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1 Integration of Uncertainty Information into Power System


Integration of Uncertainty Information into Power System Operations
Yuri V. Makarov, Senior Member, IEEE, Shuai Lu, Member, IEEE, Nader Samaan, Member, IEEE, Zhenyu Huang, Senior Member, IEEE, Krishnappa Subbarao, Senior Member, IEEE, Pavel V. Etingov, Member, IEEE, Jian Ma, Senior Member, IEEE, Ryan P. Hafen, Ruisheng Diao, Member, IEEE, and Ning Lu, Senior Member, IEEE
Abstract -- Contemporary power systems face uncertainties coming from multiple sources, including forecast errors of load, wind and solar generation, uninstructed deviation and forced outage of traditional generators, and unscheduled loss of transmission lines. With increasing amounts of wind and solar generation being integrated into the system, these uncertainties have been growing significantly. It is critically important to build the knowledge of major sources of uncertainties, learn how to model them, and then incorporate this information into decision-making processes and power system operations, for better reliability and efficiency. This paper gives a comprehensive overview on the sources of uncertainties in power systems, their important characteristics and models, and approaches for integrating uncertainty information into system operations. It is primarily based on previous works conducted at the Pacific Northwest National Laboratory (PNNL). Index Terms—Uncertainty, Power System Operations, Power System Balance, Unit Commitment, Economic Dispatch, Wind Generation Forecast Error, Solar Generation Forecast Error, Load Forecast Errors, Uninstructed Deviations.

NCERTAINTIES in forecasting the output of intermittent resources such as wind and solar generation, as well as system loads are not reflected in existing tools used for generation commitment, dispatch and market operation. The same is true for other sources of uncertainties, such as uninstructed deviations of conventional generators from their set points, generator forced outages and failures to start up, losses of major transmission lines, and frequency variations. These uncertainties can cause deviations in the system balance which require inefficient and costly last-minute solutions in the near real-time conditions. Major unexpected variations in wind power, unfavorably combined with load forecast errors and forced generator outages could cause significant power mismatches, which could be essentially unmanageable without knowing these variations in advance
This work was supported by the Energy Efficiency and Renewable Energy Office of the U.S. Department of Energy, California Energy Commission, California Independent System Operator, Bonneville Power Administration, ColumbiaGrid Corporation, and Laboratory Directed Research and Development (LDRD) grants at PNNL. The Pacific Northwest National Laboratory (PNNL) is operated by Battelle for the U.S. Department of Energy under Contract DE-AC0576RL01830. Y.V. Makarov, S. Lu, N. Samaan, Z. Huang, K. Subbarao, P.V. Etingov, J. Ma, N. Lu, R. Diao, and R.P. Hafen are with Pacific Northwest National Laboratory, Richland, WA 99354, USA. (Contact email: yuri.makarov@pnl.gov).

U

I. INTRODUCTION

[13]. In extreme cases, dispatch decisions could not be feasible because of the generators’ start up, ramping, and capacity constraints. With the growing penetration of intermittent resources, the uncertainties could pose serious risks to control performance and desired operational characteristics, as well as the reliability of a power grid. Without knowing the risks caused by uncertainties, i.e. the probability, timing and magnitude of potential system imbalances, system operators have only very limited ability to evaluate the potential problems and find solutions to mitigate their adverse impacts. Some important questions need to be addressed in counteracting the impact of uncertainties. For instance, when should one start more units to balance against possible fast ramps in the future over a given time horizon? Would the available online capacity be sufficient to balance against variations of uncertain parameters on the intra-hour and minute-to-minute basis? The need to evaluate uncertainties associated with wind and solar generation and to incorporate the knowledge into the algorithms and operating practices is well understood already. Some wind forecast service providers offer uncertainty information for their forecasts. For instance, AWS Truepower [1] and 3TIER [2] companies developed wind generation forecasting tools with built-in capability to assess wind generation uncertainty. Similar tools have been developed in Europe. In the context of the European Union project, ANEMOS, a tool for on-line wind generation uncertainty estimation based on adaptive re-sampling or quantile regression has been developed [3]. A German company, Energy and Meteo Systems, developed a tool for forecasting wind generation, assessing the uncertainty ranges associated with wind forecast, and predicting extreme ramping events [4]. Reference [5] discusses a wind generation interval forecast approach using the quantile method. Reference [6] used statistical analysis based on standard deviation to predict wind generation forecast errors. Work is underway to incorporate these uncertainties into power system operations [7], [8]. Unfortunately, in many cases these efforts are limited to wind generation uncertainties only; they ignore additional sources mentioned at the beginning of the paper1. Moreover, these approaches, while considering the megawatt imbalances, do not address such essential characteristics as ramp rate (megawatts per

1 An exception is the comprehensive tool developed by Red Eléctrica de Espa?a (REE), the Spanish Transmission System Operator [9].

978-1-4577-1002-5/11/$26.00 ?2011 IEEE

minute) and ramp duration uncertainties (minutes), required by the generators participating in the balancing process. This paper gives an overview of the sources of uncertainties in power systems, their important characteristics, models, and approaches for integrating information into system operations. The key discussion points are as follows: ? Variable generation and system load are far from being the only sources of uncertainties. Additional uncertainty is introduced by uninstructed deviations of conventional generators from their set points, forced outages of conventional generation, accidental load drops, major intermittent loads, unscheduled loss of major transmission lines, frequency variations, and other sources influencing power balance in a control area. All these sources must be accounted for. ? All sources of uncertainty interact in a complicated statistical way, so that most of the time their combined impact is reduced when compared to the sum of impacts of individual sources. At the same time, from time to time, collective impact of uncertainties can add up in an unfavorable way creating extreme system imbalances (socalled “tail events”). ? The statistical model of the uncertainties could be more complicated than it appears. It includes continuous unpredicted parameter variations and discrete sudden events (such as forced generation outages), nonparametric statistical distributions, non-stationary time dependent processes, tail events, autocorrelation and cross-correlation moments, and other external factors (such as temperature forecast errors or wind ramp prediction errors). ? Besides the capacity in megawatts, uncertainty prediction should include additional dimensions such as ramps (megawatts per minute), ramp duration (minutes), and cycling characteristics of conventional generators and energy storage facilities. Thus, the uncertainty analysis becomes multidimensional. These characteristics form a performance envelope necessary to successfully balance the system in view of uncertainties. ? Three modes of uncertainty integration are proposed: “passive,” “active,” and “proactive.” “Passive” integration is the first level of integration, which brings awareness of uncertainties into control center software tools through information visualization and alarming. “Active” integration uses the uncertainty information to modify existing grid operation functions such as unit commitment. “Proactive” integration develops new grid operation functions enabled by the uncertainty information. The paper is primarily based on the works conducted at the Pacific Northwest National Laboratory (PNNL) and is organized as follows. Section II describes power system balancing processes providing a background for the later sections. Section III discusses the sources of uncertainties, their characteristics and impacts on balancing requirements. Models that have been developed at PNNL to acquire knowledge of these uncertainties to assist operations are presented in Section IV. Three levels of integration of

uncertainty knowledge into power system operations are discussed in Section V. Section VI describes a new methodology for analyzing low probability high magnitude imbalances (tail events). Section VII summarizes the paper. II. SYSTEM POWER BALANCE AND PROCEDURES TO ACHIEVE IT An interconnected power system usually consists of one or multiple Balancing Authorities (BAs). Each BA must maintain a balance between its generation, load, interchange, and losses. The system’s conventional generation is committed and dispatched to meet the total balancing requirement, BR [MW]. BAs’ performance is judged based on control performance standards (CPS) requiring a certain degree of success in keeping the system imbalance within certain (sometimes statistically defined) bounds. The system balancing requirement BR, which is the same as net load, is the balancing job required from conventional generators, power exports, and energy storage facilities, is expressed by the formula: BR = ΔL ? ΔWG ? ΔSG ? FO ? UD + VDE ? 10 ? B ? ΔF ? TE (1) where Δ – denotes the difference between the actual and the pre-scheduled values for dispatch intervals; L – is the system load; WG – is wind generation; SG – is solar generation; FO – reflects imbalances caused by forced generation outages; UD – is the total uninstructed deviation of conventional generation units (including failure to start up); VDE – is the variability errors within a dispatch interval (or difference between the block-energy schedules and the continuous actual variation of BA, e.g., the difference between the hour ahead schedule and actual generation in Fig. 1); 10 ? B ? ΔF – is a frequency dependent term; and TE – is the time error correction term. Depending on the context and information available, the meaning and the actual presence of the terms in (1) can change. For instance, in the scheduling and real time dispatch procedures, the uninstructed deviation term, frequency dependent term, and time error term are omitted and the actual values are replaced by their forecasts. The balance in a BA is achieved through several processes including scheduling (day-ahead and hour-ahead block hour scheduling), real-time dispatch (load following or real-time scheduling), and regulation. Fig. 1 shows the relationship between these processes 0.
MW Hour-Ahead Schedule Actual Generation And Load Following Hour-Ahead Regulation Adjustment

Load Following Day-Ahead Schedule Hour-Ahead Schedule

Operating Hour

t

Fig. 1. Relationship between the scheduling, load following and regulation processes

The scheduling and real-time dispatch processes deal with certain dispatch intervals (e.g., one clock hour, 30minute or 5-minute intervals), where the power output is assumed to be equal to the average power requirement within the interval. The overall variability around these

values is caused by the forecasted values for these average requirements as well as shown in Fig. 2 0, [11], [12].
Net Load, MW 20 Minute Ramps Forecast Error Average Actual Net Load Hour Ahead Load Schedule Actual Load

Fig. 4. Impact of temperature forecast error on load [16]

t

Fig. 2. Overall uncertainty within a dispatch interval

Operating Hour

Another important issue with forecast is the forecast bias (systematic overestimation and underestimation of the system load). 2. Wind Power Forecast Errors The issues related to the statistical characteristics of wind forecast error attract significant interest from researchers and practicing engineers. A good review of the state-of-theart in this area is given in [18]. The wind power forecasting errors are sometimes simulated using the truncated normal distribution whose characteristics are determined by curve fitting 0. The truncation process is applied to reflect the natural limits posed by minimum (zero) and maximum (installed capacity minus the capacity of offline units) wind generation. This model cannot be accepted without caution because of indications that the wind power forecast error does not actually follow the normal law. There are efforts in place to propose better approximations for the error. For instance, Beta distribution has been used in [17]. In [14], a different approach based on experimental probability density functions (PDFs) is proposed instead. The approach is suitable for handling non-parametric distributions, so that no hypothesis is needed regarding the wind generation error distribution law. There is a continuing effort in place aiming to improve the accuracy of wind generation forecasting algorithms. For example, in Germany the day-ahead wind generation forecast error has been reduced to 4.5% [19]. Nevertheless, significant challenges remain with the very short term forecasts (the na?ve persistence forecast model frequently demonstrated better performance then more scientific approaches) and prediction of wind generation ramps. Wind generation forecast errors are sensitive to multiple external factors. Accurate modeling of these errors requires models that reflect these factors. In [20], a Bayesian network model is developed that reflects the influence of external factors on the wind and load forecast errors. 3. Solar Generation Forecast Errors Characteristics of the solar power forecast errors have not been sufficiently well studied yet. Nonetheless, practical needs require these models. In [21], a new model has been proposed and has been used in the California ISO 20% and 33% penetration of renewable generation studies. Variation of Solar Generation. Solar radiation (or solar irradiance) that determines the level of solar energy production at any specific location is neither completely random nor completely deterministic. Extraterrestrial (above clouds) solar radiation can be confidently predicted for any place and time interval. Solar radiation shows both yearly and daily variation. The area’s atmospheric conditions

The transitional ramps shown between the dispatch intervals in Fig. 1 and Fig. 2 are used in some systems, e.g., Western Interconnection in the U.S., to minimize the discretization errors as well as reduce generator ramping requirements. These 20-minute ramps are also applied to balancing areas’ interchange schedules. III. SOURCES OF UNCERTAINTIES AND THEIR CHARACTERISTICS Multiple sources of uncertainty contribute to the overall uncertainty facing the system. There are uncertainties contributing to the system balancing requirement BR (right hand side of equation(1)), and uncertainties associated with the market-driven generation dispatch and caused by limited knowledge of future generation bids. This paper concentrates on uncertainties of the first kind. A. Sources of Uncertainties 1. Load Forecast Errors Load forecast errors, ΔL in equation (1), contribute significantly to the overall uncertainty of the system’s balancing requirement BR. In operations, load forecasts are usually provided for the next operating day (hourly block energy schedules), next operating hour (hourly block energy schedules or schedules for average load for smaller dispatch intervals), and in real time forecast (average load for withinhour dispatch intervals, e.g., 5-10-15 minutes). The dayahead mean absolute percent error (MAPE) usually stays within 1-3% of the maximum load – see Fig. 2. It is important to mention that instantaneous error values can significantly exceed MAPE from time to time contributing to so-called tail events [15].

Fig. 3. Example of load forecast error in a real system [15].

Load forecast errors depend upon multiple factors including temperature and humidity forecast errors. This sensitivity is changing with the observed air temperature – Fig. 4 [16].

(clouds, dust storms, etc) determine the randomness of solar radiation at the ground level (also called global solar radiation). The ranges of yearly variation can be described by monthly maximum solar radiation for a sunny day, and the minimum solar radiation for a total cloudy day. The maximum and minimum solar radiation levels can be used for solar power generation forecasting as well as for forecast error simulations. The effect of clouds and other factors on solar generation varies. Minute-to-minute global solar radiation measurements Rm show that radiation varies within a one-hour period around an hourly mean value Rh. A statistically varying term (εR) can be used to simulate this process. This statistical term was set to have the same distribution as short duration variations seen in actual measurements. Therefore, it can be evaluated by using actual measurements, (2) Clearness Index (CI). If the sky is clear, solar radiation and solar power production are predictable based on the annual and daily extraterrestrial pattern. Thus, solar forecast errors are small. Solar radiation and generation forecast errors are mostly caused by clouds and other factors. These factors include: clouds (depth, water or ice concentration, and types of water particles or ice crystals), water vapor amount, and aerosol type and amount (column). The Clearness index is an index indicating what percentage of the sky is clear. High CI could mean higher global solar radiation (i.e., global solar radiation levels being closer to their extraterrestrial values) and lower forecast errors. CI is used for solar power generation forecasting. Clearness index for a given period is obtained by dividing the observed global radiation Rg by extraterrestrial global irradiation R: (3) where Rg is horizontal global solar radiation and R is horizontal extraterrestrial solar radiation. Simulating the Solar Generation Forecast Errors. Statistical characteristics of hour-ahead and real-time solar generation forecast errors are complex and depend on various factors including the extraterrestrial solar radiation annual and daily patterns, hour-to-hour clearness index, dynamic patterns of the cloud systems, types of solar generators (PV, concentrated thermal, etc.), geographical location and spatial distribution of solar power plants, and other factors. Upper and Lower Limits for Solar Generation Forecast Errors. Unlike wind generation, solar generation is limited by extraterrestrial solar irradiance levels, which change over a day. The maximum possible generation can be achieved at CI = 1, and this maximum value Pmax(t) also changes over a day following a similar mostly deterministic pattern (note that there is also an annual component in this process). Variances of the generation under these conditions can be only caused by diffused solar irradiance and ambient temperature variations. Assuming that these variances are also included in Pmax(t), the maximum solar generation during day time can be described as a function of time, and is always less than the total capacity, i.e.: (4)

where Pmax(t) is the maximum solar generation capacity. Solar power generation forecast f(t) has the relation of : (5) where the minimum solar power capacity Pmin(t) could be assumed to be zero; the maximum capacity of solar farm generation Pmax(t) is a function of time. During the night time, (6) From (5), we have, (7) may be negative or zero. where Standard Deviation of Solar Forecast Error Evaluated Using CI. Different solar generation patterns in day time and night time need to be taken into account into solar forecast errors. At night time, solar irradiance and solar generation are zero, thus solar forecast errors are zero. Sunrise and sunset time are different in different seasons at different regions. Then daily patterns of the clearness index are different. Previous years’ information regarding this matter can be categorized and used for solar forecast error evaluation. Depending on the time period and weather conditions, solar forecast errors can show different patterns, such as: 1) forecast error is zero, ε = 0, at night time; 2) forecast error is small or close to 0, ε→0, on sunny days, that is when CI→1; 3) forecast error is limited or close to zero under heavily clouding conditions, that is when CI→1;, and 4) forecast error varies in a wide range for the intermediate values of CI. Thus, the standard distribution of solar forecast errors can be described as a function of CI, eg., 0 ≤ stdmin ≤ std(ε) = f(CI) ≤ stdmax. Fig. 4 shows a possible distribution of the standard deviation of solar forecast errors depending on CI.
std(ε)

stdmax

0.5

1

CI

Fig. 4. Distribution of the standard deviation of solar forecast errors depending on the clearness index.

On a sunny day, the variation of solar forecast errors is shown in Fig. 5(a). The forecast error can be predominantly negative. If the sky is completely covered with clouds, the distribution of solar forecast error could be like in Fig. 5(b). The forecast error can be predominantly positive. Real-time Solar Forecast Model. A persistence model is used for the real-time wind forecast, but for solar generation, there are obvious incremental patterns during morning hours just after sunrise, and decremental patterns during evening hours just before sunset. Solar generation could increase or decrease dramatically in a very short time in the sunrise or sunset hours. This will cause significant ramp rate increase during these hours. The persistence model cannot address this concern. Therefore a new model based on CI is proposed in [21] to simulate real-time solar forecast.

(a) A sunny day (b) A very cloudy day Fig. 5. Distribution of solar forecast error in a very cloudy day and a very sunny day.

4. Forced Generation Outages Generator outages are addressed by specially procured contingency reserves. The imbalances caused by forced generator outages are initially mitigated by the system governor response and the AGC system, and then by committing and dispatching generation resources suitable for the intra-hour balancing purposes, or by applying load reduction schemes. It could take 5-15 minutes or even more to activate them. As a result, the system imbalances caused by forced generation outages could last for about 5-15 minutes. A schematic model for the balancing requirement is shown in Fig. 6.
AGC

- Probability of having any online unit forced out any minute. - Contingency reserve activation model. 5. Uninstructed Deviation Errors The impacts of uncertainties caused by forced generation outages, failures to start up, and uninstructed deviations of conventional generators on the system balancing requirements are frequently neglected. Nevertheless, the total uninstructed deviations resulting from their inability to follow the set points precisely could reach several hundred megawatts and may have a profound impact on the system balancing requirements. 6. Discretization Errors Discretization errors are the difference between the scheduled values within a dispatch interval and the actual minute-by-minute variations of the balancing requirement (discretization errors) – see Fig. 1. Unlike the forecast errors, the discretization errors are functions of the variability of balancing requirement and the size of dispatch intervals. B. Statistical Characteristics of Forecast Errors and Balancing Requirement Statistical characteristics of uncertainties are complicated (e.g., they can form non-parametric distributions or represent non-stationary processes) and depend upon the forecast horizon, other forecast errors (cross-correlation), time of the day, external factors, such as air temperature (load and wind forecast errors) and humidity (wind forecast errors), clearness index (solar forecast error), net load ramps, and forced generation outages (uninstructed deviations), and others. 1. Non-parametric nature Nonparametric distributions cannot be described using a standard probability distribution, e.g. the normal distribution, although sometimes they can be approximated using a standard distribution or a combination of distributions with certain limited accuracy. For instance, in 0, [11], [12], the truncated normal distribution was used to approximate distributions of wind and load forecast errors. The balancing requirement distribution could be a more difficult case. Sometimes its shape becomes essentially nonparametric. Fig. contains an example of real-life histograms of a non-parametric distribution of the regulation requirement in a large BA with wind and solar variable energy resources.

MW

Balancing Requirement

Contingency Reserve

AGC

Frequency Response

5 Minutes

10

15

Fig. 6. Balancing requirement caused by forced generation outage.

After a forced outage, the corresponding control area is subjected to a sudden imbalance which depends on the level of generation on the tripped unit, system inertia, and available frequency (governor) response. The part of the initial imbalance addressed by governor response does not contribute to the balancing requirement because of the fast recovery process involved (seconds). The AGC system in the affected control area starts to move regulating units to cover another part of the system imbalance (minutes). The job done by regulating units is addressing a part of the overall balancing requirement. Frequently the available regulating reserve is not sufficient to completely restore the system balance. The imbalance stays in the system for about 10 minutes until the contingency reserve units are started, synchronized with the system (non-spinning reserve), and dispatched to the desired level (both non-spinning and spinning reserves). Then the AGC units are moved back to completely restore the balance. A model for forced generation outage and contingency reserve activation has been developed by Washington University under a contract with PNNL [15]. The model is based on the following elements: - Forced outage rates and dispatch level of online units.

Fig. 7. Hourly regulation requirement for a real BA.

Frequently, the BR distribution exhibits long “heavy” tails indicating certain limited probability of major imbalances caused by unfortunate combinations of random factors contributing to BR in (1). Normally, the central part

of the BR distribution could be or should be balanced using the existing balancing reserves, whereas the tail part could create infrequent but significant problems because it requires balancing reserves that are not normally procured in the system. 2. Autocorrelation of Wind and Load Forecast Errors The load and wind forecast errors usually exhibit strong autocorrelation between the subsequent forecasts. Autocorrelation means that, for example, if for certain operating hour, a large positive forecast error is observed, it is likely that a similar error would be observed for the next hour. The autocorrelation can be observed not only between the subsequent errors (i-1, i), but also between errors (i-2, i), (i-3, i), …. Fig. 8 and Fig. 9 illustrate autocorrelation coefficients of real load and wind generation forecast errors (as well as those generated from error models) [14].

TABLE I CROSS-CORRELATION OF WIND FORECAST ERRORS CrossCorrelation B D E B 1 0.6047 0.1194 D 0.6047 1 0.2121 E 0.1194 0.2121 1

4. Non-stationary Nature In non-stationary processes, statistical characteristics of the process change over time. Fig.11 shows real-life examples of such processes for wind and load forecast errors [14]. The original distribution is the actual distribution of the load and wind forecast errors. It demonstrates nonstationary patterns in terms of its moving average as well as the variance. The “new sequence” results represent an effort to simulate the same distribution using a stationary model and a specially designed random number generator. The model was successfully reproducing the mean, standard deviation, and even autocorrelation of the original series, but fails to reproduce the time-varying components.

Fig. 8. Autocorrelation of original and generated load forecast errors.

Fig. 9. Auto-correlation of original and generated wind forecast errors.

3. Cross-correlation Wind forecast errors also show correlations between wind farms at different locations. A study was performed to evaluate the performance of three wind farms that are geographically far are located as shown in Fig.10 [14]. The comparison of the statistics is conducted in Table I.

Fig. 11. One-year hourly original and simulation results for load and wind forecast errors.

IV. MODELS OF BALANCING REQUIREMENT UNCERTAINTY Probability-based uncertainty analyses, so much needed in view of increasing penetration of renewables in the system, would require multiple simulations (e.g., Monte Carlo runs) with various possible combinations of the forecast errors. Existing methods of simulating forecasts errors normally assume that the forecast errors follow either a normal distribution or a truncated normal distribution [1][3], but this assumption is not necessarily true in practice. In addition, different simulated time series are generated independently from each other, without considering the correlation among them. The correlation of forecast errors for two geographically close wind farms is normally larger than the one for two faraway wind farms due to similar geography and weather patterns. The uncertainty models

Fig. 10. Geographical locations of the wind farms.

must reflect mutual dependencies and statistical interplays of various forecasts, distributed over a large territory (for instance, in various zones of an interconnection). The simulated forecasts could be misleading, if their nonparametric nature, variances, autocorrelations, and crosscorrelation moments are not reflected correctly. This deficiency could result in significant overall modeling errors while evaluating wide-area impacts of variable renewable generation on the system processes and characteristics, such as congestion, loop flows, generation reserves, system reliability (security margin), control performance (system imbalances and frequency variations) and others. A. Analytical Models vs. Empirical Distribution Models Two methods, i.e., the distribution fitting method and the empirical distribution method, are used to evaluate forecast uncertainties. 1. Distribution Fitting Method The distribution fitting method includes a hypothesis regarding a standard probability distribution of the forecast error and a fitting procedure used to find its parameters. For example, when the load and wind generation forecast errors are assumed to follow the truncated normal distribution (TND), the probability density function (PDF) of the TND is:
1 ? x?μ ? PDFN ? ? σ ? σ ? PDF TND ( x; μ , σ , a , b ) = ? b ? μ ? ? CDF ? a ? μ ? CDFN ? ? ? N ? ? σ ? ? σ ?

(3)

where ? is the mean value of non-truncated normal distribution; σ is standard deviation of non-truncated normal distribution; a and b are upper and lower limits of truncated normal distribution, i.e., x ∈ (a, b), -∞ ≤ a < b ≤ ∞; PDFN(?) is probability density function of standard normal distribution; and CDFN(?) is cumulative distribution function (CDF) of standard normal distribution. PDFN(?) and CDFN(?) are defined as:
PDF N ( x; μ , σ ) = 1

σ 2π

e

?

( x?μ )
2σ 2

2

features of the previously observed error sequences such as their standard deviations and autocorrelation functions, as well as to preserve cross-correlation properties with error series, coming from nearby regions. (Note that the observed sequences are not sufficient themselves for multivariate studies such as the Monte Carlo method simply because they are usually too short.) Starting with a known time series of error values, derived from the observed differences of the actual and forecasted values, we arrange the errors in an empirical distribution with a discrete, finite series of “bins” or equipartitions of the error values. Next, each of the error values is assigned a “state” representing the number of partition it belongs to. The time series of errors is then treated as a series of state transitions. The conditional probabilities of the state transitions are calculated and are used to generate new error sequences. To preserve cross-correlations between errors in the model, we apply the state-space methodology to several geographically contiguous error time series simultaneously. A drawback to this approach is that when two or more error time series are combined, the number of states increases proportionally to nK , where n is the number of partitions, and K is the number of simultaneously simulated error series. However, through the judicious selection of groups of the cross-correlated parameters, and sparse data structures, the algorithm can be made computationally acceptable and even effective. 1. Geometrical Illustration Fig. 12 illustrates the geometrical idea behind the model using an example of three cross-correlated forecast errors. Every state, shown as a small cube in this 3-dimensional space, is formed by combining and partitioning three distinct sequences with an equal number of points into the same number n of partitions or bins. The next state i +1 is selected based on a random draw and the conditional probabilities of transitions (defined from transition probability matrix T) from the current state.

(4)

1? ? x ? μ ?? 1 + erf ? (5) ? ?? 2? ? 2 σ ?? The analytical statistical models can then be used in (1) to simulate the resulting distribution of the system balancing requirement BR. 2. Empirical Distribution Method When the data do not follow a known probability distribution, nonparametric distribution models become more appropriate. These models make no assumptions about the form of underlying distribution and are based on empirical probability distribution. An empirical CDF approach assigns probability 1/n to each of n observations in the analyzed dataset. The CDF for any given value is calculated by adding all probabilities for the observations with values smaller than the given one in the dataset. CDFN ( x, μ , σ ) =

x(2)

xi( 2 )i

y

x (3)
i z (3)

xi

x xi(1)

i

x (1)

Fig. 12. An illustration of the state space method for three cross-correlated forecast errors.

B. State-space Load and Wind Forecast Error Models Probability-based system analyses, unless they are analytical, require multiple simulations (e.g., Monte Carlo runs) with various possible combinations of the forecast errors. In this connection, our goal is to reproduce statistical

An example is shown in Fig.12, where points in state w have a distinct probability of moving into any of the states r, u, or t, or perhaps cycling back into state w. The probabilities of moving into either state are labeled next to the arrows. Naturally, the sum of probabilities of transitions from state w into any other of the possible states will be

equal to 1. We use the transition probability matrix T to create multiple generated error sequences. These sequences will be distributed similarly to the original distribution, and will have close mathematical expectation, autocorrelation and cross-correlation moments among the errors. 2. Mathematical Formulation A more rigorous mathematical formulation for the proposed approached can be given as follows. Let { X i , i = 1,...L} be a multivariate stochastic process, where
X i is a vector of K values, X i = xi , xi ,...xi
(1) ( 2)

encountered in the balancing process. These four metrics are schematically illustrated in Fig. 13 and defined below.
MW

Capacity, MW

Ramp Rate, MW/min

Time
Energy, MWh

(

(K )

).

Suppose
Ramp Duration, min Net Load OR Load Following OR Regulation Curve

that each element of X i has a finite set of n states it can belong to 1, …, n . In our setting, the error series data is continuous. We discretize the values of each series into a finite set of states by partitioning the range of each series into n equal intervals. The intervals each form a set whose elements are used to create a Cartesian product of all the interval sets. Each element of the Cartesian product set is labeled from 1 to n and these labels form the set of states in the state space. If we treat X i as a Markov process, then the transition probabilities for transitioning from state vector

Fig. 13. Schematic illustration for the four performance metrics.

?

?

s p to state
?

vector sq is p pq (i ) = P( X i = sq | X i ?1 = s p ) . If we assume that the process is stationary, then the transition probabilities do not depend on time, p pq (i ) = p pq (i ? 1) for all i .Thus, we can discuss transition probabilities simply as p pq . For stationary Markov processes, estimation of the transition probabilities has been well-studied. For an observed sample series X i , i = 1, …, L , define
?1 (i ) δ pq =? ?0
(i)

?

if X i = q and X i ?1 = p else

(6)

Capacity (π) indicates the required minute-to-minute amount of generation or change in generation output, either up or down, to meet variations in net balancing requirements. The capacity requirement metric is calculated separately for positive and negative generation changes needed to balance the system. Ramp rate (ρ) is the slope of the ramp. This indicates the needed ramping capability of on-line generating units to meet the net balancing requirements. If the ramping capability is insufficient, extra generating units are needed to be online. Ramp duration (δ) is the duration of a curve’s ramp along the time axis. The ramp duration shows for how long the generators should be able to change their output at a specific ramp rate. Energy (?) is the integration of capacity over time and can be calculated as the area between the analyzed curve and the time axis. This indicates the energy needed to meet the net balancing requirements (either positive or negative).

Thus δ pq is the indicator that the transition from time t ? 1 to time i was from state p to state q. Then the estimate for transition probability p pq is
? pq = p

∑δ L
i =1

1

L

(i ) pq

,

(7)

which is simply the proportion of transitions from state p to state q observed in the sample. In our setting, with K series each having n states to visit, there are ( nK ) states and
( nK ) transition probabilities to estimate. Experimental results demonstrate very good accuracy of the approach. The future work should address the important case of non-stationary errors, as well as the issue of dimensionality of the problem to be solved [14].
2

1. Key Elements of Multivariate Performance Analysis To illustrate the idea of our approach, we choose any three of the four dimensions and plot them in a threedimensional space. For example, Fig. 14 illustrates a plot of three dimensions (?, π, δ) associated with each first performance envelope. Such three dimensional plots can be applied to all other combinations as well, such as ?-π-δ, ?π-ρ, ?- δ-ρ, and π- δ- ρ (π: capacity, δ: ramp duration; ρ: ramp rate; ?: energy), but the actual analysis is conducted in the four-dimensional space.

(Energy)

C. Multidimensional Uncertainty Analysis In the existing approaches, uncertainty analysis is frequently limited to just one dimension of the uncertainty problem – balancing capacity. But the capacity is not a single and sufficient descriptor of the problem. Operational performance of a power system can be demonstrated through four basic metrics, which we refer to as the “first performance envelope,” and relate to the evaluation of capacity, ramp duration, ramp rate and energy of ramps

Fig. 14. Graphical representation of (?, π, δ) dimensions in a first performance envelope.

The plot shown in Fig. 14 facilitates the following steps. a. First, choose some percentile threshold, say P%. This means that P% of all conditions will fall within the first performance envelope.

b. For each performance envelope (including net load, load following, and regulation) to be estimated, construct a bounding box such that P% of all the points in the plot are within that box as shown in Fig.14. Some percentage of the points, 100 - P%, will be left outside the box. This would mean that we are not going to balance against certain percentage of extreme situations where the components of the performance envelope exceed certain values. We then determine the dimensions of the bounding box, for instance, ?π, ?δ and ?? also shown in Fig. 14. The dimensions reflect the capacity, ramp, ramp duration, and energy requirements needed for each type of service (that is, for the net energy, load following, and regulation services). D. “Flying Brick” Method The “flying brick” method is proposed to analyze timevarying extreme requirements (the worst cases within the uncertainty range) of the look-ahead generation capacity, ramping capability, and ramp duration. The worst combinations of these parameters are found at the vertices of the flying brick. The objective is to include the capacity, ramp rate, and ramp duration requirements simultaneously and directly into the generation scheduling and dispatch processes. Fig. 15 illustrates the idea of the flying brick method. Three uncertainty ranges, i.e., capacity, ramp rate and ramp duration requirements, are represented as a threedimensional probability box, i.e. the flying brick. The blue curve shows the generation requirements that meet the expected net load. The red curve refers to actual net load, which can deviate from its expected values. Suppose t0 is the current moment. At this point, the multivariable statistical analysis is applied to forecast errors for different look-ahead intervals. The worst combinations of the three requirements shown by vertices of the probability box provide generation characteristics needed to meet system requirements with a certain level of confidence. For each subsequent time interval, the probability box is built based on threedimensional CDFs.

Building the resulting uncertainty characteristics is a repetitive process. Generation schedules, load and wind generation forecasts, and statistical characteristics of the retrospective data are updated every hour. A sliding window with a 1-hour refreshment rate is used to acquire continuously updated statistical information.

Fig. 16. Evaluation of capacity requirements.

V. EMS INTEGRATION OF UNCERTAINTIES In this section, an approach to integrate the uncertainty model with an existing EMS is presented. The integration has three levels as described below. A. Framework of Probabilistic Tool Integration Fig. 17 shows the concepts of the three levels of integration, namely passive integration, active integration and proactive integration. 1. Passive Integration (Level I) Passive integration is the initial step and the simplest way of integration to bring awareness of wind and load forecast uncertainties into a control center through visualization and alarming. In passive integration, displays with look-ahead capacity and ramping requirements are provided to the realtime operators for better situational awareness. They help operators assess balancing needs and take preventive actions to mitigate potential balancing energy deficiencies. 2. Active Integration (Level II) Active integration is more comprehensive than passive integration. Active integration uses uncertainty information to re-run existing grid operation functions such as unit commitment (UC) and economic dispatch (ED) processes for the worst-case combination of uncertainties within the specified confidence level. The tool displays warning messages about potential threats to the power system if the UC or ED procedures cannot find solutions for the worst cases. It also provides operators with advisory information regarding the actions that could be taken to avoid potential problems. Active integration does not modify the UC and ED procedures. Instead, it uses existing processes in the EMS system to check the sufficiency of balancing resources within the range of uncertain system requirements.

Fig. 15. The idea of the “flying brick” method.

E. Evaluation of Balancing Requirements Evaluation of generation requirements includes capacity requirements and ramp rate requirements. Take 1-hour resolution as an example, in Fig. 16, the blue curve corresponds to the generation schedule. The uncertainty ranges are calculated for each scheduling interval using individual statistical characteristics obtained by analyzing retrospective information for this time of the day.

1. Data Export from the EMS Environment. The probabilistic tool reads load forecast data directly from the EMS. The load forecast contains hourly mid-term load forecast data and short-term load forecast data. Wind forecast data can be accessed directly from the EMS or from a third-party forecast service external to the EMS. Actual generation and load data are also exported from the EMS.

Fig. 19. Integration of the probabilistic tool with an EMS system.

VI. TAIL EVENTS ANALYSIS [20], [22]
Fig. 17. Flowchart of EMS integration for the uncertainty model.

3. Proactive Integration (Level III) Proactive integration is the most comprehensive level of EMS integration, because it not only interacts with UC, ED and other applications in the EMS system, but also modifies these algorithms. New constraints based on uncertainty evaluations are incorporated into the UC and ED processes. For example, the uncertainty ranges of the capacity and ramping requirements can be incorporated in the UC process as part of reserve requirements as shown in Fig. 18.

Tail events refer to the situation in a power system when the imbalance between pre-scheduled generation and actual load becomes very significant. Tail events can be caused by unfavorable combination of load and wind forecast errors, and unpredicted fast load and wind ramps. These types of events occur infrequently and appear on the “tails” of the probability distribution of system imbalance; therefore, they are referred to as tail events. In this section, sources of uncertainties in system operations are modeled using an approach called Bayesian networks (BN). It is expected to be able to help system operators to determine the sufficiency of balancing capacity in the system in real time, as well as the probability of tail events. A. Bayesian Network Models Bayesian network (also called Bayes net) models are used to represent relationships among uncertain variables. The graphical representation of a BN consists of nodes and directed links or arrows. The nodes represent variables, and the arrows show the inter-dependencies between these variables. Arrows point from parent nodes to their child nodes; they show the direction of conditional dependence. Child nodes are conditionally dependent on their parents and are conditionally independent of their non-descendents given their parents. B. Building a Bayesian Network Model The proposed BN model, as shown in Fig. 20, forecasts the value or state of system imbalance (SI) for future time intervals depending on system parameters observed in the current and future intervals. The nodes representing variables with uncertainty are depicted in Fig. 20 as ovals. The model may also include decisions that might be made depending on the forecasted system imbalance. These decisions can be curtailment operations, for example, shown as rectangles. The model could also identify potential

Fig. 18

Flowchart of proactive integration.

B. EMS Integration Design A framework for integrating the uncertainty analysis tool as a standalone module into the EMS environment is shown in Fig. 19. The integrated probabilistic tool requests data from the EMS and uses the obtained uncertainty information to drive UC and ED processes. This process consists of four elements as detailed below.

outcomes resulting from the predicted system imbalance, e.g., transmission congestion problems and control performance standard (CPS) violations, which are shown as hexagons in Fig. 20. As can be seen in Fig. 20, system imbalance in the model has three primary causes: load forecast error (LFE), wind forecast error (WFE), and generator scheduling control error (SCE) (same as uninstructed deviations above, but kept as SCE here to be consistent with the original term used in [20] and [22]). All three are stochastic and have strong serial correlation. The LFE at time (t+1) is forecasted from the LFE at time (t) as well as from load, temperature and wind information at that time step. LFE is affected by diurnal and seasonal cycles, as well as meteorological events such as the passage of fronts (storms) with associated rapid changes in temperature and wind velocity. WFE at time (t+1) is predicted from WFE, wind power, and storm information at time (t). SCE is also predicted from relevant variables in a previous time step as shown in Fig. 20.
LFE: Load Forecast Error WFE: Wind Forecast Error SCE: Generation Schedule Control Error

between 700 MW to 900 MW. Therefore, the probability of a tail event is calculated by accumulating the probabilities including and beyond these two bars.
Hour 1 System Imbalance Prediction
0.35 0.30 0.25 Probability 0.20 0.15 0.10 0.05 0.00 -800 -600 -400 -200 0 200 MW 400 600 800 1000 1200

Tail event

Tail event

Fig. 21. Prediction of system imbalance for the next hour.
Hour 2 System Imbalance Prediction
0.25 0.20 Probability 0.15 0.10 0.05 0.00 -800 -600 -400 -200 0 200 MW 400 600 800 1000 1200

Tail event

Tail event

Fig. 22. Prediction of system imbalance for the second hour.

Fig.20. Bayes net model for predicting system imbalance.

The hierarchy in Fig. 20 shows the dependency relationship between nodes. Nodes for load, load forecast error, temperature, wind power, and wind forecast error observed at the current time step (t), are used in forecasting LFE and WFE in the next time step (t+1). The forecasted values of LFE, WFE and SCE are used in turn to forecast system imbalance at time (t+1). In application, observed values at the current time step (t) are entered into the DBN model to generate forecasted values for LFE, WFE and SCE at the next time step (t+1). Forecast for future system imbalance at (t+n) can be generated using the Markov model as needed. C. Bayesian Network Model Output As a proof of concept, the model shown in red in Fig. 20 has been implemented and tested [20], [22]. The focus was put on these nodes because they have the greatest impact on system imbalance and should serve well as an initial test of the feasibility of this modeling approach. Because hourly data were used to generate the model, prediction results have a time step of one hour. Fig. 21 and Fig. 22 show the predicted probability distribution of system imbalance in the next 1 and 2 hours, respectively. If the system is assumed to have a 500 MW upward regulating reserve and 700 MW downward regulating reserve, then system imbalance lower than -500 MW and higher than 700 MW indicates a tail event. In Fig. 21 and Fig. 22 the bar at -600 MW represents the interval between -500 MW and -700 MW, and the bar at 800 MW represents the interval

Fig. 21 shows that during the next operation hour, the probability of generation shortage in the provided example is 0.16%, and the probability of over generating is 1.95%. The most likely state of system imbalance is 200 MW, representing the interval between 100 MW and 300 MW. Fig. 22 shows that during the second operation hour, the probability of generation shortage is 0.08%, and the probability of over generating is 6.95%. The most likely state of system imbalance is 0 MW, representing the interval between -100 WM and 100 MW. The actual system imbalance observed in the system for this example was: Hour 0 (current hour) = 132 MW, Hour 1 = 333 MW, Hour 2 = -162 MW. VII. CONCLUSION This paper gives a comprehensive overview on the sources of uncertainties in power systems, their important characteristics and models, and approaches for integrating uncertainty information into system operations. It is primarily based on previous works conducted at the Pacific Northwest National Laboratory (PNNL). Sources of uncertainties in power systems include forecast errors of variable generation (wind and solar) and system load, uninstructed deviations of conventional generators from their set points, forced outages of conventional generation, accidental load drops, major intermittent loads, unscheduled loss of major transmission. These uncertainties are interacting in a complicated statistical manner. This paper presents modeling approaches to characterize these uncertainties and their interaction. Based on the developed uncertainty models, integration methods are developed and implemented as prototype tools for power system operations.

VIII. REFERENCES
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IX. BIOGRAPHIES
Yuri V. Makarov (SM’99) received a M.Sc. degree in computers and a Ph.D. degree in electrical engineering from the Leningrad Polytechnic Institute (now St. Petersburg State Technical University), Leningrad, Russia. From 1990 to 1997, he was an Associate Professor in the Department of Electrical Power Systems and Networks at St. Petersburg State Technical University. From 1993 to 1998, he conducted research at the University of Newcastle, University of Sydney, Australia, and Howard University, Washington, DC. From 1998 to 2000, he worked at the Transmission Planning Department, Southern Company Services, Inc., Birmingham, AL, as a senior engineer. From 2001 to 2005, he occupied a senior engineering position at the California Independent System Operator, Folsom, CA. Now he works for the Pacific Northwest National Laboratory (PNNL), Richland, WA, as a chief scientist – power systems. His activities are around various theoretical and applied aspects of power system analysis, planning, and control. Shuai Lu (M’06) received his B.S. and M.S. in electrical engineering from Tsinghua University, China, in 1999 and 2002, and Ph.D. in electrical engineering from the University of Washington in 2006. Dr. Lu is a senior research engineer at the Pacific Northwest National Laboratory (PNNL). He has broad interests in a number of areas of power engineering and has led or contributed to research and development projects at PNNL on power system security analysis, integration of renewable resources, generation dispatch optimization and PHEV and demand response technologies. Prior to joining PNNL, Dr. Lu conducted research on the design of an undersea power and communication network (NEPTUNE) to be deployed on the sea floor of northeast Pacific Ocean from Vancouver B.C. to northern California. Dr. Lu also designed digital controllers and models for static VAr compensation (SVC) devices. Nader Samaan received his BS and MS degrees in electrical engineering from the University of Alexandria, Egypt, in 1996 and 1999 respectively. He received his Ph.D. degree in electrical engineering from Texas A&M University, USA in 2004. He is currently a senior power systems engineer at PNNL, Richland, WA, USA performing research in the area of renewables integration. Prior to that, he was a power systems engineer at EnerNex Corp. for four years, performing wind integration studies, harmonics and transient analysis for wind power plants, and wind turbine dynamics molding. He was a visiting Assistant Professor at the department of electrical and computer engineering, Kansas State University during the academic year 2004-05. His research interests include renewables integration studies, power system reliability, extreme events and cascading failure analysis, harmonics and transient analysis for wind power plants, wind turbine dynamics modeling, distributed generation, artificial intelligence and intelligent optimization techniques application to power systems. Dr. Samaan is a registered professional engineer in the state of Ohio, a member of CIGRE and the IEEE Power Engineering Society where

[6] [7] [8] [9]

[10] [11]

[12] [13]

[14]

[15]

[16]

[17]

[18]

he is a member of the wind power coordinating committee and the Vice Chairman of the wind plant collector system design working group. Zhenyu Huang (M'01, SM’05) received his B. Eng. from Huazhong University of Science and Technology, Wuhan, China, and Ph.D. from Tsinghua University, Beijing, China, in 1994 and 1999, respectively. From 1998 to 2002, he conducted research at the University of Hong Kong, McGill University, and the University of Alberta. He is currently a staff research engineer at the Pacific Northwest National Laboratory, Richland, WA, and a licensed professional engineer in the state of Washington. His research interests include power system stability and control, high-performance computing applications, and power system signal processing. Kris Subbarao received his Ph.D. in Physics from Princeton University. After working in Cornell University, University of California, and the National Renewable Laboratory, he worked with several start-up companies, and most recently was a professor at Texas A&M University. His primary area of interest in recent years has been in diagnostics and optimal control of large numbers of buildings. Currently he is a Senior Research Engineer at PNNL working, among others, in the area of integration of renewables into the electricity grid and development of associated software. Pavel V. Etingov (M’05) was born in 1976 in Irkutsk, Russia. He graduated with honors from Irkutsk State Technical University specializing in electrical engineering in 1997. P.V. Etingov received his Ph.D. degree in 2003 from the Energy Systems Institute of the Russian Academy of Sciences, Irkutsk, Russia. He is currently a senior research engineer at Pacific Northwest National Laboratory (PNNL), Richland, WA. He is a member of the IEEE Power & Energy Society (PES). His research interests include stability analysis of electric power systems, emergency control, FACTS devices, wind power generation, and application of artificial intelligence to power systems. Jian Ma (S’05, M’08, SM’10) received the B.E. and M.S. degrees in 1996 and 1999, respectively. He received his Ph.D. degree in Electrical Engineering from University of Queensland, Brisbane, Australia in 2008. He was a research engineer at electric power R&D companies in China, and conducted research work at the School of Mechanical and Aerospace Engineering at Nanyang Technological University, Singapore. He is currently a research engineer at Pacific Northwest National Laboratory (PNNL), Richland, WA. His research interests include renewable energy integration, power system stability and control, phasor measurement technology, artificial intelligence and its application in power systems. Ruisheng Diao (S’08, M’09) received his B.S., M.S. degree in the Department of Electrical Engineering from Zhejiang University, Hangzhou, China, in 2004 and 2006 respectively, and the Ph.D. degree in the Department of Electrical Engineering from Arizona State University, Tempe, U.S., in 2009. Dr. Diao is currently with Pacific Northwest National Laboratory (PNNL), Richland, WA, USA, as a power system research engineer. His research interests include power system stability and control, online security assessment, integration of renewable energy and power system dynamic behavior. Ryan Hafen received his B.S. in Statistics from Utah State University in 2004, M.Stat. in Mathematics from University of Utah in 2006, and Ph.D. in Statistics from Purdue University in 2010. He currently works at Pacific Northwest National Laboratory (PNNL) as a statistical scientist. His research interests include statistical model building, visualization, massive data, time series, and nonparametric statistics.

Dr. Ning Lu (M’98-SM’05) received her B.S.E.E. from Harbin Institute of Technology, Harbin, China, in 1993, and her M.S. and Ph.D. degrees in electric power engineering from Rensselaer Polytechnic Institute, Troy, New York, in 1999 and 2002, respectively. Dr. Lu joined the Laboratory in 2003. She is a senior research engineer with the Energy and Environment Directorate, Pacific Northwest National Laboratory, Richland, WA. Her research interests are in modeling and analyzing power system load behaviors with a focus on implementing smart-grid technology on power system distribution grids. Dr. Lu has managed research projects in smart house technology development, modeling climate impacts on residential and commercial building energy consumption, wind integration studies, and system modeling for wide-area energy storage management. Dr. Lu is a senior member of the IEEE.


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