Volume 5/Number 3, Fall 2012

Forecasting transmission congestion Anders Løland Norwegian Computing Center, PO Box 114 Blindern, 0314 Oslo, Norway; email: [email protected]

Egil Ferkingstad Norwegian Computing Center, PO Box 114 Blindern, 0314 Oslo, Norway; email: [email protected]

Mathilde Wilhelmsen Norwegian Computing Center, PO Box 114 Blindern, 0314 Oslo, Norway; email: [email protected]

In electricity markets, transmission congestion creates area price differences. Therefore, good forecast models for congestion are valuable in themselves, and may also improve price predictions. We study a combined measure of the net capacity utilization for one single price area, and provide hourly day-ahead forecasts for the NO1 area of Nord Pool using several different prediction models. These forecasts are combined, using average forecasts and two different types of weighted averages of forecasts. We compare the performance of the different models. In most cases, the combined transmission congestion forecasts outperform the other methods for data from 2003 to 2009.

1 INTRODUCTION Due to capacity constraints, the Nordic electricity spot power market, Nord Pool, is divided into several price areas (Benth et al (2008); Weron (2006); and Kristiansen (2004)). Area prices differ when the capacity is insufficient. Congestion occurs when the available electricity cannot be delivered to all loads due to transmission limitations. Such bottlenecks are of interest both for transmission system operators (TSOs) and for players in the markets, since capacity forecasts can improve spot price predictions. The Norwegian TSO, Statnett, defines the price areas. If there is limited transmission capacity, Nord Pool’s implicit capacity auction on the interconnectors between the bidding areas may lead to price differences. We thank Statnett and Norsk Hydro for supplying the data, and Rønnaug Sægrov Mysterud and Stefan Erath in particular for useful discussions. This work was carried out as part of (sfi)2 Statistics for Innovation. 65

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Transmission congestion is often studied for pairs of neighboring markets, ie, price differences between neighboring areas A and B, A and C, B and C, etc, are investigated (Haldrup et al (2010)). However, we study a combined measure of the net (Elspot) capacity utilization for one single area (Løland and Dimakos (2010)). We focus on forecasting this net capacity utilization for the NO1 (Oslo) price area, which, for most of the data period (January 2003 to March 2009) covered southern Norway, the price area in Norway with the highest production. Our suggested approach should be applicable to any Nord Pool price area. Nord Pool system prices have been studied quite extensively (Benth et al (2008); Erlwein et al (2010); and Botterud et al (2010)). There has been less focus on Nord Pool area prices, with some noteworthy exceptions (Kristiansen (2004); Løland and Dimakos (2010); Marckhoff and Wimschulte (2009); Haldrup et al (2010); Haldrup and Nielsen (2006); and Fridolfsson and Tangerås (2009)). Attempts to forecast transmission congestion are, to the best of our knowledge, scarce, with the possible exception of some related approaches for North American electricity markets (Min et al (2008); Li et al (2006); and Zhang et al (2009)). Løland and Dimakos (2010) use daily prices, reservoir levels and transmission congestion data to model the daily NO1 price. There is, however, a substantial literature on the theoretical aspects of zonal pricing and transmission congestion (Oren (1997); Bjørndal and Jørnsten (2001); and Hamoud and Bradley (2004)). Also, there have been studies of the implications of congestion on cost and voltage profile management (Singh et al (1998); Yamin and Shahidehpour (2003); and Hadsell and Shawky (2006)). In this paper we study the performance of six separate forecast models (Weron (2006) and Weron and Misiorek (2008)) applied to hourly data. We also combine these forecast models using two standard methods (Clemen and Winkler (1986) and Timmermann (2006)) as well as one new approach. We proceed as follows. In Section 2 we present historical prices and explanatory variables, and the net capacity utilization is defined. In Section 3 we define the problem and explain the basic models, estimation and model combination. In Section 4 our results are presented, while we conclude and discuss the method in Section 5.

2 DATA Nord Pool is divided into several price areas. Due to Statnett’s price area policy, there have been several changes to the price areas over time, both in size and number, and this complicates the modeling. NO1 is the area that has seen the fewest changes, which is the primary reason for choosing this area for our study. With insufficient capacity, prices in neighboring areas will be different. In theory, prices will only be different when all the import or export capacity is used. However, The Journal of Energy Markets

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due to so-called ramping restrictions,1 the real capacity might be less than the quoted Elspot capacity. Hence, price differences might occur even without the Elspot capacity being fully exploited. We study hourly data from January 1, 2001 to March 11, 2009, ie, 71 808 hours of data.

2.1 Net capacity utilization For each of the connections with a neighboring price area, we have hourly values for flow and capacity. These are the Elspot quantities. In the Elspot market, hourly power contracts are traded daily for physical delivery in the next day’s twenty-fourhour period. The Elspot flow, in particular, might therefore deviate from the intraday physical flow. in in out and ch;a be the flow and capacity from area a into NO1, and let fh;a Let fh;a out and ch;a be the flow and capacity from NO1 into area a. We define the net capacity utilization nh;a between NO1 and area a in hour h as in Løland and Dimakos (2010): in out fh;a fh;a nh;a D ka max in ; out (2.1) ch;a ch;a Here, ka D 1 (respectively, ka D 1) if the maximum is obtained by export (respectively, import). Furthermore, we define weights: 0 in out D ch;a C ch;a wh;a

and these are normalized: wh;a

0 wh;a DP 0 a wh;a

The hourly net capacity utilization is then defined as: X nh;a wh;a nh D a

Thus, nh defines to what degree a price area is a net importer or exporter. If there is maximum export, then nh D 1, and if there is maximum import, then nh D 1. If exports and imports both occur in hour h, the direction that is most strained defines nh;a . Other definitions are also possible. Alternatively, we could have considered nh;a of (2.1), and forecasted nh;a for every neighboring price area a. Figure 1 on the next page displays nh for three example weeks. Example week 33 in 2004 shows the typical pattern: nh is negative during the night (import) and positive during daytime (export). 1

See the Nord Pool Spot website. URL: www.nordpoolspot.com.

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1.0

0.5

nh

Week 7, 2003 Week 33, 2004 Week 23, 2007

0

−0.5

−1.0 Mon

Tue

Wed

Thu

Fri

Sat

Sun

Figure 2 on the facing page shows nh for hours 1, 8 and 16 for the whole prediction period, where we have used moving averages with a window length of thirty days. We see the tendency to import during the night and to export during the day. During 2007 and 2008, there were long periods with almost continuous export. Note that nh is bounded by 1 and 1, will sooner or later return to zero, and is stationary in that sense. Some seasonality is expected of nh , for example, a higher probability of export (nh D 1) during the late spring, due to large amounts of inflow of melting water. However, this seasonality is not systematic enough to improve the in out and ch;a are typically almost constant day-ahead predictions of nh . The capacities ch;a in long periods, except when there are problems with a major connection. The spot price is typically higher during the winter than it is in the summer, since nearly all electricity is generated by hydro power, and it is often considered to be stationary (Ferkingstad et al (2011)).

2.2 Data used for prediction When we predict the net capacity utilization for hour h on a given day, we condition on the following explanatory variables: yesterday’s electricity spot price, yesterday’s total capacity into and out of NO1, the semi-naive predictor of the net capacity The Journal of Energy Markets

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Forecasting transmission congestion FIGURE 2 The net capacity utilization for hours 1, 8 and 16 for the whole prediction period (January 1, 2003 to March 11, 2009) as moving averages with a window length of thirty days.

1.0

Hour 1 Hour 8 Hour 16

0.5

nh

0

−0.5

−1.0 2003

2004

2005

2006

2007

2008

2009

Each point is plotted in the middle of the moving average period.

utilization (see Section 3.3.1), and dummy variables for Monday, Saturday and SunP in day. The total capacity into NO1 is given as a ch;a , whereas the total capacity out P out of NO1 is given as a ch;a . In addition to these explanatory variables, we consider others that are important in the Nord Pool system as well. These include water reservoir levels, inflow to water reservoirs, electricity consumption, temperature and wind speed. However, since we are predicting the day-ahead nh , many of these variables are not available at the time of forecasting. Day-ahead weather forecasts could potentially be used for congestion prediction. For example, the difference between the observed temperature today and a temperature forecast tomorrow could be used as an explanatory variable, basically indicating whether tomorrow’s weather is likely to be “unusual”. Unfortunately, we have not been able to obtain day-ahead temperature and wind speed forecasts for our study period (2003–9). In addition, conditioning on tomorrow’s consumption (assuming that this was known) did not seem to improve our predictions. Consumption was therefore left out as an explanatory variable. Finally, changes in water reservoir levels or inflow from one day to the next would probably have little impact on the transmission congestion the next day. Research Paper

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3 METHODS We have used several forecast models to predict the net capacity utilization. Each hour is modeled and predicted separately, so we have twenty-four different prediction models. Two naive predictors are used: yesterday’s value (naive 1) and last week’s value (naive 7) of the net capacity utilization. The other prediction models we have used are discussed briefly in Section 3.3.

3.1 Transformations The capacity utilization is bounded: nh 2 Œ1; 1. The standard practice when confronted with bounded data is to transform the data to R, hope that the implicit or explicit assumption of normality can be justified (Box et al (1978)), do the modeling there, and then back-transform to the original scale. We have considered two approaches. (1) Using the shifted logit transform: p D 12 .nh C 1/ p z D log 1p model z, and back-transform to nh . (2) Model nh directly, and round off the predicted value if it is outside the boundary Œ1; 1. It turned out that the second approach clearly gave better results than the first approach. Therefore, we decided to model the untransformed nh directly.

3.2 Inclusion of covariates For each of the prediction methods described in Sections 3.3.2–3.3.4, covariates are included in the analysis in the following way. First, a simple linear regression model is fitted, with capacity utilization as the dependent variable, and the covariates listed in Section 2.2 as independent variables. Next, the prediction model is fitted using the residuals from the estimated linear model. Finally, forecasts from the prediction model on the residuals are combined with the estimated linear model to produce forecasts that incorporate the covariates. This is a standard approach for linear models (Cowpertwait and Metcalfe (2009)). The Journal of Energy Markets

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3.3 The separate forecast models 3.3.1 Semi-naive predictor: naive 137 Let X t be the observed value of the net capacity utilization for a certain hour of the day t , and let X td be the net capacity utilization for the same hour d days earlier. We define the semi-naive predictor X t137 to be: 8 ˆ ˆ

3.3.2 Exponential smoothing: expsm Let X be a time series without any systematic trend or seasonal component. Assume that: (3.1) Xt D t C wt where t is the mean at time t and w t is white noise with standard deviation . Simple exponential smoothing estimates the level t by: a t D ˛X t C .1 ˛/a t 1

(3.2)

where ˛ is a smoothing parameter and we set a1 D X1 . The current estimated mean is then a weighted average of the current observed value and the previously estimated mean. At time t , the predicted value at time t C k is simply a t for any lead time k. The model in (3.2) can easily be generalized to include a trend and a seasonal component. We have used the “ets” function in the R (R Development Core Team (2011)) package “forecast” (Hyndman and Khandakar (2008)) for our calculations. In this package, fifteen different exponential smoothing methods are defined, according to whether a trend and/or seasonal component is fitted, whether the trend is damped (see below) and whether the trend/seasonal error component is additive or multiplicative. All of these are fitted, and the optimal method according to the Akaike information criterion is chosen. For our data, models with seasonal components and multiplicative errors were never selected, so, for simplicity, we omit the description of these. The linear exponential smoothing model with damped trend is defined by the following recursive equations: a t D ˛X t C .1 ˛/.a t 1 b t 1 / b t D ˇ.a t a t 1 / C .1 ˇ/b t 1 Here, a t is again the estimated level, b t is the estimated trend, and is the damping coefficient. For 0 < < 1, the trend is damped, meaning that the forecasts approach Research Paper

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the horizontal straight line a t C b t .1 /. For D 1 we have a simple linear trend. The predicted value at t for time t C k is a t C k b t , where k D C 2 C C k (note that k D k for D 1).

3.3.3 Autoregressive integrated moving average: ARIMA 200, ARIMA 210, ARIMA The autoregressive integrated moving average (ARIMA) model is generally called an ARIMA.p; d; q/ model, where p, d and q are integers greater than or equal to zero, and refer to the order of the autoregressive, integrated and moving average parts of the model, respectively. Let X t be the time series of data. Then: 1

p X

q X i Li .1 L/d X t D 1 C i Li t

iD1

iD1

Here, L is the lag operator, and i and i are parameters. We calculated the Akaike information criterion for all possible combinations of p 6 2, d 6 2 and q 6 2. The two best choices using this approach were .2; 0; 0/ and .2; 1; 0/. Note that the former is a stationary model, while the latter is nonstationary. We also used an autoregressive model of order one (ARIMA.1; 0; 0/) without covariates.

3.3.4 Threshold autoregression The piecewise linear threshold autoregression (TAR) model suggested by Tong and Lim (1980) provides a useful way of specifying a nonlinear time series prediction model. In the two-regime TAR model, there is a regime switching between two autoregressive processes: ( 1;0 C 1;1 X t1 C C 1;p1 X t p1 C e t if Y t 6 r Xt D 2;0 C 2;1 X t1 C C 2;p2 X t p2 C e t if Y t > r for some threshold r and observable process Y t . We have chosen Y t D X t 1 , giving the self-exciting TAR model with delay 1. The threshold parameter r can be estimated using a profile likelihood method (see Cryer and Chan (2008, Section 15.7) for details). For estimation, we have used the function “tar” in the “TSA” R package (Cryer and Chan (2008)).

3.4 Methods for combining forecasts Let X t be the observed value of the time series X at time t , and let XO t.1/ ; XO t.2/ ; : : : ; XO t.n/ be predictions from n different models of X at time t . The n predictions can be The Journal of Energy Markets

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combined into one prediction, XO t , by taking a weighted average: XO t D

n X

wi XO t.i /

iD1

In this paper, we use three different methods to determine the weights. For more information on combining methods see, for example, Clemen and Winkler (1986), Timmermann (2006) and Palm and Zellner (1992).

3.4.1 Average: CB1 The simplest combination method is to take the average of the n predictions: 1 ; i D 1; : : : ; n n Previous studies have shown that this method works surprisingly well and is quite robust (Timmermann (2006)). wi D

3.4.2 Weighted average with respect to estimated prediction error variances: CB2 A more sophisticated combining method is to let the weights be given as: w D .w1 ; : : : ; wn / D

uT ˙O 1 uT ˙O 1 u

where u D .1; 1; : : : ; 1/T and ˙O is the estimated covariance matrix of the forecast errors e D .XO .1/ X; : : : ; XO .n/ X /. This puts lower weights on models with highly variable prediction errors. The covariance matrix at time t is estimated based on the d previous forecast errors. We assume zero correlation between the forecast errors. This is a common assumption (Timmermann (2006)), since the number of parameters will be large compared with the covariance estimation period d . Also, studies have shown that the prediction results rarely improve when correlations are estimated. We tried different values for d , and d D 56 days (8 weeks) gave best prediction results in terms of mean absolute error (MAE) (4.1).

3.4.3 Weighted average with respect to previous performance: CB3 Instead of weighting with respect to previous forecast error variances, we can weight with respect to previous performance of the prediction models. Here, we suggest letting the MAE of the prediction models decide the weights. For prediction model i , i D 1; : : : ; n, the weight is given as: wi0 D

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The first term in the denominator is the MAE of predictions the previous d days. The second term, , is a small number equal to 106 , which is added to the MAE to avoid P zeros in the denominator. The weights are then normalized, wi D wi0 = niD1 wi0 . Here we also tested different values for d , and d D 7 days gave the best prediction results in terms of MAE. This combination is inspired by a proposal by Bates and Granger (1969), who used the mean squared error (MSE) instead of MAE to define the weights. We suggest MAE, which puts less emphasis on large deviations.

4 RESULTS For each hour, nh was modeled and predicted separately. We used a fitting period of 365 days, and predicted one day ahead. In this paper we only present the prediction results from January 1, 2003 onward. For evaluating the out-of-sample performance, we focus particularly on two measures: root mean squared error (RMSE) and mean absolute error (MAE). Both RMSE and MAE are commonly used error measures. Both are on the same scale as the original measurements (this is the reason for using RMSE instead of simply MSE), easing interpretation. If the prediction is unbiased, then the RMSE is identical to the standard error. The RMSE can be sensitive to outliers, while MAE has the effect of downweighing the most extreme results. Since nh is often close to or exactly zero, we cannot use relative error measures such as the mean absolute percentage error. Let X t be the observed value of nh for a certain hour of the day t , t D 1; : : : ; T , and let XO t.i/ be the prediction for that hour and day using model i . The RMSE for model i is then: v u T u1 X t .XO .i / X t /2 RMSE D T t D1 t Similarly, the MAE for model i is: MAE D

T 1 X O .i / jX X t j T t D1 t

(4.1)

Table 1 on the facing page summarizes the performance of the prediction models. The first four rows show the mean RMSE over all hours and for different periods. The next four rows show the mean MAE over all hours and for the same periods. The next five rows show the MAE for five example hours for the whole prediction period, while the last five rows show the same for the last prediction period. The last prediction period is of particular interest, since there are more occurrences of extreme values during these last few years. For each case (row), the best models are in bold, The Journal of Energy Markets

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TABLE 1 Summary of the out-of-sample performance of the prediction models.

Hour

Measure

Naive 1

Naive 7

Naive 137

Expsm

TAR

ARIMA 200

ARIMA 210

ARIMA

CB1

CB2

CB3

2003–9 2003–4 2005–6 2007–9

1–24 1–24 1–24 1–24

RMSE RMSE RMSE RMSE

0.41 0.37 0.44 0.40

0.41 0.36 0.44 0.42

0.34 0.28 0.37 0.35

0.31 0.26 0.34 0.32

0.32 0.27 0.34 0.33

0.30 0.26 0.32 0.30

0.30 0.27 0.31 0.31

0.38 0.34 0.41 0.38

0.30 0.26 0.32 0.30

0.29 0.25 0.31 0.30

0.29 0.25 0.31 0.30

2003–9 2003–4 2005–6 2007–9

1–24 1–24 1–24 1–24

MAE MAE MAE MAE

0.26 0.25 0.30 0.23

0.28 0.27 0.32 0.25

0.22 0.20 0.26 0.20

0.21 0.19 0.24 0.20

0.22 0.20 0.25 0.21

0.21 0.19 0.24 0.20

0.21 0.20 0.23 0.20

0.28 0.26 0.32 0.26

0.20 0.19 0.23 0.20

0.20 0.18 0.23 0.19

0.20 0.18 0.23 0.19

2003–9 2003–9 2003–9 2003–9 2003–9

1 8 12 16 20

MAE MAE MAE MAE MAE

0.17 0.38 0.31 0.33 0.22

0.27 0.29 0.28 0.29 0.26

0.21 0.22 0.21 0.24 0.22

0.20 0.22 0.21 0.23 0.21

0.20 0.23 0.21 0.24 0.21

0.18 0.23 0.21 0.24 0.21

0.17 0.24 0.22 0.24 0.20

0.18 0.42 0.33 0.35 0.24

0.18 0.23 0.21 0.23 0.20

0.17 0.22 0.20 0.23 0.20

0.17 0.22 0.20 0.23 0.20

2007–9 2007–9 2007–9 2007–9 2007–9

1 8 12 16 20

MAE MAE MAE MAE MAE

0.21 0.35 0.21 0.27 0.14

0.30 0.23 0.21 0.25 0.20

0.24 0.18 0.17 0.20 0.16

0.22 0.20 0.17 0.21 0.16

0.24 0.21 0.17 0.21 0.16

0.22 0.21 0.18 0.22 0.15

0.21 0.22 0.18 0.21 0.15

0.23 0.40 0.25 0.31 0.17

0.22 0.21 0.17 0.21 0.15

0.21 0.21 0.17 0.21 0.14

0.21 0.20 0.16 0.20 0.14

The following abbreviations are used: naive 1 (yesterday’s value), naive 7 (last week’s value), naive 137 (semi-naive), expsm (exponential smoothing), TAR (piecewise linear threshold autoregression), ARIMA 200 (ARIMA(2,0,0)), ARIMA 210 (ARIMA(2,1,0)), ARIMA (naive ARIMA(1,0,0)), CB1 (combination model 1; simple average), CB2 (combination model 2; weighted sum with respect to estimated prediction error variances), CB3 (combination model 3, weighted sum with respect to previous performance).

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FIGURE 3 The out-of-sample performance (RMSE) of the prediction models in each hour, together with summarizing values of the absolute value of the net capacity utilization. [Figure continues on next page.]

(b)

(a)

nh

Naive 1

ARIMA

ARIMA 210

CB2

Naive 7

Expsm

TAR

CB3

Naive 137

ARIMA 200

CB1

0.6

0.6

0.5

0.5

RMSE

RMSE

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0.4

0.4

0.3

0.3

0.2

0.2

1.0

1.0

0.8

0.8

0.6

nh

0.6

0.4

0.4

0.2

0.2 0

0 5

10 15 Hours

20

5

Mean |nh|

20q |nh|

Median |nh|

80q |nh|

10 15 Hours

20

Mean export

(a) Whole period. (b) 2003–4.

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FIGURE 3 Continued.

(d)

nh

Naive 1

ARIMA

ARIMA 210

CB2

Naive 7

Expsm

TAR

CB3

Naive 137

ARIMA 200

CB1

0.6

0.6

0.5

0.5

RMSE

RMSE

(c)

0.4

0.4

0.3

0.3

0.2

0.2

1.0

1.0

0.8

0.8

0.6

nh

0.4 0.2

0.6 0.4 0.2

0

0 5

10 15 Hours

20

5

Mean |nh|

20q |nh|

Median |nh|

80q |nh|

10 15 Hours

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Mean export

(c) 2005–6. (d) 2007–9.

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while the worst models are in italics. The three model combinations outperform the other models in most cases. Furthermore, the naive autoregressive model of order one, together with the two naive predictors (yesterday’s and last week’s value), perform the worst in all cases. The RMSE for all prediction models, for all hours and for different periods, is presented in Figure 3 on page 76. For each period, a summary of the absolute value of the net capacity utilization is shown. The performances of the two combination models that use weighted sums (CB2 and CB3 ) are practically identical. Hence, there must be a high correlation between the variances and the absolute levels of the forecast errors. These two combination models perform better than all other models, except for the ARIMA.2; 1; 0/ model for hours 1–5 and hours 23–24. The combination model that uses a simple average (CB1 ) performs slightly worse than the two other combination models. This is what we would expect, since the other two combination models use more sophisticated combining methods. The two ARIMA models also give good prediction results.

4.1 Classification It is of particular interest to see how the prediction models perform when nh is very low or very high. To achieve this, we use a classification approach where we divide into “export”, “import” or “between” days. We use the concepts of recall and precision (Berry and Linoff (1997)). Recall is defined as the percentage of days that are predicted to belong in class X among days truly belonging to class X. Thus, recall for “export” days is the percentage of true “export” days that are predicted to be in the “export” class. Conversely, precision is defined as the percentage of days that truly belong to class X among the days that are predicted to be in class X. Thus, precision for “export” days is the percentage of predicted “export” days that truly are “export” days. Informally, “export” recall measures the ability to detect “export” days, while “export” precision measures the ability to trust in an “export” prediction. Table 2 on the facing page, Table 3 on the facing page and Table 4 on page 80 show the classification results for the prediction period 2003–9 for three of the prediction models: yesterday’s value as naive predictor (naive 1), the semi-naive predictor (naive 137) and the combination model where we take the weighted sum with respect to previous performance (CB3 ). We have used a cutoff of 0.8, meaning that nh above 0.8 corresponds to export, and nh below 0.8 corresponds to import. The first data column in these tables shows how many of the observed high values of nh are predicted as export (above 0.8), import (below 0.8) or in between. The second and third data columns show the same for the observed values. The bottom row shows the recall percentages, which are given by dividing the diagonal values by the values The Journal of Energy Markets

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Forecasting transmission congestion TABLE 2 Out-of-sample performance of the naive 1 (yesterday’s value) model, using a cutoff of 0.8. ‚ Export 8 ˆ < Export Predicted Import ˆ : Between Total Recall (%)

True …„ Import

ƒ Between

Total

Precision (%)

11 130

28

3 389

14 547

77

126

4 865

2 208

7 199

68

3 302

2 306

26 934

32 542

83

14 558

7 199

32 531

54 288

79

76

68

83

79

TABLE 3 Out-of-sample performance of the naive 137 (semi-naive) model, using a cutoff of 0.8. ‚ Export 8 ˆ < Export Predicted Import ˆ : Between Total Recall (%)

True …„ Import

ƒ Between

Total

Precision (%)

11 458

2

3 083

14 543

79

13

5 181

2 005

7 199

72

3 087

2 016

27 443

32 546

84

14 558

7 199

32 531

54 288

81

79

72

84

81

in the “Total” row. Similarly, we get the precision percentages in the last column by dividing the diagonal values by the values in the “Total” column. The recall and precision percentages are better for export than import for all three models. One reason for this might be that there are fewer observations of imports than of exports in our prediction period 2003–9. Furthermore, the combination model has lower recall percentages than the two naive models, especially when it comes to import. However, our combination model has the best precision percentages. In general, the classification results are highly dependent on the cutoff. Figure 4 on the next page shows how the precision and recall vary with cutoff for the combination model. The recall percentages decrease as the cutoff increases. This is also the case for the other two naive models. The precision percentages for the combination model are less dependent on the cutoff level. When it comes to the naive models, the precision percentages are the same as the recall percentages for all cutoff levels. Hence, for all three models, there is no trade-off between precision and recall, in contrast with other studies (Buckland and Gey (1994)). Research Paper

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A. Løland et al TABLE 4 Out-of-sample performance of the CB3 (combination model 3; weighted sum with respect to previous performance) model, using a cutoff of 0.8. ‚ Export 8 ˆ < Export Predicted Import ˆ : Between Total

True …„ Import

10 501

0

ƒ Between

Total

Precision (%)

1 817

12 318

85

1

4 250

800

5 051

84

4 056

2 949

29 914

36 919

81

14 558

7 199

32 531

54 288

82

72

59

92

82

Recall (%)

FIGURE 4 The recall and precision (%) for export and import in the combination model CB3 , when the cutoff varies.

100

80

60

% 40

20

Recall, export Recall, import Precision export Precision import

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5 DISCUSSION We have built and applied models for forecasting hourly day-ahead transmission congestion in the southern Norway (NO1) price area of the Nord Pool system. Using the net capacity definition (nh ) for a price area as a whole, we avoid restricting our analysis to pairs of neighboring price areas. Our method can also be applied for other price areas, and for two neighboring price areas, if this is of interest. The fact that the The Journal of Energy Markets

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Nord Pool price area definitions change over time emphasizes the need for a flexible model. With our semi-naive predictor, the out-of-sample results are quite good, even though they vary between hours and throughout the historic period. Of the parametric models, ARIMA and exponential smoothing performed best. The results were enhanced further by adaptive model combinations. The best combination method was to take the weighted sum with respect to previous performance. For high absolute values of the net capacity utilization, our combination method is not superior to the other methods. Our approach is a first attempt at predicting transmission congestion, and there are plenty of opportunities for future work. First, we have only provided point forecasts. These should be accompanied by proper uncertainty measures. Second, we would like to predict beyond one day ahead. One way of extending the prediction horizon is to include more explanatory variables. The inclusion of electricity consumption data did not improve the results, but including more data from the neighboring price areas should help. Since price differences between NO1 and surrounding price areas are only present when the net capacity utilization is very high in absolute value, we are particularly interested in good prediction performance at the extreme values of nh . Classification methods such as support vector machines, as employed by Zareipour et al (2011) on future electricity prices, could be used to predict whether nh is above or below a certain threshold. However, applying support vector machines or linear discriminant analysis to our problem did not improve the results in terms of classification (results not shown). Finally, predicting transmission congestion is interesting in itself, but we would also like to improve spot price predictions by utilizing the transmission congestion predictions. Li et al (2007) worked along these lines by conditioning on known day-ahead transmission constraints when forecasting day-ahead prices. For the Nord Pool market, the transmission constraints are not known before the prices are set, so we must rely on transmission congestion predictions. Our preliminary investigations suggest that including congestion forecasts in simple price models does not improve price predictions. It seems clear that more sophisticated price forecast models are needed to take advantage of the congestion forecasts, which is an interesting area for future research.

REFERENCES Bates, J. M., and Granger, C. W. J. (1969). The combination of forecasts. Operational Research Quarterly 20(4), 451–468. Benth, F. E., Benth, J. S., and Koekebakker, S. (2008). Stochastic Modelling of Electricity Markets. Advanced Series on Statistical Science and Applied Probability, Volume 11. World Scientific. Berry, M. J. A., and Linoff, G. (1997). Data Mining Techniques. Wiley. Research Paper

www.risk.net/journal

81

82

A. Løland et al Bjørndal, M., and Jørnsten, K. (2001). Zonal pricing in a deregulated electricity market. Energy Journal 22(1), 51–73. Botterud, A., Kristiansen, T., and Ilic, M. D. (2010). The relationship between spot and futures prices in the Nord Pool electricity market. Energy Economics 32(5), 967–978. Box, G. E. P., Hunter, W. G., and Hunter, J. S. (1978). Statistics for Experimenters. Wiley. Buckland, M., and Gey, F. (1994). The relationship between recall and precision. Journal of the American Society for Information Science and Technology 45(1), 12–19. Clemen, R. T., and Winkler, R. L. (1986). Combining economic forecasts. Journal of Business and Economic Statistics 4(1), 39–46. Cowpertwait, P. S. P., and Metcalfe, A. V. (2009). Introductory Time Series with R. Springer. Cryer, J., and Chan, K. (2008). Time Series Analysis: With Applications in R. Springer. Erlwein, C., Benth, F. E., and Mamon, R. (2010). HMM filtering and parameter estimation of an electricity spot price model. Energy Economics 32(5), 1034–1043. Ferkingstad, E., Løland, A., and Wilhelmsen, M. (2011). Causal modeling and inference for electricity markets. Energy Economics 33(3), 404–412. Fridolfsson, S. O., and Tangerås, T. P. (2009). Market power in the Nordic electricity wholesale market: a survey of the empirical evidence. Energy Policy 37(9), 3681–3692. Hadsell, L., and Shawky, H. (2006). Electricity price volatility and the marginal cost of congestion: an empirical study of peak hours on the NYISO market, 2001–2004. Energy Journal 27(2), 157–180. Haldrup, N., and Nielsen, M. O. (2006). Directional congestion and regime switching in a long memory model for electricity prices. Studies in Nonlinear Dynamics and Econometrics 10(3), 1–22. Haldrup, N., Nielsen, F. S., and Nielsen, M. O. (2010). A vector autoregressive model for electricity prices subject to long memory and regime switching. Energy Economics 32(5), 1044–1058. Hamoud, G., and Bradley, I. (2004). Assessment of transmission congestion cost and locational marginal pricing in a competitive electricity market. IEEE Transactions on Power Systems 19(2), 769–775. Hyndman, R., and Khandakar, Y. (2008). Automatic time series forecasting: the forecast package for R. Journal of Statistical Software 27(3), 1–22. Kristiansen, T. (2004). Congestion management, transmission pricing and area price hedging in the Nordic region. International Journal of Electrical Power and Energy Systems 26(9), 685–695. Li, G., Liu, C.-C., and Salazar, H. (2006). Forecasting transmission congestion using dayahead shadow prices. In 2006 IEEE/PES Power Systems Conference and Exposition, Volumes 1–5, pp. 1705–1709. IEEE, New York. Li, G., Liu, C.-C., Mattson, C., and Lawarree, J. (2007). Day-ahead electricity price forecasting in a grid environment. IEEE Transactions on Power Systems 22(1), 266–274. Løland, A., and Dimakos, X. (2010). Modeling Nord Pool’s NO1 area price. The Journal of Energy Markets 3(1), 1–20. Marckhoff, J., and Wimschulte, J. (2009). Locational price spreads and the pricing of contracts for difference: evidence from the Nordic market. Energy Economics 31(2), 257– 268. The Journal of Energy Markets

Volume 5/Number 3, Fall 2012

Forecasting transmission congestion Min, L., Lee, S. T., Zhang, P., Rose, V., and Cole, J. (2008). Short-term probabilistic transmission congestion forecasting. In 2008 Third International Conference on Electric Utility Deregulation and Restructuring and Power Technologies, Volumes 1–6, pp. 764–770. IEEE, New York. Oren, S. (1997). Economic inefficiency of passive transmission rights in congested electricity systems with competitive generation. Energy Journal 18(1), 63–83. Palm, F. C., and Zellner, A. (1992). To combine or not to combine? Issues of combining forecasts. Journal of Forecasting 11, 687–701. R Development Core Team (2011). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna. Singh, H., Hao, S., and Papalexopoulos, A. (1998). Transmission congestion management in competitive electricity markets. IEEE Transactions on Power Systems 13(2), 672–680. Timmermann, A. G. (2006). Forecast combinations. In Handbook of Economic Forecasting, Elliott, G., Granger, C. W. J., and Timmermann, A. G. (eds), pp. 135–194. Elsevier. Tong, H., and Lim, K. (1980). Threshold autoregression, limit cycles and cyclical data. Journal of the Royal Statistical Society B 42(3), 245–292. Weron, R. (2006). Modelling and Forecasting Electricity Loads and Prices: A Statistical Approach. Wiley. Weron, R., and Misiorek, A. (2008). Forecasting spot electricity prices: a comparison of parametric and semiparametric time series models. International Journal of Forecasting 24(4), 744–763. Yamin, H., and Shahidehpour, S. (2003). Transmission congestion and voltage profile management coordination in competitive electricity markets. International Journal of Electrical Power and Energy Systems 25(10), 849–861. Zareipour, H., Janjani, A., Leung, H., Motamedi, A., and Schellenberg, A. (2011). Classification of future electricity market prices. IEEE Transactions on Power Systems 26(1), 165–173. Zhang, G., Zhang, B., Sun, H., and Wu, W. (2009). Ultra-short term probabilistic transmission congestion forecasting considering wind power integration. In Advances in Power System Control, Operation and Management (APSCOM 2009), 8th International Conference, pp. 1–6. Curran, Norwich, UK.

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