Application of probabilistic wind power forecasting in electricity markets


Correspondence: Z. Zhou, Decision and Information Sciences Division, Argonne National Laboratory, Argonne, IL, USA.



This paper discusses the potential use of probabilistic wind power forecasting in electricity markets, with focus on the scheduling and dispatch decisions of the system operator. We apply probabilistic kernel density forecasting with a quantile-copula estimator to forecast the probability density function, from which forecasting quantiles and scenarios with temporal dependency of errors are derived. We show how the probabilistic forecasts can be used to schedule energy and operating reserves to accommodate the wind power forecast uncertainty. We simulate the operation of a two-settlement electricity market with clearing of day-ahead and real-time markets for energy and operating reserves. At the day-ahead stage, a deterministic point forecast is input to the commitment and dispatch procedure. Then a probabilistic forecast is used to adjust the commitment status of fast-starting units closer to real time, on the basis of either dynamic operating reserves or stochastic unit commitment. Finally, the real-time dispatch is based on the realized availability of wind power. To evaluate the model in a large-scale real-world setting, we take the power system in Illinois as a test case and compare different scheduling strategies. The results show better performance for dynamic compared with fixed operating reserve requirements. Furthermore, although there are differences in the detailed dispatch results, dynamic operating reserves and stochastic unit commitment give similar results in terms of cost. Overall, we find that probabilistic forecasts can contribute to improve the performance of the power system, both in terms of cost and reliability. Copyright © 2012 John Wiley & Sons, Ltd.


Driven by increasing prices on fossil fuels and concerns about greenhouse gas emissions, wind power, as a renewable and clean source of energy, is rapidly being introduced into the existing energy supply portfolio throughout the world. The U.S. Department of Energy has analyzed a scenario in which wind power meets 20% of the U.S. electricity demand by 2030, which means that the US wind power capacity would have to reach more than 300 GW. [1] The European Union is pursuing a target of 20/20/20, which aims to reduce greenhouse gas emissions by 20%, increases the amount of renewable energy to 20% of the energy supply, and improve energy efficiency by 20% by 2020 as compared with 1990. [2] China claims it will have 100 GW of wind power capacity by 2020. [3]

Traditionally, power system uncertainty arises from load fluctuation and system contingencies. Additional generation capacity is reserved to hedge against the risk of this uncertainty. However, the large-scale integration of wind power into power systems gives rise to new challenges since the uncertainty from wind power adds to the overall level of uncertainty in the system. Hence, wind power forecasting (WPF) therefore is a critical task for power system operations. There are two main types of WPF methods: point forecasting and uncertainty forecasting. Point forecasting provides single-value estimation of wind power at given points in time. [4-6] The main drawback of point forecasts is that no information is provided about forecast errors magnitude. Uncertainty forecasts [7-12] can estimate the uncertainty of the wind power, in the form of probability density functions (pdfs), scenarios, quantiles, and so forth, which thereby provide more information compared with point forecasts. In terms of the time horizon, there are very short-term (intraday), short-term (next 1–2 days ahead), and long-term WPF. For power system operation, day-ahead and hours-ahead WPF are useful for day-ahead and intraday market operations. For a comprehensive overview of WPF methods, please refer to Monteiro et al. [13] and Giebel et al. [14]

Misestimating the wind power can make the power system unreliable. For example, if the wind power is underestimated, the system operator could commit more thermal units than necessary and cause wind power curtailment and increased generation costs. If the wind power is overestimated, the system will experience a power supply shortage. [15] Furthermore, misestimating the change of wind power output can also make the system unreliable since there could be insufficient ramping capacity available to account for the fluctuations.

Even if we have perfect forecasting of wind power, because the variability of wind power and constraints on system operation (such as ramping constraints on thermal units or transmission constraints), with a large wind power penetration, there may be situations where wind power cannot be dispatched at its maximum available output although it is free in terms of fuel costs. Therefore, besides WPF techniques, the inherent variability, uncertainty, and limited controllability of wind power also require industrial practitioners to rethink the existing power system modeling techniques, such as unit commitment (UC) and economic dispatch (ED), to account for large amounts of wind power generation. Alternative UC models and reserves strategies for wind power must be designed to guarantee operational reliability and minimize costs. Stochastic UC has been proposed as one approach to better handle the wind power uncertainty in market operations. In Barth et al., [16] the unit commitment problem is modeled as a stochastic linear programming model, and the wind power forecasts are presented by a scenario tree. Bouffard et al. [17] propose stochastic secure-economic short-term forward market-based scheduling of generation, load and reserves for power system with uncertainty from wind and load, in which the load and wind power forecasting errors are assumed to be normally distributed. Wang et al. [18] present a mixed integer programming unit commitment model with transmission constraints and wind penetration, in which wind power point forecasts are used at the UC stage, and wind power scenarios are used in the dispatch stage. Binlinton et al. [19] evaluate the unit commitment risk of a system with wind power. The distribution of wind power is estimated by using autoregressive moving average time series models. Makarov et al. [20] evaluate the impact of wind on the load following and regulation requirements of the California Independent System Operator system. The wind is forecasted by adding historical forecast error to expected wind power. Both Pappala et al. [21] and Li et al. [22] use an adaptive particle swarm optimization (PSO) method to solve the stochastic UC problem for a wind integrated power system, whereas in Li et al., [22] wind power scenario generation and reduction are also performed by the PSO. Tuohy et al. [23] examines the effects of stochastic wind and load on the UC and ED of power systems with high levels of wind power by using the WILMAR model. [24] Constantinescu et al. [25] present a computational framework for a stochastic UC/ED formulation with the integration of a numerical weather prediction model of wind forecasts. Wang et al. [26] propose a two-stage stochastic unit commitment model with a scenario representation of wind power uncertainty to handle the uncertainty in wind power forecasts. Moreover, new approaches to calculating operating reserve requirements considering wind power uncertainty have also been studied. [27-31]

In this paper, by extending our previous work, [13, 26] we take the Illinois power system as a case study and model a power market with a high wind power penetration by a hybrid model of probabilistic WPF and stochastic unit commitment. Specifically, this paper addresses three research questions: (i) how probabilistic wind power forecasts can be used in electricity market operations; (ii) how stochastic UC and its associated commitment strategies impact the power system compared to deterministic UC; and (iii) how different WPFs and operating strategies impact the Illinois power system operations. The paper contributes in the following aspects:

  • The paper proposes a framework for modeling a two-settlement power market with both day-ahead (DA) and real-time (RT) markets to co-optimize energy and operating reserves, which is close to real-world practice in the USA. The unit commitment decisions in the DA market are adjusted in the RT market by committing/de-committing fast start units based on an updated (more recent) wind power forecast.

  • This paper applies an advanced probabilistic wind power forecasting model with a quantile-copula estimator (QCE), which can forecast the pdf of hourly wind power for different forecast horizons. This is the first time such a kernel density forecasting (KDF) method has been applied to stochastic scheduling approaches, and this represents and important advance to the current state of the art.

  • This paper evaluates the impact of a large-scale wind power expansion in the current Illinois power system with the proposed forecasting and operational methods. The comprehensive analysis and observations provide valuable insights to policy makers and market regulators on efficient integration of renewable energy into the electric power grid.

The rest of the paper is organized as follows: Section 2 describes the models used for this study, including the WPF and scenario generation/reduction method, and the two-settlement power market model. Section 3 presents the results and analysis. Section 4 concludes the study and discusses future work.


In this paper, the uncertainty of wind power is represented and forecasted by a KDF model. The KDF model produces a pdf of wind power for each look-ahead hour, from which quantiles and scenarios of wind power can be derived. To account for the multiple wind power scenarios in system operation, a stochastic unit commitment model is used as part of a two-settlement electricity market with scheduling and dispatch of energy and operating reserves. Alternatively, a deterministic unit commitment with operating reserves derived from the forecast quantiles can also be used. The overall procedure, from the probabilistic wind power forecast to commitment and dispatch of generating units, is described in the following section.

2.1 Probabilistic wind power forecasting

2.1.1 Uncertainty forecasting

From the aforementioned discussion, it is clear that additional information on the uncertainty associated with future wind power predictions is valuable information for the system operator as well as market participants. Recent research efforts have focused on associating uncertainty estimates with point forecasts. Uncertainty information can take the form of probabilistic forecasts, risk indices or scenarios of short-term wind power generation. [13] The most common approaches are to represent WPF uncertainty as quantiles (or interval forecasts) and scenarios (Figures 1 and 2). An advantage of the scenario approach is that the scenarios can represent the temporal dependence of forecast errors, and they can be used in stochastic unit commitment models to cover different wind power output possibilities.

Figure 1.

Examples of quantile (or interval) forecast of wind power generation.

Figure 2.

Examples of scenarios of wind power generation.

In this paper, we use the wind power point forecast as input to a probabilistic model to compute the wind power uncertainty forecast, as illustrated in Figure 3. The probabilistic model used here is a statistical uncertainty forecast method based on conditional kernel density estimation. To evaluate different uncertainty estimation procedures, we use the metrics of calibration and sharpness. [32] Results so far indicate that the approaches based on kernel density estimation outperform the performance of the quantile regression method, which is the current state of the art, particularly in the calibration measure. [33, 34]

Figure 3.

The probabilistic forecast framework with scenarios generation.

2.1.2 Kernel density forecasting with time-adaptive quantile-copula estimator

We apply the KDF method proposed in Bessa et al., [33, 34] which is briefly outlined in the following. The problem consists of forecasting the wind power pdf at time step t for each look-ahead time step t + k of a given time-horizon knowing a set of explanatory variables (e.g. point forecasting and measured wind power output). Mathematically, this can be formulated as follows:

display math(1)

where pt + k is the variable of the forecasted wind power for look-ahead time t + k, xt + k | t are the explanatory variables forecasted for look-ahead time step t + k given available information at time step t, fP,X(pt + k,xt + k | t) is the joint density function, fX(xt + k | t) is the marginal density of X and fp is the conditional density function of wind power at time step t + k .

The QCE was introduced in Faugeras [35] and further enhanced with a time-adaptive version in Bessa et al. [33, 34] The basic idea is to represent the joint density function fP,X(pt + k,xt + k | t) in equation (1) by a copula function which models the dependency structure among the explanatory variables and the target variable. The modified equation (1) is as follows:

display math(2)

where c is a copula density function, and u and v are quantile transforms of the data − u = FP(p) and v = FX(x), respectively.

For computing the density function in equation (2), it is necessary to use nonparametric estimators such as the classical kernel density estimator (KDE). Thus, equation (2) (i.e. QCE method) with the nonparametric KDE estimator is represented as follows:

display math(3)

where N is the number of samples; hp, hu and hv are the bandwidth parameters; u and v are a quantile transform of the data u = FP(p) and v = FX(x); Ui and V i are the data transformed by the empirical cumulative distribution function, i.e. inline image and inline image; and K( ⋅ ) are the kernel functions for random variables p, u and v.

The next step is the selection of kernel functions, which depends on the type of the random variable. Our WPF problem needs two types of kernels corresponding to two types of variables in consideration: (i) a kernel for variables bounded between 0 and 1, i.e. after normalizing, wind power and the quantile transforms variables u and v, for which we use the beta kernel proposed in Chen; [36] and (ii) a circular kernel for the hour of the day, for which we use the von Mises distribution. [37]

To better account for evolving data streams of wind power data, we developed in Bessa et al. [34] a time-adaptive version of KDE for equation (3), where the KDE estimator is updated in the following way:

display math(4)

where λ is the forgetting factor and controls how fast the model learns from the new data.

The output is a pdf for wind power generation for each look-ahead time step of a specific time horizon. Figure 4 depicts what is called in Hyndman et al. [38] a stacked conditional plot. This plot represents the information contained in equation (3). It allows seeing the changes in the wind power density function (z-axis) conditioned on different values of the point forecast (ranging from 0 to 1, with increments of 0.02 on the y-axis). Each density function is a possible output of the KDF method, conditioned on the point forecast value. The conditional densities for intermediate values of the point forecast are broad (more uncertainty in the nonlinear part of the power curve), and there is a higher concentration of density in the tails for lower and higher values of point forecast (flat part of the power curve).

Figure 4.

Stacked conditional plot for wind power and point forecast. x-axis, range of wind power values (between 0 and 1); y-axis, point forecasts (explanatory variable); z-axis, conditional density.

As depicted, for low-point forecast values, the density functions have more than one mode, indicating multimodal distributions for the wind power uncertainty. This illustrates one advantage of the KDF methods. Because these methods provide the full distribution, it is possible to detect and compute more than one mode in case of multimodal distributions, instead of just computing the expected value. Another advantage is that the pdf representation is general, from which several uncertainty representations can be derived (e.g. quantiles and scenarios as illustrated in Figures 1 and 2).

For more details on the implementation of the KDF WPF methods, we refer to Bessa et al. [34] The forecasted pdfs for each look-ahead time step are the input of the scenario generation method. In the case study in Section 3, we compare the results from the use of probabilistic forecasts and point forecasts in power system operations.

2.1.3 Scenario generation and reduction

The forecasted pdfs express the probability distribution of the wind power forecast for a specific point in time. However, for unit commitment problems, it is important to take intertemporal relationships in the forecast uncertainty into account. This makes scenarios a more appropriate representation of the uncertainty. We use the WPF scenario generation approach described in Pinson [41] to convert pdfs to forecast scenarios. In this approach, scenario sampling is performed using a Monte Carlo approach. The dependency structure is a Gaussian copula, and the covariance matrix is estimated on the basis of historical forecasting errors and updated with new coming information, and used to represent intertemporal (hour-to-hour) correlations.

At the same time, to reduce the computational complexity of the decision problem, it may be necessary to use a limited set of scenarios as input to the stochastic UC model. In this paper, we use the scenario reduction method from Gröwe-Kuska et al., [42] which is common within the power systems domain. Here, a reduced scenario set is derived from the Kantorovich distance between the original and reduced sets of scenarios, taking scenario probabilities and distances of scenario values into account. A comparison of different scenario reduction methods, and their implications for the unit commitment problem, is presented in Botterud et al. [39]

2.2 Electricity market model

2.2.1 Market structure and clearing processes

The overall structure of the simulated market operation resembles US electricity markets with a DA market and an RT market, which are cleared in sequence. In the DA market, the market is cleared by a traditional least-cost UC and an ED optimization model. A DA wind power point forecast is used for wind power input, and the DA schedule and clearing prices are determined for the next 24 h. The operating reserve requirement can be set to account for the uncertainty from wind power, load fluctuations and outages. The model co-optimizes the energy, spinning reserves and nonspinning reserve simultaneously. The output includes the hourly prices, unit commitment and generation schedules.

After the DA market, the system operator performs a revised commitment procedure focusing on the reliability of the power system. The procedure is called Reliability Assessment Commitment (RAC). The RAC procedure is performed after the DA market, and we assume that a 4-h-ahead wind power forecast is used for this purpose. * Some fast-starting units may be committed/de-committed in the RAC to accommodate unexpected events, whereas the commitment of slow-starting base load units is fixed in the DA market. At the RAC stage, we assume that the wind power can be a point forecast or scenarios generated from the forecasted pdf, as input to a deterministic or stochastic UC model. After the RAC procedure, we run the ED with the realized availability of wind power generation. In the real-time market, the ED determines the hourly dispatch results and energy prices. The market settlement is based on the hourly RT deviations from the DA schedules. Figure 5 illustrates the clearing procedure of the simulated DA and RT markets.

Figure 5.

Market clearing procedure.

2.2.2 Stochastic unit commitment

We improve the model proposed in Wang et al. [26] to analyze the scheduling of energy and operating reserves in the electricity market with wind power integration. In addition to the representation of a more detailed market structure, as outlined earlier, the model improvements mainly focuses on two aspects: (i) an implementation of an improved representation of the binary commitment variables for thermal units, which is proposed in Rajan and Takriti, [43] and (ii) introduction of both spinning and nonspinning reserves.

The objective of the UC model can be expressed as follows:

display math(5)

The model minimizes three types of costs: the expected sum of fuel costs from thermal units FC, the expected cost of reserve not served C(RNS) and the cost of energy not served C(ENS) over a set of wind power scenarios s, and the thermal units start-up costs SC. The problem is assumed to have an hourly time resolution, and the model is solved to determine the commitment in both the DA and RT markets. The load and reserve (including spinning and nonspinning reserve) demand balance in all time periods and scenarios are represented as follows:

display math(6)
display math(7)
display math(8)

where inline image is the amount of generation provided by thermal unit i at time t in scenario s. inline image is the amount of wind power at time t in scenario s. inline image is the amount of energy not served at time t in scenario s. inline image is the amount of spinning/nonspinning reserve available from thermal unit i at time t in scenario s. αsr is the ratio of spinning reserve requirement to the amount of total reserve requirement. inline image is the amount of spinning/nonspinning reserve not served at time t in scenarios s.

Both thermal generation and wind power contribute to meet the load, but we assume that only thermal generation can supply the reserve requirement. We consider both spinning and nonspinning reserves, which are assumed to be provided by committed thermal units only, except that a fast start unit can be scheduled to provide nonspinning reserve when it is off. Moreover, we differentiate between reserve requirement to meet uncertainty in thermal generation and load (ORreg) and wind power uncertainty (ORwind). If there are scenarios with scarcity where operating reserves or load is not being served, this is penalized in the objective function. The traditional UC constraints are also included, such as ramping constraints and minimum up and down constraints. Note that the model becomes deterministic simply by assuming that there is only one scenario. For more details on the formulation of the model, we refer to Wang et al. [26]

In this analysis, we focus on the impact of wind power uncertainty and the corresponding reserve strategies to hedge the risk from this uncertainty. The system's ability to handle RT deviation of wind power from forecasts will depend on both the UC strategy and the operating reserve policy. DA UC schedules the thermal generating units for the next day 1 day ahead, and the RAC process adjusts the commitment decisions (intraday commitment) based on updated wind power forecasts by commitment/de-commitment of fast-starting units, taking intertemporal constraints into account. Operating reserves impose additional constraints on unit scheduling for each individual hour. The wind power uncertainty can be accommodated by using a stochastic UC, which considers a range of scenarios for wind power generation for a given time horizon. If the scenarios give a good representation of the wind power uncertainty, it should not be necessary to impose an additional operating reserve for wind, ORwind. Alternatively, one could use a deterministic UC model and handle the wind power uncertainty through the additional reserve requirement. In this case, ORwind could be derived from the probabilistic WPF. With the first approach, additional reserves are implicitly added through the stochastic UC, as opposed to the second approach that uses an explicit reserve requirement for wind power. In either case, the performance of the scheduling decisions would depend on the quality of the probabilistic forecast.


The purpose of the case study is to test out the different unit commitment models and operating reserve strategies on a two-settlement electricity market. The commitment schedule for thermal units is first determined on the basis of a DA wind power point forecast in the DA market, using deterministic UC. In the RAC process, the commitment of fast-starting units may be adjusted on the basis of a 4 h-ahead probabilistic forecast, using stochastic or deterministic UC. The RT ED is then solved with the realized availability of wind power. We focus solely on the impact of WPF uncertainty. Other uncertainties, like generator outages and load forecast errors, are therefore omitted. We first discuss the performance of the applied probabilistic wind power forecasts. Then we look at the impact on unit commitment and dispatch results in the Illinois power system.

3.1 Settings and assumptions

3.1.1 The Illinois power system

The Illinois power system is used to study the effects of wind power uncertainty and reserve margin requirements to accommodate wind power variability and uncertainty under a large-scale expansion of wind power. The Illinois system consists of thermal generation and wind power, with only a negligible amount of hydro generation. In the case study, load and thermal generation correspond to the situation in Illinois in 2006, whereas we assume that additional wind power is added to the system, as further described in the following.

  • Illinois thermal generation units

    The data for thermal generators were first prepared for the study in Cirillo et al. [44] but has been updated to reflect the historical situation in 2006. The Illinois system in 2006 consisted of 210 thermal plants (some of them are aggregate units) with a total capacity of 41,380 MW. The distribution of generation capacity by technology is illustrated in Figure 6. The thermal unit data, which includes the maximum/minimum output, initial state, minimum on/off hours, the cost of cold/warm start, fuel cost and heat rate curves (four curve segments for each plant), are used as input to the unit commitment problem.

  • Wind power

    We assume that the total installed capacity of wind power in the system is 14,000 MW. The wind power generation fluctuates between 0 and 12,680 MW. Under this assumption, wind power provides 20% of the in-state load and 16.6% of the total load including export in the period of July–October.

    The wind power data correspond to wind power forecasts and realized wind power generation for 15 hypothetical locations in the state of Illinois for 2006. Time series of wind power generation for the 15 sites were obtained from the Eastern Wind Integration and Transmission Study (EWITS) study of National Renewable Energy Laboratory. [45] This data were produced by combining a weather model with a composite power curve for a number of potential sites for wind power farms. The forecasts were generated on the basis of observed forecast errors from four real wind power plants. [46] The wind power data for the 15 sites were aggregated into one time series. In this analysis, we used the DA point forecast from the EWITS study in the DA market clearing. Then we used the 4-h-ahead point forecast from the EWITS to generate a probabilistic forecast for the reliability unit commitment. The accuracy of the wind power point forecasts from the EWITS vary from day to day. The mean absolute errors * for the DA and 4hr-ahead point forecasts are 8.5% and 7.3%, respectively, for the 4 month simulation period.

  • Illinois load and export

    The hourly load profile corresponds to the historical data from two large utilities (ComEd and Ameren) in the state of Illinois for the months of July–October 2006 (Figure 7). Illinois is typically a net exporter of electricity. In 2006, the average hourly export was 4020 MW. In this case study, we assume a fixed export schedule for each day, with higher exports at night and lower exports during the day. The export is added to the load within the state of Illinois. The resulting maximum and minimum loads, including export, in this period are 37,419 and 16,979 MW, respectively. The hourly time series for wind and load are shown in Figure 7.

  • Other assumptions

    We assume perfect information about the load, i.e. the forecasted load is equal to the actual load. The reason for this assumption is to isolate the effects of wind power uncertainty from load uncertainty. Outages of thermal plants and wind farms are also not simulated. Thus, the results of the simulated cases show the effects of WPF uncertainty only. Still, through the operating reserve requirements, we do take into account that a certain amount of reserve capacity (equal to the single largest contingency) must be maintained to accommodate uncertainty in load and thermal plants. In scarcity situations, the cost of reserve curtailment (both spinning and nonspinning reserve) is set to $1100 MWh  − 1, and the cost of unserved energy is $3500 MWh  − 1, on the basis of current practice at the Midwest ISO. [47] The percentage of spinning reserve is set to 50% of the total operating reserve requirement, i.e. αsr = 0.5. The wind power plants do not provide operating reserves. With these assumptions, the total installed capacity of the thermal units is 10.6% higher than the peak load.

Figure 6.

Thermal generation capacity distribution of the Illinois system.

Figure 7.

Wind and load time series data in the 4 month test period (July–October 2006).

3.2 Test cases

To study the impact of different WPF and corresponding reserve strategies to electricity market operations, eleven cases are developed (Table 1). The DA UC and ED, followed by the intraday RAC process and the RT ED, are run in sequence. All cases share the same DA market clearing mechanism (except case P1, in which forecasts are assumed perfect in both DA and RT markets) with the same reserve requirement (20% of the DA wind power point forecasts plus ORreg,t) but are different in the RAC process in terms of reserve strategies, UC formulation (deterministic or stochastic) and wind power forecast. Their description is listed in Table 1. The first case (P1) assumes a perfect wind power forecast and serves as a reference case. The forecast (PF) cases all use a point forecast equal to the median (i.e. the 50% quantile) of the probabilistic wind power forecast as input to a deterministic unit commitment in the RAC stage. However, the cases differ in terms of the operating reserve strategy. PF–F1 to PF–F3 use a fixed level of additional reserves for all hours of all days, whereas PF–D1 to PF–D3 use dynamic reserve requirements derived from the forecast quantiles for each operating day. Finally, the Scenario Forecast (SF) cases insert “F-S0 to SF-S2” use stochastic unit commitment based on 10 forecast scenarios, and the amount of additional operating reserve is determined by a percentage of the wind power in the forecast scenarios for each hour.

Table 1. Simulated cases
CaseAdditional reserve: ORwind,t*ForecastUC strategy at RAC stage
  • UC=unit commitment, RAC=Reliability Assessment Commitment, DA=day-ahead, RT=real time, PF=power forecast.
  • *

    This additional reserve is applied at the RAC stage only to handle wind power uncertainty. All cases use a regular reserve, ORreg,t, equal to the largest contingency (1146 MW).

P1NonePerfect in both DA and RTDeterministic
PF–F0None50% quantileDeterministic
PF–F1Fixed: average 50–10% quantile50% quantileDeterministic
PF–F2Fixed: average 50–5% quantile50% quantileDeterministic
PF–F3Fixed: average 50–1% quantile50% quantileDeterministic
PF–D1Dynamic: 50–10% quantile50% quantileDeterministic
PF–D2Dynamic: 50–5% quantile50% quantileDeterministic
PF–D3Dynamic: 50–1% quantile50% quantileDeterministic
SF–S0None10 ScenariosStochastic
SF–S110% of wind scenario10 ScenariosStochastic
SF–S220% of wind scenario10 ScenariosStochastic

3.3 Results and analysis

We first present an analysis of the performance of the probabilistic wind power forecast. Then the dispatch results for a period of 4 months (from 1 July 2006 to 31 October 2006) are presented to study the impact of different wind power forecasts and scheduling strategies. Detailed results from two selected days (large over-forecast or under-forecast) are also presented to illustrate the system performance when the wind power is misestimated.

The probabilistic wind power forecasting and scenario generation are implemented in R and conducted on a laptop with Intel i7 2.67 GHz CPU, 8.0 GB memory. This takes 90 s for each day on average. The unit commitment and economic dispatch process is implemented in AMPL and Cplex 12.0. The two-settlement market clearing simulation is conducted on a Linux server with 2 quad-core, 2.66 GHz CPU, 32.0 GB memory. It takes 48 min on average to get the dispatch results for each day (including DA UC, EA ED, RAC and RT ED).

3.3.1 Evaluation of probabilistic wind power forecasts

First, we evaluate the performance of the KDF model with QCEs. The training set is selected from 1 January 2006 to 30 June 2006, and the data from 1 July to 31 October 2006 are used as test set. The explanatory variables for the probabilistic forecast are the wind power point forecasts and the hour of the day. The value of the forgetting factor (λ) in equation (3) is set to 0.999.

The performance of the KDF model is evaluated by two metrics: [32] calibration and sharpness. Calibration is a measure of how well the forecasted quantiles match the observed values. For instance, the wind power generation should be below the 5% quantile only 5% of the time. If the realized wind power generation is below the 5% forecast more frequently, there is a positive bias in the forecast, measured in terms of deviation from perfect calibration. The sharpness metric represents the tendency of the probabilistic forecast towards discrete forecasts, measured by the mean size of the forecast intervals (i.e. the distance between quantiles). Hence, sharpness is a measure of the width of the forecast distribution.

Figure 1 shows an example of a probabilistic forecast obtained with QCE in the form of a set of forecasted quantiles. The kernel size is 0.001 for both realized and forecasted wind power.

Figure 8 illustrates the calibration (left) and sharpness (right) diagram of the probabilistic kernel density forecasts averaged over the 4 months data in the test set. The average deviation from the ‘perfect calibration’ is less than 4% for all quantiles. The positive deviation corresponds to an underestimation of the forecasted quantiles. In the right part of Figure 8, the x-axis is the nominal coverage of the forecast interval ( 1 − α), and the y-axis is the average size of the intervals. In this case, what is desired is to have intervals with smaller size for all coverage rates. The flat curve shows that the mean length of the forecast intervals increases in a close to linear manner when the nominal coverage rate increases. For a more detailed analysis of the performance of KDF with the QCE compared with the more traditional quantile regression for probabilistic forecasting, we refer to Bessa et al. [33] The results in Bessa et al. [33] show that KDF tends to give better performance in terms of calibration, whereas quantile regression gives sharper forecasts.

Figure 8.

Calibration (left) and sharpness (right) diagrams of kernel density forecasting with quantile-copula estimator.

In cases PF–D1 to PF–D3, we use the probabilistic forecasts directly to calculate hourly dynamic operating reserve requirements as the difference between the median (i.e. the point forecast) and a lower quantile. The choice of the lower quantile determines the likelihood of scarcity and the overall level of reliability in the system. For instance, if the reserve requirement is equal to the difference between the median and the 5% quantile, the likelihood of curtailment of either operating reserves or load should be close to 5%. Of course, this requires that the probabilistic forecast has a good calibration. At the same time, the level of hedging in the commitment decisions will also depend on the penalties for curtailment in the objective function.

The stochastic unit commitment takes a set of scenarios as input, and the scenarios also capture intertemporal constraints in the scheduling problem. The procedure outlined in Section 2.1.3 is used to generate a reduced set of scenarios. Previous results have shown that scenario reduction tends to reduce the variance in the reduced scenario set compared with the original scenarios, and this may reduce the level of hedging in the stochastic commitment decisions. [39] Additional reserves may therefore be required, also with stochastic unit commitment. This is further explored in the operational analysis.

Metrics for evaluating these scenarios are being presently studied by other researchers. [40]

3.3.2 Operational performance evaluation

In the analysis of commitment and dispatch decisions, we use the total system cost including fuel cost, start-up cost and the costs of unserved load and operating reserve as the main performance criteria. We also look at other performance metrics, such as the amount of curtailment and the electricity prices. In general, the results depend on not only the scheduling strategy for energy and operating reserves but also the quality of the wind power forecast. We first present aggregate results for the four month period, and then we look at more detailed results for two selected days.

  • Results of a 4 months period

Figure 9 shows the total costs of the power system in the 4 months simulation period for each case. As the benchmark case, case P1 has the lowest costs because of the perfect forecasts. Case PF–F0 has the highest costs because it does not schedule any additional reserve for wind power uncertainty, which causes a high penalty from load and reserve curtailment. In comparison, all the other PF and SF cases have much lower total costs than PF–F0. Hence, there is clearly a need to schedule more capacity, either explicitly through additional operating reserves or implicitly through stochastic unit commitment. The cases using stochastic unit commitment have the lowest total costs. It means that the stochastic unit commitment model structure can partially handle the wind power uncertainty. However, the results indicate that additional reserves are also needed with stochastic unit commitment, as SF–S2 (with the highest level of additional operating reserve) has the lowest total costs. Similar results were found in Wang et al. [26] and Ruiz et al. [48] Cases using fixed operating reserve requirements have slightly higher costs than cases with dynamic reserve requirements, although the average level of reserve is the same for the two categories of cases. This result shows that dynamic operating reserves derived from probabilistic forecasts are better than fixed reserves at capturing the uncertainty of the wind power. Overall, more operating reserves lead to lower costs within the same categories, as illustrated in Figure 9 within cases PF–F0 to PF–F3, cases PF–D1 to PF–D3 or cases SF–S0 to SF–S2. This result indicates that the cost of procuring additional reserve is low compared with the cost of load and reserve curtailments, under the current assumptions. For instance, Figure 10 shows that cases using the stochastic UC model (SF–S0 to SF–S2) have slightly higher generation costs (fuel and start-up costs) because they commit more generation capacities to provide sufficient ramping capacity to account for the different forecast scenarios. However, the additional generation costs are more than offset by the reduced curtailment costs. Another interesting observation from Figure 10 is that the cases with dynamic reserves (PF–D1 to PF–D3) have significantly lower generation costs compared with the cases with fixed reserves requirements (PF–F1 to PF–F3).

Figure 9.

Total cost over 4 months.

Figure 10.

Total generation costs over 4 months.

Figure 11 shows that cases using wind power scenarios and stochastic unit commitment have more start-ups, which means that they require more status changes on generation units to cover the complex wind scenarios. Fixed operating reserve requirements also require more start-ups than dynamic reserve requirements. Not surprisingly, when comparing the number of start-ups within the same case categories, the higher the reserve requirement, the more start-ups are scheduled.

Figure 11.

Total number of start-ups over 4 months.

We use the curtailment of load and reserve to evaluate the reliability of different the different strategies. As illustrated in Figure 12, the curtailment of both load and spinning reserve show the same trend. As a reference case, P1 does not have any load curtailment and very limited reserve curtailment because of the perfect wind power forecast. There are more load curtailments in cases with fixed reserve strategies (PF–F1 to PF–F3) but more spinning reserve curtailment in cases with dynamic reserve strategies (PF–D1 to PF–D3). A potential reason that there are more hours of reserve curtailment in the cases with dynamic reserve is that there are many days when the dynamic reserve requirement is lower than the fixed reserve. This also occurs during days with overestimation of wind power when curtailment typically occurs. However, in terms of total cost, the cases PF–D1 to PF–D3 perform better since the penalty of load curtailment is much higher than that of the reserve curtailment. The cases using stochastic unit commitment with wind power scenarios have the least curtailment on both load and spinning reserve.

  • Results of selected days

Figure 12.

Total curtailment on load and spinning reserve over 4 months.

In this section, 2 days with large over-forecast and under-forecast of wind power is selected to demonstrate how the wind uncertainty and forecasting errors are handled in more detail with a smaller time granularity.

  1. Over-forecasted day (19 October 2006)

    The bias of the wind power forecast is positive compared with the realized data on 19 October 2006, as illustrated on Figure 13. It means that there is much less wind power in real time than what was forecasted 4 h ahead.

    Because of the over-forecasts of wind power, the system does not commit enough generation and reserve capacity to meet the demand requirement, which causes a large volume of load and reserve curtailment, as shown in Figures 14 and 15. In Figure 14, besides the models with perfect forecasts (case P1), only the stochastic model SF–S2 commits enough generation capacity to meet the load in most hours (except hours 7 and 9) to account for the unexpected lower realized wind power. As a consequence, the RT energy and spinning reserve prices are skyrocketing in cases with large curtailment (see Figure 16). Comparably, the prices stay relatively lower in cases SF–F2 than in PF–F2 and PF–D2. It means that stochastic models with wind power scenarios perform better for this day when wind power is overestimated. When comparing the fixed and dynamic operating reserve requirement, the fixed reserve strategy gives slightly less curtailment than the dynamic one.

    The RT prices in Figure 16 show extreme values equal to the marginal cost of unserved load and reserve for many hours due to all the curtailments occurring on this day. For comparison, the price in the DA market is much lower, as there is no curtailment in the DA market since the same forecast is used in both UC and ED stages in the DA market. Note that the simulation setup does not capture all the flexibilities that the system operator has in the RT dispatch. For instance, the final commitment decisions are based on a 4-h-ahead forecast, whereas the system operator can commit resources closer to real time. Furthermore, there is no demand response, which is an additional source of flexibility. The impacts of system flexibility on DA and RT prices will be studied in more detail in future work.

  2. Under-forecasted day (22 September 2006)
Figure 13.

Four-hour-ahead wind power forecasts (point and scenarios) and realized wind power on 19 October 2006.

Figure 14.

Real-time hourly load curtailment on 19 October 2006.

Figure 15.

Real-time hourly spinning reserve curtailment on 19 October 2006.

Figure 16.

Real-time hourly energy prices on 19 October 2006.

The bias of the wind power forecast is negative before the 16th hour and positive afterwards compared with the realized data on 22 September 2006, as illustrated in Figure 17. It means that there is more wind power in real time than what is being forecasted for the first part of the day.

Figure 17.

Wind power forecasts (points and scenarios) and realized wind power on 22 September 2006.

There is no load curtailment in any hour on this day because the actual available wind power is greater than the forecast for most of the day. This makes the total available generation greater than the demand for energy and reserves (except in hour 20, which has the highest over-forecast). In fact, there is an over-commitment of thermal generation in the first part of the day, leading to curtailment of wind power in all the cases, as shown in Figure 18. The stochastic case SF–S2 has the most curtailment of wind power in this day. The over-commitment also results in very low prices, with the RT energy price dropping to zero when there is wind power curtailment (Figure 19). Note that in hour 20, the realized wind power dropped dramatically to a very low level, and the energy price at this hour is high because of reserve curtailment.

Figure 18.

Real-time hourly wind curtailment on 22 September 2006.

Figure 19.

Real-time hourly energy prices on 22 September 2006.


A large-scale expansion of wind power clearly gives rise to new challenges in the operation of the electric power grid. In this paper, we have analyzed the potential use of probabilistic WPF as a means to address variability and uncertainty from wind power in electricity market operations. We applied an advanced probabilistic density forecasting approach with a QCE to generate pdfs for the forecasted wind power. We model a framework of a two-settlement power market with both DA and RT markets to co-optimize energy and operating reserves, which is close to real-world practice in the USA. To make the proposed framework work for a much larger and more realistic power system, we adopted an efficient UC formulation and the use of a state-of-the-art solver. Furthermore, we analyzed how the probabilistic forecasting quantiles can be used to calculate dynamic operating reserve requirements and how scenarios of wind power can be used as input to a stochastic unit commitment model. Overall, the methodologies proposed in this paper contribute to the development of more advanced tools for operation of power systems with a large share of wind power and other renewable generation.

In a case study of the power system in Illinois, we simulated the operation of the market, where the DA market is cleared on the basis of a point forecast, and an updated probabilistic forecast is used to adjust the commitment of thermal units ahead of the RT dispatch. The results show that the re-commitment decisions are very important for the cost and reliability of the system and that additional reserves are required to handle the uncertainty in wind power. When basing the re-commitment decisions on a deterministic unit commitment model, the use of dynamic operating reserves derived from the probabilistic forecast gives better performance than fixed reserve requirements, since the operating reserve levels are better aligned with the wind power forecast uncertainty. A stochastic unit commitment model with wind power scenarios leads to different re-commitment decisions and RT dispatch. However, dynamic operating reserves and stochastic unit commitment give similar results in terms of total system cost. The detailed results also illustrate that inaccurate wind power forecasts can lead to over-commitment and under-commitment of generation capacity and therefore have large implications for system reliability and RT prices.

The results of our analysis clearly show the potential value of probabilistic WPF in power system operations. We have shown that such forecasts can be incorporated into deterministic UC through probabilistic reserve requirements or can provide scenarios as input to a stochastic UC. In general, advantages of the deterministic approach with dynamic reserves include that this is more aligned with current operating procedures. Furthermore, from a computational perspective, it is a simpler UC model with less computational burden. However, this approach does not capture the effect of temporal correlation in WPF errors. It also does not consider uncertainty and its cost in the objective function. In contrast, stochastic UC does address temporal variability through the scenario representation of uncertainty. Moreover, the total cost, including the expected cost of scarcity, is explicitly taken into account in the objective function. However, the switch to stochastic UC involves a more radical departure from current practice, and it may run into computational constraints in large systems; our results show that the benefits in terms of cost savings may be limited.

Probabilistic WPF is still at an early stage, and more work is needed to improve the quality of such forecasts. However, just as for point forecasts, there will always be inaccuracy in the probabilistic forecasts. This calls into question the overall risk paradigm under which scheduling decisions are made. In stochastic UC, with the minimization of expected cost, one could adjust the risk level by changing the costs of unserved energy and reserves in the objective function. However, other approaches based on different decision paradigms, such as utility theory, value at risk, robust optimization or regret, should also be considered. The optimal decision strategy does depend not only on the risk preferences of the system operator but also on the quality of the probabilistic forecast.

As future improvements, we will first introduce the demand response into the current UC/ED model, which will give a more realistic representation of real-world power systems, particularly with the advent of a smarter grid. Another direction is to explore the possibility of using wind power to meet parts of the operating reserve requirements. On the system analysis side, we plan to investigate in more detail the commitment and dispatch of different generation technologies and their impacts on the whole system, including the impact on DA and RT prices. We will also analyze the overall impacts on power systems emissions in Illinois from a large penetration of wind power.


The authors gratefully acknowledge the anonymous reviewers for their constructive comments.

The authors acknowledge the U.S. Department of Energy, Office of Energy Efficiency and Renewable Energy through its Wind and Water Power Program for funding the research presented in this paper. The submitted manuscript has been created by UChicago Argonne, LLC, Operator of Argonne National Laboratory (‘Argonne’). Argonne, a U.S. Department of Energy Office of Science laboratory, is operated under Contract No. DE AC02-06CH11357. The U.S. Government retains for itself, and others acting on its behalf, a paid-up nonexclusive, irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public and perform publicly and display publicly, by or on behalf of the Government.

The work of Hrvoje Keko was supported by Fundação para a Ciência e Tecnologia (FCT) PhD Scholarship SFRH/BD/43087/2008. The work of Jean Sumaili was supported by FCT within the program ‘Ciência 2008’.

The work of R. J. Bessa was supported by Fundação para a Ciência e Tecnologia (FCT) Ph.D. Scholarship SFRH/BD/33738/2009.

  • *

    In this paper, point forecasts are used in the day-ahead stage, and probabilistic forecasts are used in the reliability assessment commitment (RAC) stage. The day-ahead market is cleared on the basis of the bids from the market participants, and we assume that the point forecast represents the aggregate bid from the wind power producers. From the perspective of the system operator, the focus is on the system reliability. In the RAC, which takes place before the real-time dispatch, most US system operators already use their own forecast of wind power (and loads) to ensure system reliability. We therefore see it as more likely that a probabilistic forecast could be applied by the system operator at the RAC stage to adjust the commitment from the day-ahead market if needed.

  • *

    The mean absolute errors are calculated as follows: inline image where e is the forecast error and N is the number of samples.