Estimating Wind Forecast Errors and Quantifying Its Impact ...€¦ · a) predicted wind output of 50 MW, base case; b) 30 MW deviation from forecast, xed participation factor; c)

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.


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.


Figure 1.2: Wind speed map of Ontario at a height of 100 meters [2]


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.


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


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).


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.


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


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.


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


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.


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.


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


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.


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


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).


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


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.


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


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.


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


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)


GEN1

225 MW 5 MW 250 MW 20 MW

Bus 2Bus 1

GEN2

(c)


GEN1

230 MW 20 MW 250 MW 0 MW

Bus 2Bus 1

GEN2

(d)


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),


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.


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.


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].


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.


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)


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)


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


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


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)


∆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


W∆w −∆fmax � 0


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


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


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)


W∆w −∆fmax + z4 = h4= 0 (5.33)


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).


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,


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)


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-


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


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-


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.


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.


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


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


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.


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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