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TRANSCRIPT
Estimating Wind Forecast Errors and Quantifying ItsImpact on System Operations Subject to Optimal Dispatch
by
Xiaoguang Li
A thesis submitted in conformity with the requirementsfor the degree of Master of Applied Science
Graduate Department of Electrical and Computer EngineeringUniversity of Toronto
Copyright c© 2011 by Xiaoguang Li
Abstract
Estimating Wind Forecast Errors and Quantifying Its Impact on System Operations
Subject to Optimal Dispatch
Xiaoguang Li
Master of Applied Science
Graduate Department of Electrical and Computer Engineering
University of Toronto
2011
Wind power is being added to the supply mix of numerous jurisdictions, and an increasing
level of uncertainties will be the new reality for many system operators. Accurately
estimating these uncertainties and properly analyzing their effects will be very important
to the reliable operation of the grid. A method is proposed to use historical wind speed,
power, and forecast data to estimate the potential future forecast errors. The method
uses the weather conditions and ramp events to improve the accuracy of the estimation.
A bilevel programming technique is proposed to quantify the effects of the estimated
uncertainties. It improves upon existing methods by modeling the transmission network
and the re-dispatch of the generators by operators. The technique is tested with multiple
systems to illustrate the feasibility of using this technique to alert system operators to
potential problems during operation.
ii
Acknowledgements
I would like to express my sincere gratitude to my supervisor, Prof. Z. Tate, for his guid-
ance and support throughout my time here. This thesis would not be possible without
his advice, help, and patience.
I would also like to thank my committee: Prof. R. Adve, Prof. R. Iravani, and Prof. P.
Lehn.
Finally, I would like to thank my parents for their continual love and support.
iii
Contents
1 Introduction 1
1.1 Background & Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Thesis Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
I Estimating Wind Forecast Error 7
2 Wind Forecasting 8
2.1 State of Wind Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Wind Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Wind Speed Versus Wind Power . . . . . . . . . . . . . . . . . . . . . . . 10
2.4 ARMA Forecasting Program . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 Estimating Wind Uncertainties 18
3.1 Analyzing Forecast Error . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2 Refining Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.1 Weather Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.2 Ramp Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
iv
II Quantifying Impact of Wind Forecast Errors 28
4 Modeling System Response to Forecast Errors 29
4.1 Overview of Power System Operations . . . . . . . . . . . . . . . . . . . 29
4.2 Illustration of System Security Issues . . . . . . . . . . . . . . . . . . . . 30
4.3 Review of Current Methods . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.4 Bilevel Programming Method . . . . . . . . . . . . . . . . . . . . . . . . 34
5 Mathematical Model Description 35
5.1 Nomenclature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
5.2 Overview of Bilevel Programming . . . . . . . . . . . . . . . . . . . . . . 36
5.3 Bilevel Programming Formulation . . . . . . . . . . . . . . . . . . . . . . 37
5.3.1 Power Flow Equation . . . . . . . . . . . . . . . . . . . . . . . . . 37
5.3.2 Objective Function . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.3.3 Full BLP Formulation . . . . . . . . . . . . . . . . . . . . . . . . 42
5.4 Solving Bilevel Programming . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.4.1 Replacing Follower Optimization with KKT Conditions . . . . . . 44
5.5 Final MILP Form . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6 BLP Case Study 49
6.1 Experimental Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.2 Case Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.3 Results & Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
7 Performance of BLP 56
7.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7.2 Constant System Size, Varying Number of Wind Farms . . . . . . . . . . 57
7.3 Varying System Size, Constant Number of Wind Farms . . . . . . . . . . 58
v
8 Conclusion 61
8.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
Bibliography 63
vi
List of Tables
2.1 Wind Forecast Time Horizon Categories . . . . . . . . . . . . . . . . . . 9
2.2 RMSE of Different ARMA(p, q) for a one hour ahead forecast . . . . . . 17
3.1 RI vs modified method: percentage of measurements that are outside a
95% CI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6.1 Results for 37-bus System Study . . . . . . . . . . . . . . . . . . . . . . . 51
6.2 Wind Forecast Error Resulting in the Largest Line Overloads for ±30 MW
Forecast Error Bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
7.1 Computation Time Comparison Between CPLEX’s Branch-and-Cut and
Vertex Enumeration [s] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
7.2 Performance of Solver Using Multiple Processors . . . . . . . . . . . . . . 59
vii
List of Figures
1.1 Aggregate Ontario wind output from January 2010. Each line represents
one day in January [1] . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Wind speed map of Ontario at a height of 100 meters [2] . . . . . . . . . 4
2.1 Typical wind turbine power curve . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Estimated power curve from historical data using nonlinear least square
fitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3 Example of diurnal wind speed pattern . . . . . . . . . . . . . . . . . . . 13
2.4 Example of Weibull wind speed distribution . . . . . . . . . . . . . . . . 14
2.5 PMF of wind speed data before (left) and after (right) transformation and
standardization process . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6 Wind forecast process using ARMA . . . . . . . . . . . . . . . . . . . . . 16
3.1 Forecast error histograms for four different forecast time horizons . . . . 19
3.2 Empirical CDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.3 Determining CI for a 90% confidence level . . . . . . . . . . . . . . . . . 21
3.4 Example of wind power measurements, predictions, and CIs for a 90%
confidence level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.5 Power measurement and predictions from different forecast horizons . . . 23
3.6 During a ramp event, many measurements are outside of a 95% CI esti-
mated using RI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
viii
3.7 95% confidence level for the same ramp event as Figure 3.6 using the
modified method. The CI noticeably increases at the 1h mark to account
for the predicted ramp . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.8 Wind forecast CI estimation process . . . . . . . . . . . . . . . . . . . . 27
4.1 Two-bus system to demonstrate the analysis of impact of forecast errors.
a) predicted wind output of 50 MW, base case; b) 30 MW deviation from
forecast, fixed participation factor; c) 30 MW deviation from forecast,
optimal dispatch; d) 50 MW deviation from forecast, optimal dispatch . . 31
5.1 Convex, piecewise-linear penalty function to model line violations . . . . 40
5.2 Representation of the piecewise-linear objective using λ formulation, M is
a constant associated with the maximum overload to be considered . . . 41
5.3 Representation of the piecewise-linear objective using epigraph formulation. 41
6.1 37-bus system from [3] with eight wind farms introduced. . . . . . . . . . 51
6.2 Number of wind scenarios that cause zero line violation with optimal re-
dispatch (gray line) and fixed participation factor (black line) for the 37-
bus system in Figure 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.3 Improvement in objective when using optimal dispatch over fixed partici-
pation factor dispatch for the 37-bus system study with ±30 MW forecast
error bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
6.4 Worst wind outputs and line overloads for ±30 MW forecast error case . 55
7.1 Computation time statistics for the 37-bus system study. . . . . . . . . . 57
7.2 Computation time statistics for the 118-bus system study. . . . . . . . . 60
ix
Chapter 1
Introduction
1.1 Background & Motivation
The electricity sector is undergoing major changes to its energy supply mix, with numer-
ous jurisdictions in North America adding renewable generation as part of its Renewable
Portfolio Standards (RPS) [4]. For example, the government of Ontario is encouraging
renewable generation through the feed-in tariff (FIT) program, which pays a fixed price
for energy production from renewables with a twenty year contract period [5]. As a re-
sult of the FIT program, the Independent Electricity System Operator (IESO) of Ontario
expects 10,700 MW of renewable generation to be added by 2018, and a large portion
of that is wind generation [6]. The addition of a large quantity of wind generation will
challenge IESO’s ability to maintain the grid, in terms of reliability and efficiency [6].
Wind as a source of electricity has several characteristics that differentiates it from
conventional power plants. For one, wind is not naturally dispatchable, which means
the output of the wind turbines cannot be changed in the same manner as conventional
power plants. It can be made to be dispatchable by curtailing the outputs of the turbine,
but this is not desirable since it forgoes some of the energy production. In addition,
considering some of the states’ RPS require a certain amount of energy to be from
1
Chapter 1. Introduction 2
renewables, forgoing energy becomes even less of a desirable option.
The inability to be dispatched is not a difficult problem to solve in and of itself. For
example, nuclear power in Ontario also has very little capability to change its output,
and renewables such as tidal power can be easily integrated since its power output can
be accurately predicted. It is the intermittent and variable nature of wind, on top of its
inability to be dispatched, that create problems for its integration into the grid. Figure
1.1 shows the aggregate Ontario wind output from the month of January 2010 on an
hourly time granularity [1], which helps to illustrate the difficulties of integrating wind
power into the power system. The outputs vary from day to day and could change
drastically over the course of a day. During a ramp event, which is a large change in the
wind power output over a short period of time, the outputs could even experience large
change within a few hours.
Figure 1.1: Aggregate Ontario wind output from January 2010. Each line represents one
day in January [1]
Wind is also a location dependent resource that is abundant in only specific areas.
Chapter 1. Introduction 3
Figure 1.2 is a wind speed map of Ontario at a height of 100 meters, obtained from
Ontario Ministry of Natural Resources [2]. The same map also marks the existing trans-
mission network in Ontario, which shows limited transmission lines connecting to the
wind resources in northern Ontario.
This map illustrates one of the problems with wind power. Some of the great wind
locations, for example in northern Ontario, are very far away from the load centers in
southern Ontario. Unfortunately, there are also very limited transmission lines connecting
to these locations. Adequate transmission connections are needed to bring the power
production to southern Ontario, and to bring power up north when the wind outputs
drop below the demand in the north.
However, it is actually uneconomical to construct transmission capacities to accom-
modate the rated wind power output. Wind power in most locations only has a capacity
factor of 20%–40% [7], which means the realizable energy production is 20%–40% of the
maximum energy production possible at that site. Thus it is highly likely that the trans-
mission capacity will be unused most of the time if it is built to match the rated power
output. A statistical analysis could be conducted to reduce the transmission capacity
to some value that makes the most economical sense, but this would mean transmission
constraints will be a regular occurrence on the system.
These are difficult problems facing wind integration, but in terms of safely and reliably
operating the grid, some of the problems can be greatly reduced with accurate wind
forecast. An accurate wind forecast can help to reduce the level of uncertainties in the
wind production, and allow system operators to plan ahead for the varying wind outputs.
The forecast is not always accurate, and although the forecast error will reduce as the
forecast improves, it will never be perfect, and some uncertainties will always remain.
For example, in a report to IESO [8], the uncertainties associated with wind is shown to
be higher than the uncertainties of the load alone. It is this increased uncertainty that
system operators must deal with during operations.
Chapter 1. Introduction 4
Figure 1.2: Wind speed map of Ontario at a height of 100 meters [2]
Chapter 1. Introduction 5
The Ontario system is an example of the types of problems that many other juris-
dictions will face as well as they add wind generation to their systems. Given a future
with increasing uncertainties, a tool that can identify potential problems to the system
operators would help tremendously. To do that, the tool needs to achieve two objectives.
One, it needs to estimate the uncertainties, or the errors in the wind forecasts; especially
during periods of ramp events, when the forecast errors are known to be greater [9]. Two,
once the uncertainties are estimated, it needs to accurately quantify the possible effects
of the uncertainties on the system.
1.2 Thesis Objective
This thesis proposes a method to estimate the wind forecast errors, with a modification to
especially address the large potential forecsat errors during ramp events. In addition, it
proposes a method to evaluate whether the uncertainties could potentially cause problems
on the system, and identify the worst problem if problems exist.
1.3 Thesis Outline
This thesis is divided into two major parts. The first part describes the procedure to
estimate wind forecast errors. It begins by briefly reviewing the current forecasting
methods, and then it describes in detail the implementation of one of the forecasting
methods. It then describes a method to estimate the error from the forecast, with a
novel technique to improve the accuracy of the estimation during wind ramp events.
The second part describes the procedure to quantify the impact of the forecast errors.
It begins by briefly reviewing how forecast error is accommodated during real-time power
system operations. An optimization model is then developed to quantify the impact of
the forecast errors, subject to optimal dispatch. Finally, the performance of the method
is analyzed using various power system test cases.
Chapter 1. Introduction 6
The entire work is summarized in the conclusion, with remarks about future works.
Part I
Estimating Wind Forecast Error
7
Chapter 2
Wind Forecasting
2.1 State of Wind Forecasting
Before the wind forecast errors can be estimated, it is important to have an understanding
of the existing wind forecasting techniques. Wind forecasting is one method to manage
the variability of wind; by knowing the wind forecast for the near future, the system
operators can plan to accommodate the wind through adjustments in unit commitment
and dispatch. In North America, the trend appears to be a movement towards centralized
forecasting [10]. Centralized forecasting shifts the responsibility of wind prediction from
each individual wind generator owner to one central entity, usually the independent
system operator (ISO). This allows the ISO to place more trust in the forecasts, since
the wind generator owners have an economical incentive to sometimes distort the wind
forecast results; but this is no longer an issue if the task is under the control of the ISO.
In addition, the ISO has the incentive to purchase a more accurate but potentially more
expensive forecasting program, and it is able to spread the cost of the program over all
the wind generators on the system. Due to these benefits, IESO of Ontario has decided
to implement centralized forecast and plans to have it in operation by 2012 [11].
The various forecasting techniques are usually categorized by their intended forecast-
8
Chapter 2. Wind Forecasting 9
Table 2.1: Wind Forecast Time Horizon Categories
Time horizon Very short-term Short-term Medium-term Long-term
Range Less than 30 min-
utes
30 minutes to
6 hours
6 hours to 1
day
More than 1
day
ing time horizon [12], as shown in Table 2.1.
In this thesis, the time horizon of interest is several hours into the future, which
reside within the short-term forecasting category. For this time horizon, techniques
based on statistical analysis and machine learning have performed much better than
other techniques [12]. Statistical analysis techniques are based on some variations of
time-series models, such as autoregressive moving average (ARMA) or autoregressive
with exogenous inputs (ARX) and machine learning techniques are based on artificial
neural networks. Detailed review of either technique can be found in [12].
For medium-term and long-term forecasts, numerical weather prediction (NWP) pro-
grams are usually used. NWP is a simulation of the weather system; it uses the initial
weather conditions and physical models to calculate the expected weather conditions
hours into the future. The simulation is fairly computationally intensive, which is why
NWP is not practical for use in the very short-term and short-term forecasts.
2.2 Wind Data
One problem with analyzing forecast errors is the difficulty of obtaining real world data
of synchronized forecast and measured wind outputs. Forecasting product vendors try
to withhold that data, so their competitors cannot analyze the performance of their
products. An in-house forecasting program using ARMA is developed to predict wind
speed, and the prediction is compared to the measured data to generate the forecast error
data. It should be noted that the goal of the forecasting program is not to have the best
Chapter 2. Wind Forecasting 10
prediction performance, but simply as a way to generate the forecast error data that is
needed for analysis. Forecast error data generated using other forecasting programs can
be analyzed in the same manner.
The data used in thesis were from the National Renewable Energy Laboratory (NREL)
Eastern Wind Dataset [13]. The dataset contains both wind speed and wind power data
at a height of 100 meters and at 10 minute intervals. This is the height and time interval
used for all examples in the remainder of this thesis. For example, an one hour ahead
forecast would be referring to six data points into the future.
2.3 Wind Speed Versus Wind Power
A subtle but very important distinction must be explained before the ARMA forecasting
program is described. Most of the wind forecasting programs predict wind speed and not
wind power for two reasons. First, different wind turbines have different characteristics
and controls that would generate different amount of power for the same wind speed. A
forecasting program would need to know the characteristics of each turbine in order to
predict the power output; but on the other hand, wind speed is universal for all turbines.
Second, the typical wind turbine power curve looks like Figure 2.1. The wind power
output remains constant for wind speeds below the cut-in speed, above cut-out speed,
and the region of rated power output in between. A forecasting program looking only at
the power output would have difficulty identifying at which point of the power curve the
wind turbine is currently operating at. Instead, forecasting programs predict the wind
speed, and then use the power curve to obtain the power output from the predicted wind
speed. This is the procedure that is used for the in-house implementation of the ARMA
forecasting program.
The power curve is estimated using historical data from the wind turbine and modeled
based on the sigmoidal function [14], which is shown in (2.1).
Chapter 2. Wind Forecasting 11
Figure 2.1: Typical wind turbine power curve
P =c
1 + e−b(v−a)(2.1)
where P and v are the wind power and wind speed, respectively. a, b, and c are parameters
of the function which are estimated using historical wind speed and wind power data. A
nonlinear least square fitting tool from MATLAB is used to fit the parameters a, b, and
c to the data, and the resultant power curve is shown in Figure 2.2. A post-processing
step is added to keep the power output between zero MW and the rated power.
2.4 ARMA Forecasting Program
A (p, q) order ARMA model is a linear regression model that uses p autoregressive (prior
state) terms and q moving average (white noise error) terms to estimate the state:
Xt =
p∑i=1
φiXt−i +
q∑j=1
θjεt−j
Xt = Xt + εt
(2.2)
where φ1, . . . , φp, θ1, . . . , θq are the ARMA parameters, X is the predicted state, X is the
observed state, and ε is the error term.
Chapter 2. Wind Forecasting 12
Figure 2.2: Estimated power curve from historical data using nonlinear least square
fitting
When using ARMA, an implicit assumption is that the process being forecast is
stationary and has independent identically-distributed error terms [15]; this is not true
for wind speed data. Wind speed time series have been shown to exhibit a diurnal pattern
(Figure 2.3) and seasonal pattern [16].
In addition, the distribution of wind speed data is Weibull instead of Gaussian.
Weibull distribution is positively skewed, which means that the tail of the distribution
on the right side is longer than the left side and most of the distribution is on the left
side of the mean (Figure 2.4). The error terms will not be the same as the terms that
are from a Gaussian distribution. If ARMA is directly applied to the wind speed data,
these factors will cause the predictions to be less accurate.
A transformation and standardization technique was developed by [16] to remove the
diurnal non-stationarity of the wind speed data and transform the distribution of the
Chapter 2. Wind Forecasting 13
Figure 2.3: Example of diurnal wind speed pattern
data from Weibull to one that is shaped closer to Gaussian. Seasonal non-stationarity
could be removed by dividing the data into months of a year and repeating this technique
for each month.
The technique starts with a power transformation to change the shape of the distri-
bution to resemble a Gaussian distribution.
Mn = (Un)m (2.3)
where Mn is the transformed wind speed data and Un is the original wind speed data.
The value for m can be selected by using the skewness statistic, Sk, which is a mea-
sure of the symmetry of a distribution. Sk is zero when the distribution is completely
symmetrical, and it can be calculated with equation (2.4).
Sk =N∑
n=1
(Mn−Ms
)3
N(2.4)
where M is the mean of Mn, s is the standard deviation of Mn, and N is the number
of data points. Since Gaussian distribution is symmetrical, m can be selected by finding
the skewness that is closest to zero.
Chapter 2. Wind Forecasting 14
Figure 2.4: Example of Weibull wind speed distribution
Next, the diurnal non-stationarity is removed from the transformed wind speed data
Mn. The data Mn is divided into 144 bins to form Md,t, t ∈ [1, 144], for the 144 ten
minute periods in a day. The mean and standard deviation of each period is calculated,
and saved in µt and σt. Then the data has the mean removed and divided by its stand
deviation using (2.5).
M∗d,t =
Md,t − µt
σt(2.5)
Using an example dataset from NREL, a comparison of the probability mass function
(PMF) of the wind speed data before and after the process is shown in Figure 2.5.
Once the wind speed data have been standardized, it is then used to estimate the
parameters of the ARMA model. armax, a built-in tool from the MATLAB System
Identification Toolbox, is used for the parameter estimation. A different tool, predict,
from the same toolbox is then used to predict future data points. predict receives real-time
wind speed data on a ten minute basis that have also been transformed and standardized,
and uses the estimated ARMA model to forecast future data points. The forecasted
data can be converted back into real wind speed time series by applying the inverse
of the standardization technique, at which point, the wind speed is converted to wind
Chapter 2. Wind Forecasting 15
Figure 2.5: PMF of wind speed data before (left) and after (right) transformation and
standardization process
power through the power curve. Finally, the wind power forecast is post-processed to be
within the range of [0, P rated] of the wind turbine. The entire wind forecast process is
summarized in the flowchart in Figure 2.6.
The wind speed and power dataset is divided into two parts. The first part is used to
estimate the parameter of the ARMA model, and the second part is used to validate the
performance of the ARMA model. Validation is performed by comparing the predicted
wind output to the historical output and quantified using root mean square error (RMSE).
RMSE =
√∑Ni=1(Pi,measured − Pi,predicted)2
N(2.6)
The selection of the order of ARMA has intentionally been ignored until now. Nu-
merous techniques have been proposed in the literature to identify the order of ARMA
[17–19]. However, during validation of the ARMA model, it was discovered that the
order did not have a large impact on the performance of the ARMA forecasting program.
Table 2.2 shows the RMSE of different orders of ARMA for an example site from the
NREL dataset. The RMSE values are the power output error as a percentage of the rated
power for an one hour ahead forecast. The RMSE of the different orders differed by less
than 1%, and there is no substantial improvement with higher order ARMA models.
This result is support for the underlying stochastic process being a low order process.
Therefore, ARMA(1,1) model was used.
Chapter 2. Wind Forecasting 16
Historical wind speed and power data
Power transformation on wind speed data
Remove diurnal non-stationarity
Estimate arma parameters with MATLAB
Predict future data points with MATLAB
Real-time wind speed data
Power transformation on wind speed data
Remove diurnal non-stationarity
Inverse transform data back to wind speed
Wind speed to wind power conversion
Restrict data to between [0, P_rated]
Predicted wind power
Figure 2.6: Wind forecast process using ARMA
Similar results have been observed for many other sites from the NREL dataset, but
since wind profile could be very different depending on the local terrain, it is possible that
higher order ARMA can significantly outperform lower order ARMA for a particular site.
In those cases, one of the proposed techniques from literature [17–19] should be used to
identify the proper model order.
Chapter 2. Wind Forecasting 17
Table 2.2: RMSE of Different ARMA(p, q) for a one hour ahead forecast
q = 1 q = 2 q = 3 q = 4 q = 5
p = 1 10.36 10.31 10.31 10.32 10.35
p = 2 10.32 10.36 10.35 10.31 10.35
p = 3 10.31 10.34 10.35 10.32 10.33
p = 4 10.33 10.31 10.34 10.35 10.34
p = 5 10.34 10.33 10.31 10.34 10.35
Chapter 3
Estimating Wind Uncertainties
3.1 Analyzing Forecast Error
Studies in the literature have looked at statistical analysis of the wind forecast errors and
attempted to find a good estimate of the error for future predictions [20]. Some studies
have tried to model the error distribution as a Gaussian distribution, but this is a gross
approximation, as it is shown in Figure 3.1. The histograms of the wind power forecast
error is plotted for four different forecast time horizons, ranging from one hour to eight
hours ahead. It is clear from a visual examination that the shape of the distribution
is very different from a Gaussian distribution. In addition, the error distribution has a
finite support, limited by the power rating of the wind turbine, which cannot be modeled
with the infinite support of the Gaussian distribution.
Another parametric distribution that has been used to model the error distribution
is beta distribution [20]. Beta distribution is able to approximate the data more closely,
but it is not able to accurately model the tail of the distribution [20].
A different way of approaching this is to not use a parametric distribution to fit the
error data, but to generate an empirical PMF from the historical error data, as it was
done in [21]. The generated distribution can precisely model the error data as long as
18
Chapter 3. Estimating Wind Uncertainties 19
Figure 3.1: Forecast error histograms for four different forecast time horizons
sufficient amount of training data is used. It has the benefits of avoiding making an
assumption about the shape of the error distribution.
In addition, the forecast error data is dependent on the performance of the forecasting
program; as the program improves over time, the error data will also change over time. A
model that fits the data today cannot be guaranteed to still fit in the future. This issue
can be easily solved when using the empirical PMF method. By limiting the training
data set to only the more recent periods, the error distribution automatically adjusts as
the forecasting program changes.
Using the empirical PMF forecast error distribution, one can estimate the error by
associating a probability to the magnitude of the expected error. For example, if one
wants to be 90% confident that the error is within a certain range of the forecast, how
large should that range be. The probability is defined by the term confidence level and
Chapter 3. Estimating Wind Uncertainties 20
the range around the forecast is defined by the term confidence interval (CI). The CI
associated with a confidence level α is given by (3.1).
CI(α) , [PLB(α), PUB(α)] ∈ R (3.1)
such that Pr(Pmeasured − Ppredicted ≤ PLB(α)) =1− α
2
Pr(Pmeasured − Ppredicted ≤ PUB(α)) =1 + α
2
where PLB: lower bound of confidence interval
PUB: upper bound of confidence interval
Pr(x): probability of x occurring
To calculate the CI, the empirical PMF is first converted to an empirical cumulative
distribution function (CDF), as it is shown in Figure 3.2.
Figure 3.2: Empirical CDF
By restricting the confidence level α to 90% of the empirical CDF, the corresponding
CI can be quantified, as it is shown in Figure 3.3. In this particular example, the CI is
−20% to 15% of the rated power output of the wind turbine.
An example of the power measurements, predictions, and CIs in the time domain is
shown in Figure 3.4.
Chapter 3. Estimating Wind Uncertainties 21
Figure 3.3: Determining CI for a 90% confidence level
3.2 Refining Estimate
The estimation method outlined above is a very primitive first approach, further refine-
ments could be added to increase the accuracy of the estimation.
3.2.1 Weather Stability
The first refinement is using the fact that the expected forecast error is dependent on the
stability of weather conditions. When the weather is relatively stable, the expected error
is small, and when the weather is relatively unstable, the expected error is large [22]. By
distinguishing between these different weather conditions, the estimated CI can be made
more accurate.
To do this, a metric that can identify the weather condition is needed. Ideally,
the metric would be based on the predictions of multiple NWP programs. Since NWP
simulates the physical conditions of the atmosphere, when the predictions from multiple
NWP programs diverge from each other, it is usually due to the weather being unstable
and difficult to predict [23]. However, in the case multiple forecasting program are not
Chapter 3. Estimating Wind Uncertainties 22
Figure 3.4: Example of wind power measurements, predictions, and CIs for a 90% confi-
dence level
available, a metric which uses multiple forecasts from different time horizons of the same
forecasting program could be used instead [22]. The metric measures the difference
between the most recent forecast and all previous forecasts for the same time t, and
when the difference is large, the weather is assumed to be unstable.
The idea is illustrated in Figure 3.5. The predictions from time horizons 1h, 2h, and
3h ahead are fairly close at the left side of the figure, and the error between them and
the measurement is also small. The predictions are farther apart at the right side of
the figure, and the error between them and the measurement is now larger. This metric
will be used in this thesis and will subsequently be referred to as risk index (RI). RI is
calculated using the 2-norm of the differences between the most recent forecast and all
previous forecasts, see (3.2).
Chapter 3. Estimating Wind Uncertainties 23
Figure 3.5: Power measurement and predictions from different forecast horizons
RI(t) ,
√√√√ N∑i=2
(Ppredictedt−1− Ppredictedt−i
)2 (3.2)
where N is the number of forecasts for time t.
Based on its values, the RIs are divided into three categories: low risk, medium risk,
and high risk. For each category, the CI estimation method based on empirical CDF is
used.
3.2.2 Ramp Events
A ramp event such as the one depicted in Figure 3.6 tend to have many wind output
measurements outside of the CI based on RI alone. Measurements are outside of the CI
because a ramp event can be very sudden, and might not have any indication in the past
data points. Since ARMA makes predictions based on linear extrapolations of past data
points, if the past data points show no signs of a ramp event in the future, then ARMA
will fail to predict the ramp event. The ARMA forecasts from different forecast horizons
will all fail to predict the ramp events, and this means the RI metric might not reflect
a high risk period for the event. Other statistical forecast programs that uses past data
Chapter 3. Estimating Wind Uncertainties 24
points will exhibit the same problem. For example, in [24], the five algorithms tested
each have a lag between the predicted and the measured ramp, indicating they suffer the
same problem.
Figure 3.6: During a ramp event, many measurements are outside of a 95% CI estimated
using RI
To increase the accuracy of the CIs, an additional refinement is proposed in this thesis
in which ramp events are considered separately. A ramp event is deemed to have occurred
if the change in wind power output over a specified period of time is larger than a certain
threshold. In this thesis, the definition of a ramp event is based on the method proposed
in [25] and calculated using (3.3).
RAMPt = |mean{pt+h − pt+h−N}|, h = 1, . . . , N (3.3)
where pt is the power measured at time t, N is the number of power measured, and
RAMPt is the metric used to decide if a ramp has occurred. The equation calculates
how much the power measurements change in the next N data points. The values N =
12 and RAMPt > 0.4 work well for the example used in this thesis, but these values will
depend on the time granularity of the data and the wind profile of the site.
Chapter 3. Estimating Wind Uncertainties 25
The proposed method to account for ramp events is to estimate the confidence interval
of ramp events separately. When creating the empirical CDF from historical data, all
forecast errors within some time period of a predicted ramp event are marked as possible
ramp periods (PRP). The size of the time period to be marked as PRP should depend
on the typical wind profile of the wind farm site, and the value that works well can be
determined from past experience. A four hour span was found to work well for the site
used in this thesis. Once marked, all PRPs are separated from the non-PRPs, and an
empirical CDF is created for the PRPs to model the error that exists during ramp events.
Non-PRPs are processed the same way as before, using RI to divide the forecasts into
different risk categories.
During real-time operation, the most recent NWP forecast can be used to predict
ramp events; since NWP simulates the physical conditions of the atmosphere and is
capable of forecasting ramp events. Ramps predicted by NWP often have a temporal
error [9], which is accounted for by the four hour span around the ramp event.
The result of the modified method is shown in Figure 3.7, with the predicted ramp
located after the 3h mark. The figure depicts the same event as Figure 3.6. In Figure 3.7,
after the 1h mark, the CI noticeably increased to account for the predicted ramp. The
performance of the modified method versus the RI method is shown in Table 3.1. The
table contains the percentage of measurements that are outside a 95% CI with a month
of data. With the RI method, the error rate is smaller for non-PRPs and much higher
for PRPs. The modified method has a higher error rate when no ramp event is close
by, but still remain less than 5%. More importantly, it has a much more accurate error
rate around ramp events. The error rate for the PRPs is more important, because of the
large errors that are usually associated with these events. If the error estimate is wrong
for these time periods, then the operators could be misled to make incorrect decisions.
The entire error estimation process is summarized in the flowchart in Figure 3.8. This
concludes the error estimation part of the thesis. By repeating the estimation procedure
Chapter 3. Estimating Wind Uncertainties 26
Figure 3.7: 95% confidence level for the same ramp event as Figure 3.6 using the modified
method. The CI noticeably increases at the 1h mark to account for the predicted ramp
Table 3.1: RI vs modified method: percentage of measurements that are outside a 95%
CI
RI Method Modified Method
Non-PRP data points 3.99% 4.38%
PRP data points 13.41% 4.75%
at each wind site on the system, the CI of all the wind sites on the entire system can be
calculated. The next part of the thesis will propose a method of quantifying the effects
of the total uncertainties on the system.
Chapter 3. Estimating Wind Uncertainties 27
Historical power measurement and prediction data
Identify PRPs
Calculate RI for non-PRP data
Separate data into 3 RI bins and a PRP bin
Create empirical CDF for each bin
Real-time prediction with different time horizons
Identify PRPs
Calculate RI for non-PRP data
Estimate error for confidence level for each bin
Restrict CI to between [0, P_rated]
CI for wind power prediction
Use estimated error from corresponding bin to calculate CI
Divide into 3 RI bins and a PRP bin
Figure 3.8: Wind forecast CI estimation process
Part II
Quantifying Impact of Wind
Forecast Errors
28
Chapter 4
Modeling System Response to
Forecast Errors
4.1 Overview of Power System Operations
A very short introduction to power system operations is given here in order to gain
insights into the effects of forecast errors on the grid. Variable generations such as wind
are assumed to be nonexistent in the description below.
Electrical energy that is generated needs to be consumed instantaneously, because for
all intends and purposes, energy storage is nonexistent on the grid. However, some of the
power generators, such as coal or nuclear, require a long time to start or stop the units
and to change the outputs of the units. As a result, generations need to be carefully
planned to match with the daily demand curve. The entire process starts one day before
the generators are needed and ends during the real-time operation of the grid [26].
In Ontario, the generation units are planned to be on or off one day in advance in
a process called unit commitment. Using the load forecast data, the unit commitment
process determines which generator need to be on at what time and for how long. The
generator operators use the results of the unit commitment to schedule the units hours
29
Chapter 4. Modeling System Response to Forecast Errors 30
before the units are actually needed. During real-time operation, when the realized load
deviates from the load forecast, the difference is corrected through two mechanisms. The
first mechanism is operating reserve (OR), in which the dispatchable generators that
have already been committed through the unit commitment process can be dispatched
to match the realized load. Economic dispatch occurs every five minutes in Ontario, so it
can correct any generation-load mismatch on a five minute time granularity. The second
mechanism is frequency-response reserve (FRR), in which some generators are controlled
by automatic generation control (AGC) to regulate the frequency of the grid. AGC can
change the output of the generators on a time scale of seconds, thus it is able to correct
the remaining generation-load mismatch within the five minute interval.
With the introduction of wind generation, uncertainties are added to the system op-
erations. As mentioned earlier, wind forecasts could reduce the amount of uncertainties,
but never completely eliminate them. The wind forecasts could be used in the unit com-
mitment process, and any deviation from the forecast during real time operation would
be compensated by the two mechanisms mentioned earlier. This means that an increasing
amount of OR and FRR are needed to maintain the reliability of the grid [27]. Refer-
ence [27] has shown that the required increase in FRR is low compared to the required
increase in OR. In the same study, it was noted that the required increase for OR of
longer time scale is bigger. Thus, the adequacy of OR on the system will be important
for reliability and security, and that adequacy is investigated in this thesis as a way to
quantify the impact of the wind forecast errors.
4.2 Illustration of System Security Issues
Given that the OR is dispatched to accommodate the wind forecast errors during real-
time operation, any analysis attempting to quantify the impact of forecast errors must
realistically model the dispatching of the OR in order to get an accurate assessment. In
Chapter 4. Modeling System Response to Forecast Errors 31
addition, the transmission network must be modeled in order to ensure the OR will not
be constrained by the network, which are increasingly being operated closer to its full
capacity.
A two-bus system shown in Figure 4.1 is used to illustrate these two principles. The
system has one transmission line with a line rating of 225 MW and the load is fixed at
250 MW. GEN1 is a baseload unit rated at 250 MW and GEN2 is a peaking unit rated
at 20 MW. Economic dispatch of the system results in GEN1 being dispatched before
GEN2.
Limit: 225 MWLoading: 200 MW (89%)
GEN1
200 MW 0 MW 250 MW 50 MW
Bus 2Bus 1
GEN2
(a)
Limit: 225 MWLoading: 225 MW (100%)
GEN1
225 MW 5 MW 250 MW 20 MW
Bus 2Bus 1
GEN2
(c)
Limit: 225 MWLoading: 230 MW (102%)
GEN1
230 MW 20 MW 250 MW 0 MW
Bus 2Bus 1
GEN2
(d)
Limit: 225 MWLoading: 230 MW (102%)
GEN1
230 MW 0 MW 250 MW 20 MW
Bus 2Bus 1
GEN2
(b)
Figure 4.1: Two-bus system to demonstrate the analysis of impact of forecast errors. a)
predicted wind output of 50 MW, base case; b) 30 MW deviation from forecast, fixed
participation factor; c) 30 MW deviation from forecast, optimal dispatch; d) 50 MW
deviation from forecast, optimal dispatch
In the base case (Figure 4.1a), the generators are dispatched according to a predicted
wind output of 50 MW. GEN1 is outputting 200 MW, and GEN2 is outputting 0 MW
due to its higher cost.
If the realized wind output is 20 MW (i.e., there is a forecast error of −30 MW),
Chapter 4. Modeling System Response to Forecast Errors 32
then GEN1 and GEN2 must be re-dispatched to maintain power balance on the system.
Consider two different methods of modeling OR re-dispatch: fixed participation factors
[28] based on the base case generator schedules (Figure 4.1b), or optimal dispatch that
maintains power balance while minimizing transmission line loading (Figure 4.1c).
Participation factor, defined in (4.1), is a method of proportionally dispatching the
generators.
GENi,new = GENi,old + αi(NET LOAD) (4.1)
G∑i
αi = 1 (4.2)
The factor αi is fixed based on some metrics; in this example, the cost of the generators.
Since GEN1 is cheaper than GEN2, α1 would be 1 and α2 would be 0. This means GEN1
would increase its output to 230 MW and the system would have a line overload of 5
MW.
Conversely, with optimal dispatch (Figure 4.1c), GEN1 would be dispatched over
GEN2, but the transmission line constraints on the system is still respected. Optimal
dispatch more closely represents what system operators would do if this situation arises.
If instead the realized wind output is 0 MW (i.e., a forecast error of −50 MW), then
even with optimal dispatch of OR, a 5 MW line overload still occurs (Figure 4.1d), since
GEN2 can only output 20 MW.
This example illustrates that analysis of forecast error impacts should not use fixed
participation factors (Figure 4.1b), since this could overestimate line overloads. Instead,
an optimized re-dispatch of generators (Figure 4.1c) should be used. If overloads are un-
avoidable (Figure 4.1d), these situations should be identified and presented to operators
so they can better manage system security.
Chapter 4. Modeling System Response to Forecast Errors 33
4.3 Review of Current Methods
In the literature, there are several methods which can analyze the system in the presence
of uncertainties. References [27] and [29] quantify the OR requirement by calculating
the probability of events that would cause the OR to be insufficient. However, in these
methods the locational component of the OR is ignored, and the effect of the transmission
network is not studied.
Stochastic unit commitment has also been used to study OR requirements [30]. In
this approach, transmission between neighboring regions is modeled, but transmission
constraints within region are not considered. In addition, stochastic unit commitment
approaches limit the potential outcomes to a finite number of scenarios for computational
reasons.
Probabilistic load flow (PLF) [31, 32] is another method of analyzing the impact
of supply and demand uncertainty on system performance. By including the wind as
negative, variable loads, PLF can calculate the line flows for different scenarios of wind
outputs. However, the reliance of the PLF formulation on fixed participation factors [32]
has the potential to misidentify reliability problems on the system, as shown in the two-
bus example above.
Monte Carlo simulation has also been used to determine OR requirements [33]. Al-
though it was not modeled in [33], the method could be modified to include the transmis-
sion network and the optimal OR re-dispatch for each wind output scenario. However,
the large number of simulation runs needed to obtain accurate results prevents the use of
Monte Carlo analyses during online operations. Calculating how many runs are sufficient
for the simulation is also a fundamental concern in any Monte Carlo based analysis.
Interval analysis was applied in [34] to identify best- and worst-case scenarios with
system uncertainties, but only uncertainties in branch admittances were considered.
Chapter 4. Modeling System Response to Forecast Errors 34
4.4 Bilevel Programming Method
This thesis proposes to use a bilevel programming (BLP) formulation to find the impact
of wind forecast errors, taking into account both the output limits and locations of OR
resources on the system. It models the transmission network and optimally re-dispatches
OR to model actions that could be taken by system operators to accommodate wind
forecast errors (Figure 4.1c). This formulation can identify incidents such as Figure 4.1d,
thus it is able to highlight overloads that cannot be alleviated despite optimal dispatch
from system operators. The intended use is to assist operators in determining whether
the existing commitment schedule is sufficient to manage potential forecast errors. The
time scale of interest is between day-ahead unit commitment (12–36 hours) and real-time
dispatch (5 minutes). For example, the method could be used to identify potential prob-
lems with an existing commitment schedule on a 1–4 hours ahead time scale, coinciding
with multi-interval [35] or look-ahead [36] market dispatch.
Chapter 5
Mathematical Model Description
5.1 Nomenclature
Indices
W Number of wind farms
G Number of dispatchable generators
L Number of monitored transmission lines
Variables
∆w ∈ <W Deviations in wind plant outputs from forecast output
∆g ∈ <G Changes in outputs of dispatchable resources
∆f ∈ <L Changes in line flows
p,q ∈ <L Line overload penalty value in inner and outer optimizations,
respectively
Parameters
ΨW∈ <L×W Injection shift factors for wind plants
ΨG∈ <L×G Injection shift factors for dispatchable resources
Φ ∈ <L×L Diagonal matrix of line ratings
∆wmin,∆wmax ∈ <W Lower and upper bounds on ∆w
35
Chapter 5. Mathematical Model Description 36
∆gmin,∆gmax ∈ <G Lower and upper bounds on ∆g
∆fmin,∆fmax ∈ <L Lower and upper bounds on ∆f
fbase ∈ <L Line flows based on forecast wind conditions
XL Diagonal matrix containing the susceptance of the lines
I Incidence matrix or adjacency matrix
B Admittance matrix
5.2 Overview of Bilevel Programming
BLP, also known as static infinite Stackelberg game [37] or “max-min” problem in lit-
erature, is a two-level optimization problem, with a leader and a follower, as shown in
(5.1).
maxx
F (x, y) (5.1)
subject to y ∈ arg miny
f(x, y)
g(x, y) ≤ 0
x ∈X, y ∈ Y
where F : X × Y → <, f : X × Y → <, and X and Y are the domain of x and y,
respectively.
The leader maximizes the objective function F (x, y), over the domain of x, while
subject to the constraint that the decision variable y is the optimal decision for another
optimization, min f(x, y). As the variable x is being optimized in the top level optimiza-
tion, the variable y changes, or “reacts”, to the change in x. Thus, x is given the name
of leader’s variable and y the follower’s variable. A more extensive review of BLP can be
found in [38].
Chapter 5. Mathematical Model Description 37
5.3 Bilevel Programming Formulation
One way to formulate the analysis problem described in Section 4.2 is to find, within
the set of CIs, the wind outputs resulting in the worst effects on the system subject to
optimal OR dispatch. It can be written as a BLP in the form shown in (5.2)–(5.3).
max∆w ∈ W
L∑l=1
ql(∆w,∆g) (5.2)
subject to
∆g ∈ arg min
∆g
L∑l=1
ql(∆w,∆g)
∆gmin �∆g �∆gmax
h(∆w,∆g) = 0
(5.3)
where (5.2) represents the outer (or “leader”) optimization problem and (5.3) represents
the inner (or “follower”) decision made in reaction to the leader’s decision. Here, the
“leader” is nature, and the “follower” is the system operator, who reacts to the wind
forecast errors to minimize line overloads. h(∆w,∆g) = 0 represents the power flow
equations, W represents the domain of possible wind forecast errors, ∆gmin and ∆gmax
represent the lower and upper bounds on available OR, and ql is the objective function
to represent the effects on the system. The wind forecast error vector ∆w that solves
(5.2)–(5.3) is the forecast error that results in the biggest problem on the system, subject
to optimal dispatch of OR by the system operator to mitigate the impact of ∆w.
5.3.1 Power Flow Equation
h(∆w,∆g) in (5.3) models the transmission network and the steady-state power flow
equations. To obtain the most accurate representation of the physical network, the full
ac power flow equations should be used. This is a nonlinear set of equations, and it
needs to be solved iteratively, usually with methods such as Newton-Raphson or Gauss-
Seidel. When this nonlinear set of equations is used as constraints in an optimization
problem, the optimization becomes nonlinear and requires the use of a nonlinear solver.
Chapter 5. Mathematical Model Description 38
Nonlinear optimization is extremely computationally intensive to solve and very difficult
to guarantee global optimum without an exhaustive search.
The alternative is to use dc power flow equations, which uses several approximations
to make the power flow equations linear. The approximations are that transmission line
resistance is zero, the phase angles between the connected bus voltages are small, and the
voltages are 1.0 per unit [39]. The equations after the approximations are a set of linear
equations that model the power flow. Although the solutions are only approximate, the
use of dc power flow equations to reduce the computational burden is a fairly common
practice in power systems [40,41].
This thesis uses dc power flow equations, which makes the optimization model linear
and allows the application of linear programming techniques. The dc power flow is used
to model the changes on the system as linear sensitivities. Linear sensitivities is a measure
of how much one parameter on the system changes as another parameter changes. More
specifically, in this work, the sensitivity injection shift factor (ISF) [42] is used to model
the change in line flow when there is an injection change at one of the buses:
∆flow on line l = ISFl,b ×∆Pb (5.4)
ISF is calculated with (5.5).
ISF = XL−1 × I×B−1 (5.5)
To model the change in lines flows due to forecast error, first, a dc power flow is
solved with the forecast wind output. The resultant line flows, fbase, are the base case
line flows from the power flow solution. From the base flow values, limits on the decrease
and increase in line flows (∆fmin and ∆fmax) are calculated using the line ratings for
each line (5.6).
∆fminl = −fbase,l − ratingl (5.6)
∆fmaxl = −fbase,l + ratingl (5.7)
Chapter 5. Mathematical Model Description 39
The set of potential wind deviations, W , is defined by element-wise upper and lower
bounds on the forecast errors at each site (i.e., ∆wmin �∆w �∆wmax). The bounds are
from the CIs estimated using methods described in part I of this thesis. The uncertainty
in load could also be modeled the same way as wind, but in the case studies presented
here, it is assumed that the demand is fixed.
The ISF can be separated into ΨG
and ΨW
to model the impact of changes in
generation dispatch (∆g) and wind output (∆w) on line flows, see (5.8).
∆f = ΨG∆g + Ψ
W∆w =
[∆f1 · · ·∆fL
]T(5.8)
and the following equality constraint is used to ensure power balance in a lossless, dc
system model:
0 = 11×G ×∆g + 11×W ×∆w (5.9)
11×X ,
[1 1 · · · 1 1
]∈ <1×X
5.3.2 Objective Function
The objective could be modeled to include various actions that system operators might
take to re-dispatch the OR. For example, by including the cost of the generators in the
objective, a cheaper generator would be dispatched over a more expensive generator; or by
including the change in generator output, the generator movement could be minimized.
In this thesis, only the transmission line violations are included in the objective, but
other terms could be added to the objective very easily.
To model the transmission line violations, a convex, piecewise-linear penalty function
is defined for each line (Figure 5.1) [43]:
ql =
0 , flowl
ratingl∈ [−1, 1]
|flowl|−ratinglratingl
, otherwise
(5.10)
Chapter 5. Mathematical Model Description 40
Figure 5.1: Convex, piecewise-linear penalty function to model line violations
The penalty function is then changed to a form that corresponds to the representation
of the linear sensitivity formulation:
ql =
0 ,
fbase,l+∆flratingl
∈ [−1, 1]
|fbase,l+∆fl|−ratinglratingl
, otherwise
(5.11)
Due to the objective in (5.11) being a convex, piecewise-linear function, it is repre-
sented differently for maximization and minimization. For maximization, it is represented
with the λ formulation [44], and for minimization, it is represented with the epigraph
formulation [45].
λ Formulation
The objective is divided into three sections marked by {x1l , x
2l , x
3l , x
4l } (Figure 5.2), and
each xi value is associated with a weighting term λi. The weighting terms λi are subject
to:
λ1l x
1l + λ2
l x2l + λ3
l x3l + λ4
l x4l = ∆fl (5.12)
λ1l + λ2
l + λ3l + λ4
l = 1 (5.13)
λ1l , λ
2l , λ
3l , λ
4l ≥ 0 (5.14)
and for each line l, at most two elements of {λ1l , λ
2l , λ
3l , λ
4l } are greater than zero. Fur-
thermore, if two elements are greater than zero, then they must be neighboring elements
Chapter 5. Mathematical Model Description 41
Figure 5.2: Representation of the piecewise-linear objective using λ formulation, M is a
constant associated with the maximum overload to be considered
in the set (e.g., λ1, λ3 cannot both be > 0). This is enforced in commercial mixed in-
teger linear program (MILP) solvers through special ordered sets of type 2 (SOS2) [46]
constraints.
With these constraints, the objective ql can be expressed as:
ql = λ1l ·
M
ratingl
+ λ2l · 0 + λ3
l · 0 + λ4l ·
M
ratingl
(5.15)
Epigraph Formulation
Figure 5.3: Representation of the piecewise-linear objective using epigraph formulation.
The objective is represented by a variable, pl, constrained to the shaded region in
Chapter 5. Mathematical Model Description 42
Figure 5.3. The constraints are:
pl ≥ 0 (5.16)
pl ≥∆fl −∆fmax
l
ratingl
(5.17)
pl ≥−∆fl + ∆fmin
l
ratingl
(5.18)
(5.19)
In matrix form, the constraints can be written as:
−p � 0 (5.20)
−p Φ + ΨG∆g + Ψ
W∆w −∆fmax � 0 (5.21)
−p Φ−ΨG∆g −Ψ
W∆w + ∆fmin � 0 (5.22)
By placing the variable pl in the objective of the minimization, the minimization auto-
matically finds the bottom edge of the shaded region, which is the same as the convex,
piecewise-linear function.
5.3.3 Full BLP Formulation
With the modeling techniques above, the BLP from (5.2)–(5.3) can be written in full as:
max∆w ∈ W
L∑l=1
ql(∆w,∆g) (5.23)
subject to
∆wmin �∆w �∆wmax (5.24)
∆f = ΨG∆g + Ψ
W∆w (5.25)
Chapter 5. Mathematical Model Description 43
∆fl =4∑
s=1
λslxsl
ql =4∑
s=1
λsl ysl
1 =4∑
s=1
λsl
λsl ≥ 0 ∀s = 1, 2, 3, 4
x1l
x2l
x3l
x4l
,
∆fminl −M
∆fminl
∆fmaxl
∆fmaxl +M
y1l
y2l
y3l
y4l
,
Mratingl
0
0
Mratingl
∀l = 1, . . . , L,
Λl , {λ1l , λ
2l , λ
3l , λ
4l }
subject to SOS2 [46]
constraints
(5.26)
∆g ∈ arg min∆g
L∑l=1
pl
∆gmin �∆g �∆gmax
11×G ×∆g + 11×W ×∆w = 0
−p � 0
−p Φ + ΨG∆g + Ψ
W∆w −∆fmax � 0
−p Φ−ΨG∆g −Ψ
W∆w + ∆fmin � 0
(5.27)
As implicit assumption is made that for every possible wind deviation a feasible
solution exists, i.e., for each deviation ∆w ∈ [∆wmin,∆wmax], there exists a vector ∆g
such that the power balance equation (5.9) is satisfied. This is equivalent to assuming
that, without considering transmission constraints, sufficient operating reserves have been
Chapter 5. Mathematical Model Description 44
committed to compensate for any potential wind forecast deviation. If the forecast errors
are not considered during OR commitment and insufficient OR exists on the system, then
an additional constraint could be added to the outer optimization to ensure feasibility of
the solution.
5.4 Solving Bilevel Programming
Solving the BLP can be computationally expensive and, even with linear objective and
constraints, is an NP-hard problem [47]. Fortunately, the usefulness of BLPs in model-
ing two-stage optimizations has led to considerable research focused on finding efficient
solution techniques (e.g., see [38] for a detailed review). However, most of these tech-
niques require a long implementation time because they propose custom algorithms to
solve the BLP. There is one method that allows the use of existing optimization solvers
and is numerically efficient. The method [48] reformulates the linear BLP by using the
follower’s Karush-Kuhn-Tucker (KKT) conditions to replace the follower optimization
with a set of linear equality, inequality, and complementarity constraints. The converted
MILP problem can then be solved using one of many commercial grade MILP solvers,
which have been shown to perform well in other power systems applications [49,50].
5.4.1 Replacing Follower Optimization with KKT Conditions
The follower optimization is a linear programming problem that minimizes a piecewise-
linear objective. By converting the inequality constraints to equality constraints with
added slack variables, and assuming a fixed wind output ∆w, the follower optimization
Chapter 5. Mathematical Model Description 45
becomes:
min
∆g, z1, z2 ∈ <G;
p, z3, z4, z5 ∈ <L
L∑l=1
pl (5.28)
subject to
11×G ×∆g + 11×W ×∆w = h0 = 0 (5.29)
−∆g + gmin + z1 = h1= 0 (5.30)
∆g − gmax + z2 = h2= 0 (5.31)
−p + z3 = h3= 0 (5.32)
−p Φ + ΨG∆g + Ψ
W∆w −∆fmax + z4 = h4= 0 (5.33)
−p Φ−ΨG∆g −Ψ
W∆w + ∆fmin + z5 = h5= 0 (5.34)
z1, z2, z3, z4, z5 � 0 (5.35)
where z1 to z5 are the slack variables associated with each of the inequality constraints
defined in (5.30)–(5.34).
To convert the BLP into a MILP, the follower optimization is replaced with its KKT
conditions, which consist of the stationarity and complementarity conditions [51].
Stationarity Condition
At optimality, the stationarity conditions require the gradient of the Lagrangian with
respect to ∆g and p to be equal to zero. The Lagrangian of the follower optimization
problem is:
L(∆g,p, d, cmin, cmax, e0, emax, emin) = 1Tp + dh0 + cTminh1 + cT
maxh2
+eT0 h3 + eT
maxh4 + eTminh5 (5.36)
where (d, cmin, cmax, e0, emax, emin) are the Lagrange multipliers associated with the con-
straints (h0,h1,h2,h3,h4,h5) defined in (5.29)–(5.34).
Chapter 5. Mathematical Model Description 46
The gradient of the Lagrangian with respect to ∆g and p are equated to zero:
∇∆gL (·) = d11×G − cTminIG×G + cT
maxIG×G + eTmaxΨG − eT
minΨG (5.37)
= 01×G
∇pL (·) = 11×L − eT0 IL×L − eT
maxΦ− eTminΦ (5.38)
= 01×L
Complementarity Condition
At optimality, the complementarity conditions require the slack variable or the associated
Lagrange multiplier to be equal to zero for each of the inequality constraints:
0 � z1 ⊥ cmin � 0 (5.39)
0 � z2 ⊥ cmax � 0 (5.40)
0 � z3 ⊥ e0 � 0 (5.41)
0 � z4 ⊥ emax � 0 (5.42)
0 � z5 ⊥ emin � 0 (5.43)
where a ⊥ b represent ab = 0. By enforcing these KKT optimality conditions, equations
(5.29)–(5.34) and (5.37)–(5.43) can replace the follower optimization in (5.27).
5.5 Final MILP Form
With the follower optimization replaced by its KKT optimality conditions, the problem
becomes a single-level maximization subject to complementarity and linear constraints,
Chapter 5. Mathematical Model Description 47
and a MILP formulation of the BLP is obtained:
max
∆f ,p, e0, emin, emax, z3, z4, z5 ∈ <L;
∆g, cmin, cmax, z1, z2 ∈ <G;
∆w ∈ <W ; d ∈ <; Λ = {Λ1,Λ2, . . . ,ΛL}
L∑l=1
ql (5.44)
subject to
∆wmin �∆w �∆wmax (5.45)
0 = 11×G ×∆g + 11×W ×∆w (5.46)
∆f = ΨG∆g + Ψ
W∆w (5.47)
∆fl =4∑
s=1
λslxsl
ql =4∑
s=1
λsl ysl
1 =4∑
s=1
λsl
λsl ≥ 0 ∀s = 1, 2, 3, 4
x1l
x2l
x3l
x4l
,
∆fminl −M
∆fminl
∆fmaxl
∆fmaxl +M
y1l
y2l
y3l
y4l
,
Mratingl
0
0
Mratingl
∀l = 1, . . . , L,
Λl , {λ1l , λ
2l , λ
3l , λ
4l }
subject to SOS2 [46]
constraints
(5.48)
Chapter 5. Mathematical Model Description 48
01×G = d11×G − cTminIG×G + cT
maxIG×G + eTmaxΨG − eT
minΨG (5.49)
01×L = 11×L − eT0 IL×L − eT
maxΦ− eTminΦ (5.50)
z1 = ∆g − gmin (5.51)
z2 = −∆g + gmax (5.52)
z3 = p (5.53)
z4 = p Φ−ΨG∆g −Ψ
W∆w + ∆fmax (5.54)
z5 = p Φ + ΨG∆g + Ψ
W∆w −∆fmin (5.55)
0 � z1 ⊥ cmin � 0 (5.56)
0 � z2 ⊥ cmax � 0 (5.57)
0 � z3 ⊥ e0 � 0 (5.58)
0 � z4 ⊥ emax � 0 (5.59)
0 � z5 ⊥ emin � 0 (5.60)
Now the optimization problem can be solved using any commercial MILP solver.
Chapter 6
BLP Case Study
6.1 Experimental Methodology
The experimental results were obtained using CPLEX v12.2.0.0 [52] as the solver, which
uses a branch-and-cut algorithm for the solution of MILPs [53]. MATPOWER v4.0 [54]
was used to calculate the ISF matrices used in (5.8) and the base case (no forecast error)
line flows, fbase. The computer used to run the experiments is an Intel Core 2 Quad
Q9450 machine with 4 GB of RAM, running 64-bit Ubuntu Linux version 11.04.
Because the cases used for evaluation did not have wind generation sites specified,
wind generators were placed on the systems using the following method. First, the ISFs
for all generator/line combinations (ΨG
in (5.8)) was calculated. For each line, the
absolute values of the ISFs were summed over all the dispatchable generators:
SISFl =∑i∈G
|ISFl,i| (6.1)
where ISFl,i is the ISF for line l and the generator at bus i. Lines with relatively small
SISFl values indicate that the dispatchable generators have less control of the flow of
these lines. The lines were then sorted by the size of their SISFl, from the smallest to
the largest. Starting from the smallest SISFl, for each line l, a wind farm was placed at
the bus i with the maximum ISF value for line l (i.e., the bus where a change in power
49
Chapter 6. BLP Case Study 50
injection has the largest influence on this line’s flow). This was repeated for the line with
the next smallest SISFl value, and the process was repeated until the desired number
of wind generators were placed on the system. The intention of this approach was to
introduce wind farms at those buses which have the largest impact on the lines that are
least influenced by generator dispatch.
6.2 Case Description
The 37-bus system from [3] was used in the study, with loads, generator outputs, and
generator limits uniformly increased from their base values by 93%. This scaling was
done to simulate a heavily loaded system without any base case line violations. The
generators’ dispatch ranges (∆gmin, ∆gmax) were assumed to be 30% of the plant ratings,
based on the one-hour ramping capabilities of a typical coal power plant [55], subject to
the minimum and maximum generation output specified in the case. Eight wind farms
were introduced at the locations shown in Figure 6.1, using the method outlined above.
Each wind farm is rated at 60 MW, representing 18.6% of total generation capacity on
the system.
6.3 Results & Discussion
To evaluate the proposed method on the 37-bus system, six different bounds were consid-
ered for the per-site wind forecast error; 0, ±5, ±10, ±15, ±20, ±25, and ±30 MW; and
the forecast (base) output of each wind generator was set to 30 MW. Table 6.1 presents
the results of solving the worst-case line loading optimization problem (5.44)–(5.60) for
each set of forecast error bounds. For forecast uncertainty up to ±20 MW, the distri-
bution of OR resources on the system is sufficient to maintain power balance without
causing any line overloads. As the forecast error bounds are extended to ±25 MW, the
non-zero objective value indicates that it is no longer possible for the operator to simul-
Chapter 6. BLP Case Study 51
SLACK345
SLACK138RAY345
RAY138
RAY69
FERNA69
DEMAR69
BLT69
BLT138
BOB138
BOB69
WOLEN69
SHIMKO69
ROGER69
UIUC69
PETE69
HISKY69
TIM69
TIM138
TIM345
PAI69 GROSS69
HANNAH69
AMANDA69HOMER69
LAUF69
MORO138
LAUF138
HALE69
PATTEN69
WEBER69
BUCKY138SAVOY69
SAVOY138
JO138
JO345
LYNN138
Figure 6.1: 37-bus system from [3] with eight wind farms introduced.
taneously satisfy both the power balance and line flow constraints using dispatchable
generation. The difference in the objective value between the ±25 MW and ±30 MW
bounds (0.068 and 0.198, respectively) also quantifies the severity of the potential effects
of forecast uncertainty, which allows the system operators to make an informed decision
on the situation.
Table 6.1: Results for 37-bus System Study
Forecast error in MW (±) 5 10 15 20 25 30
Objective value (∑L
l=1 ql) 0 0 0 0 0.068 0.198
Referring back to Figure 4.1b and Figure 4.1c, one of the desirable features of the
analysis method is its ability to properly model operator reaction in determining the
Chapter 6. BLP Case Study 52
256 256 245
202
129
82
256 256 256 256251
217
70
120
170
220
270
5 10 15 20 25 30
Num
ber
of sc
enar
ios r
esul
ting
in n
o lin
e vi
olat
ions
Deviation from forecast [MW]
fixed participation factor optimal redispatch
Figure 6.2: Number of wind scenarios that cause zero line violation with optimal re-
dispatch (gray line) and fixed participation factor (black line) for the 37-bus system in
Figure 6.1
consequences of forecast errors. To determine the importance of modeling operator reac-
tion to forecast errors, the effects of extreme forecast deviations under fixed participation
factor and optimal re-dispatch were compared based on the number of cases with no line
violations. The fixed participation factor of each dispatchable generator was set propor-
tional to its remaining positive (negative) capacity, if the net wind deviation was negative
(positive). For each forecast error bound, 256 (28) scenarios were created by setting the
forecast error at each wind site to either the lower or upper error bound. Of the 256
total scenarios, the number of scenarios in which no line violations occurred is shown in
Figure 6.2.
The gap between the two lines reflects the number of scenarios in which fixed partic-
ipation factor dispatch resulted in additional line overloads. For example, with forecast
error bounds of ±20 MW, there were 54 (256−202) scenarios in which dispatch with
fixed participation factors resulted in line overloads versus zero scenarios in which op-
Chapter 6. BLP Case Study 53
50% 60% 70% 80% 90% 100%0
2
4
6
8
10
12
14
16
18
20
Improvement in objective over fixed participation factor dispatch
Num
ber o
f sce
nario
s
Figure 6.3: Improvement in objective when using optimal dispatch over fixed participa-
tion factor dispatch for the 37-bus system study with ±30 MW forecast error bounds
timal dispatch resulted in line overloads. As the error bounds were increased, the gap
between the two methods also increased. This suggests that when large potential forecast
errors are expected, i.e., situations that might necessitate an impact analysis of forecast
uncertainties, the BLP approach will be better at assessing potential reliability problems
on the system.
For the scenarios based on ±30 MW bounds, there are 39 wind output scenarios in
which optimal re-dispatch is unable to alleviate all line overloads. However, optimal
re-dispatch is still beneficial in these situations because it provides a better estimate of
the severity of the overloads associated with each forecast error scenario. Figure 6.3
illustrates the benefit of applying optimal re-dispatch rather than fixed participation
factor re-dispatch for these scenarios. In each of these scenarios, the reduction in the
objective function was greater than 50% in comparison to fixed participation factor re-
dispatch.
Chapter 6. BLP Case Study 54
Table 6.2: Wind Forecast Error Resulting in the Largest Line Overloads for ±30 MW
Forecast Error Bounds
Bus TIM138 FERNA69 GROSS69 HANNAH69
Error −30 MW +30 MW +30 MW −30 MW
Bus PAI69 DEMAR69 HOMER69 AMANDA69
Error −30 MW +30 MW −30 MW −30 MW
For the case of ±30 MW deviation, the wind scenario resulting in the maximum
objective value is presented in Table 6.2. For this scenario, there is a violation on the
line connecting bus PAI69 to bus GROSS69, which is loaded at 119.8% of its capacity.
An annotated one-line diagram of this scenario is presented in Figure 6.4. The yellow
highlights indicate a line is loaded at 90%–100% of its rating, and the red highlights
indicate that a line is loaded at above 110% of its rating.
Table 6.2 highlights a very important result. The worst wind scenario in this case is a
mix of positive and negative deviations at the various wind sites, with a net deviation of
−60 MW on the entire system. If all the deviations are either positive or negative, the net
deviation on the system would be ±480 MW. In this particular example, a net deviation
of −60 MW is causing a worse transmission overload than if the net deviation is ±480
MW. This shows that it is imperative to consider the transmission network in determining
which combination of forecast errors is most likely to cause reliability problems. In [56],
the reduced net forecasting error on the aggregated system level is mentioned as a benefit
for geographically distributed wind sites. However, the result from Table 6.2 indicates
that this benefit cannot be assumed without properly analyzing the transmission limits
that exist on the system. A transmission constrained system could potentially observe
worse problems with geographically distributed wind sites.
Chapter 6. BLP Case Study 55
SLACK345
SLACK138RAY345
RAY138
RAY69
FERNA69
DEMAR69
-30 MW
-30 MW
119%
99.8%97%
93%
92%
92%
99.6%
+30 MW
+30 MW
+30 MW
-188 MW
+55 MW
+127 MW
+40 MW+126 MW
-81 MW
-60 MW
+41 MW
0 MW
-30 MW
-30 MW
-30 MW BLT69
BLT138
BOB138
BOB69
WOLEN69
SHIMKO69
ROGER69
PETE69
HISKY69
TIM69
TIM138
TIM345
PAI69 GROSS69
HANNAH69
AMANDA69HOMER69
LAUF69
MORO138
LAUF138
HALE69
PATTEN69
WEBER69
BUCKY138SAVOY69
SAVOY138 JO138JO345
LYNN138
Figure 6.4: Worst wind outputs and line overloads for ±30 MW forecast error case
Chapter 7
Performance of BLP
7.1 Motivation
The computation time for the 37-bus system study is shown in Figure 7.1. The plot
shows the maximum, minimum, 25th, 50th, and 75th percentile of the computation time
from ten trials, for each set of deviation bounds. The results show a sharp increase in
computation time as the forecast error bounds are increased past ±15 MW due to an
increased number of lines with flows at or near their rating. This highlights a potential
disadvantage of the proposed method—because BLP solution is an NP-hard problem,
the worst-case computational requirements are very high. On the other hand, as shown
in Figure 7.1, the practical solution of BLPs using off-the-shelf MILP solvers can be fast
enough for operational use. To more fully explore the suitability of the proposed method
for online applications, additional experiments were conducted to test the performance
on larger and more complex systems.
56
Chapter 7. Performance of BLP 57
5 10 15 20 25 300
1
2
3
4
5
6
Tim
e [s
]
Forecast error [MW]
Maximum
25th Percentile
75th PercentileMedian
Minimum
Figure 7.1: Computation time statistics for the 37-bus system study.
7.2 Constant System Size, Varying Number of Wind
Farms
An increasing number of wind farms were added to the 37-bus system from Section 6.2,
chosen with the method outlined earlier, to determine the effect of system complexity
on solution time. The wind farms were assumed to have a forecast output of 20 MW
and forecast error bounds of ±15 MW. The branch-and-cut algorithm in CPLEX is also
compared to a direct vertex enumeration algorithm to show the efficiency of the branch-
and-cut algorithm. The direct vertex enumeration algorithm traverses through all the
vertices of the domain of ∆w and solves the follower optimization at each vertex. This
algorithm is based on the result of [57], which states that the optimal solution of a linear
BLP must be at an extreme point of the constrained domain of the decision variables.
Due to the assumption that there exists a feasible solution for every ∆w, the domain of
∆w is only constrained by the CIs, and the extreme points are the permutations of the
Chapter 7. Performance of BLP 58
Table 7.1: Computation Time Comparison Between CPLEX’s Branch-and-Cut and Ver-
tex Enumeration [s]
# of wind sites Branch-and-cut (CPLEX) Vertex enumeration
2 0.5 0.1
4 0.8 0.1
6 0.9 0.6
8 1.1 2.4
10 1.6 9.4
12 3.6 38.3
14 5.8 153.4
bounds on CI. The result of the comparison is shown in Table 7.1.
As expected, the computation time for the vertex enumeration algorithm increases
exponentially since the number of vertices to be checked is 2W . On the other hand, the
computation time when CPLEX is used to solve the MILP increases at a non-exponential
rate due to the efficiency of the branch-and-cut algorithm. It is clear that for systems
with a large number of wind farms, the branch-and-cut algorithm is the better choice;
for systems with few wind farms, the vertex enumeration method could be sufficient.
7.3 Varying System Size, Constant Number of Wind
Farms
To see how the algorithm performs with larger systems, it was tested with a modified
version of the IEEE 118-bus system [58]. A modified 118-bus system was used because the
original system model [59] does not contain transmission line ratings, which are needed
to properly assess the impact of wind forecast errors. Eight wind farms were added to
Chapter 7. Performance of BLP 59
Table 7.2: Performance of Solver Using Multiple Processors
# of processors # of nodes per second
1 6 461
2 12 831
4 21 647
the system at locations chosen with the method outlined previously, with forecast output
set to 30 MW and the deviation bounds varied from ±5 MW to ±30 MW for comparison
with the 37-bus results. The computation time is presented in Figure 7.2.
None of the cases above resulted in line violations, so to more fully explore the per-
formance range, we increased the potential deviation to +150 MW / −30 MW at each
of the wind sites. This resulted in line violations on the system and took 299 seconds to
solve, averaged over ten trials.
If faster solutions are needed, parallel processors could be added to reduce the com-
putation time. The CPLEX solver is designed to take full advantage of additional pro-
cessors and, to verify the performance improvement when additional processors are used,
the solver was tested by running with one, two, and four processors active. The number
of branch-and-cut nodes the solver iterated through per second is shown in Table 7.2, and
these results indicate that the computation time can be reduced by adding processors to
the system, assuming the node traversal strategy is not negatively impacted by parallel
processing.
Chapter 7. Performance of BLP 60
5 10 15 20 25 3030
40
50
60
70
80
90
100
110
Tim
e [s
]
Forecast error [MW]
Maximum
25th Percentile
75th PercentileMedian
Minimum
Figure 7.2: Computation time statistics for the 118-bus system study.
Chapter 8
Conclusion
As more jurisdictions add wind energy to their supply mix, system operators will have to
deal with the increased uncertainties associated with wind. Improving the performance
of wind forecasts will help to reduce the level of uncertainties, but never completely
eliminate them. Thus, a new analysis method is needed to address the potential impact
of wind generation.
This thesis presents a method of estimating the forecast error, without assuming a
particular parametric distribution, that can automatically adjusts its estimates as the
performance of the wind forecast improves. In addition, ramp events are known to
have large errors associated with them, and the estimated error for the events are very
inaccurate. A modification is proposed to estimate the errors for these events separately
and experimental result shows that the accuracy is greatly improved.
In the second part, this thesis develops a method of quantifying the impact of wind
forecast errors on system operations subject to optimal dispatch. It illustrates the impor-
tance of properly modeling the system operator’s response to forecast errors; specifically,
it shows that using fixed participation factor can overestimate the effects of the wind
uncertainties. A BLP formulation is applied to quantify the impact of wind forecast er-
rors on transmission line loading and identify the wind output scenario that would cause
61
Chapter 8. Conclusion 62
the worst transmission line overloads. This method incorporates both the transmission
network constraints on the system and the optimal operator dispatch to wind forecast
deviations, thus more accurately analyzes the impact of wind forecast errors. Studies
conducted on a 37-bus system showed that the benefit of smaller aggregated system fore-
cast errors cannot be assumed without considering the transmission network. This is an
important result that challenges previous work that claims benefit of smaller net forecast
errors from geographically distributed wind sites.
In addition, a method of solving the BLP with commercial optimization software, by
transforming it into a single-level MILP, has been presented. Experiments with 37- and
118-bus cases illustrate the feasibility of using this formulation for online operations, even
if significant forecast errors are to be considered. The ability of the MILP solver to fully
utilize additional processors, as shown in Table 7.2, suggests that a more in-depth study
of parallel solution methods [60] and MILP parameter tuning [61] could further reduce
the computation time.
8.1 Future Work
Future work in the area could focus on examining the spatial correlation of the wind
forecast errors at different geographic locations to reduce the search space for a particular
confidence level, adding N-1 contingency cases to perform a more comprehensive analysis
of the impact of wind, and adding generator ramping rate constraints to the optimization
problem. Additional ways to improve the solution accuracy could also be explored in
future work, such as moving from dc to linearized ac for the base case power flow solutions.
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