandes GAS Workshop 35 / 53
36 Introduction Figure: Random walk, autoregressive and seasonal components for the Gamma GAS model. Gilson Matos Cristiano Fernandes GAS Workshop 36 / 53
37 Table: Diagnostics (p-value) for the Gamma GAS models. Test SARIMA UCM Autocorrelation (LB) 0,70 0,71 Heteroscedasticity (LB) 0,01 0,04 Normality (JB) 0,12 0,47 Gilson Matos Cristiano Fernandes GAS Workshop 37 / 53
38 Beta GAS Models Introduction Univariate Beta GAS Models Table: Model selection for univariate Beta GAS models. Statistic SARIMA UCM Log-lik ,6-1050,8 AIC 2191,1 2129,6 BIC 2273,1 2184,3 Gilson Matos Cristiano Fernandes GAS Workshop 38 / 53
39 Table: Goodness of t for univariate Beta GAS models. Period Measures SARIMA UCM In-sample Out-of-sample (1) RMSE 5,2 4,9 MAE 4,0 3,6 MASE 0,58 0,53 smape 6,8 6,3 pseudo R 2 0,86 0,87 RMSE 7,7 7,9 MAE 5,8 6,2 MASE 0,85 0,90 smape 12,7 13,2 pseudo R 2 0,90 0,90 Note: (1) Predictions k-step ahead obtained by simulation. Gilson Matos Cristiano Fernandes GAS Workshop 39 / 53
40 Figure: 1-step ahead for wind power factor series (UC evolution). Gilson Matos Cristiano Fernandes GAS Workshop 40 / 53
41 Introduction Figure: Random walk, autoregressive and seasonal components for the Beta GAS model. Gilson Matos Cristiano Fernandes GAS Workshop 41 / 53
42 Table: Diagnostics (p-value) for the Beta GAS models. Test SARIMA UCM Autocorrelation (LB) 0,54 0,52 Heteroscedasticity (LB) 0,00 0,00 Normality (JB) 0,17 0,01 Gilson Matos Cristiano Fernandes GAS Workshop 42 / 53
43 Gamma-Beta GAS Model Bivariate Gamma-Beta GAS Model Consider the evolution equation of f t: p q f t+1 = w + A i s t i+1 + B j f t j+1 i=1 j=1 We tested some structures (AR; diagonal matrices A i and B j - SUTSE; diagonal matrices A i and B j with only B 1 and B 12 full); Best structure (forecasting/ residual autocorrelation): diagonal matrices and B 1 and B 12 full. The chosen structure allows direct communication between the conditional means, capturing short-term and seasonal dependence. Gilson Matos Cristiano Fernandes GAS Workshop 43 / 53
44 Table: Model selection by type of model. Model Measures Log-Ver. AIC BIC Bivariate -2616,1 5326,2 5509,8 Univariate -2513,6 5119,2 5298,8 By univariate model, in this context, we mean the bivariate model assuming independence between streamow and wind processes. In this case, the joint likelihood is obtained multiplying the likelihoods of the univariate models, ie: L(y 1, y 2 ; θ, f 0 ) = L(y 1 ; θ 1, f 10 )L(y 1, y 2 ; θ 2, f 20 ) Gilson Matos Cristiano Fernandes GAS Workshop 44 / 53
45 Table: Goodness of t for bivariate Gamma-Beta GAS models. Period Measures Streamow Wind In-sample Out-of-sample 1 RMSE 18,2 5,1 MAE 12,8 3,9 MASE 0,67 0,57 smape 8,9 6,6 pseudo R 2 0,72 0,86 RMSE 12,9 9,6 MAE 10,9 7,4 MASE 0,57 1,08 smape 8,9 14,9 pseudo R 2 0,85 0,89 Note: (1) Predictions k-step ahead obtained by simulation. Gilson Matos Cristiano Fernandes GAS Workshop 45 / 53
46 Table: Diagnostics (p-value) for the bivariate Gamma-Beta GAS model. Test Streamow UCM Autocorrelation (LB) 0,11 0,19 Heteroscedasticity (LB) 0,24 0,00 Normality (JB) 0,10 0,66 Gilson Matos Cristiano Fernandes GAS Workshop 46 / 53
47 Introduction Figure: 1-step ahead for stream ow and wind power series. Gilson Matos Cristiano Fernandes GAS Workshop 47 / 53
48 Figure: Conditional correlation. Gilson Matos Cristiano Fernandes GAS Workshop 48 / 53
49 Introduction Figure: Stream ow - simulated paths and k -step ahead predictions. Gilson Matos Cristiano Fernandes GAS Workshop 49 / 53
50 Introduction Figure: Power wind factor series - simulated paths and k -step ahead predictions. Gilson Matos Cristiano Fernandes GAS Workshop 50 / 53
51 Outline Introduction 1 Introduction Gilson Matos Cristiano Fernandes GAS Workshop 51 / 53
52 We specify and estimate a bivariate Gamma-Beta GAS model based on Nadjahara's distribution. The bivariate Gamma-Beta GAS model integrated the univariate models (parameterizations and marginal distributions), besides permitting the joint generation of streamow and wind power factor scenarios. Our model compares favorably to a VARX model tted to the same bivariate series (Amaral, 2011), with the additional feature that no data transformation is necessary. Forecasting obtained by simulation and diagnostics based on quantile residuals. Recursion initialization (f t) and maximum likelihood initialization need further thinking. Gilson Matos Cristiano Fernandes GAS Workshop 52 / 53
53 Bibliography Introduction AMARAL, B. M. Modelos VARX para geração de cenários de vento e vazão aplicados à comercialização de energia f. Dissertação (Mestrado em Engenharia Elétrica) - PUC-Rio, Rio de Janeiro CREAL, D., KOOPMAN, S. J. and LUCAS, A. Generalized Autoregressive Score Models with s. Journal of Applied Econometrics, v. 28, n. 5, p , KALLIOVIRTA, L. Diagnostic Tests Based on Quantile Residuals for Nonlinear Time Series Models f. Thesis (Doctorate in Economics) - University of Helsinki, Helsinki ORD, J. K., KOEHLER, A. B. and SNYDER, R. D. Estimation and prediction for a class of dynamic nonlinear statistical models. Journal of the American Statistical Association, v. 92, n. 440, p , NADARAJAH, S. A bivariate distribution with gamma and beta marginals with application to drought data. Journal of Applied Statistics, v. 36, n. 3, p , Gilson Matos Cristiano Fernandes GAS Workshop 53 / 53
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