20

Probabilistic Methods for Planning and Operational Analysis

Gerald T. Heydt

Arizona State University

Peter W. Sauer

University of Illinois at Urbana-Champaign

20.1    Uncertainty in Power System Engineering

20.2    Deterministic Power Flow Studies

20.3    Monte Carlo Power Flow Studies

20.4    Analytical Probabilistic Power Flow Studies

20.5    Applications for Available Transfer Capability (ATC)

20.6    An Example of Stochastic ATC

20.7    An Example of Expected Financial Income from Transmission Tariffs

20.8    Wind Energy Resources and Stochastic Power Flow Studies

20.9    Solar Photovoltaic Energy Resources and Stochastic Power Flow Studies

20.10  Conclusions

References

20.1  Uncertainty in Power System Engineering

Probabilistic methods are mathematical techniques to formally consider the impact of uncertainty in models, parameters, or data. Typical uncertainties include the future value of loading conditions, fuel prices, weather, and the status of equipment. Methods to consider all possible values of uncertain data or parameters include such techniques as interval analysis, minimum/maximum analysis, and fuzzy mathematics. In many cases, these techniques will produce conservative results because they do not necessarily incorporate the “likelihood” of each value of a parameter in an expected range, or they might be intentionally designed to compute “worse case” scenarios. These techniques are not discussed further here. Instead, this chapter presents two techniques that have been successfully applied to power system planning and operational analysis as noted by the “Application of Probability Methods” subcommittee of the IEEE Power Engineering Society in Rau et al. (1994). They are

•  Monte Carlo simulation: analysis in which the system to be studied is subjected to pseudorandom operating conditions, and the results of many analyses are recorded and subsequently statistically studied. The advantage is that no specialized forms or simplifications of the system model are needed.

•  Analytical probability methods: analysis in which the system to be studied is represented by functions of several random variables of known distribution.

Rau et al. (1994) also noted that probabilistic load flow methods would be well suited to evaluate loadability limits and transfer capabilities under uncertainties created by industry restructuring. They also noted the needs to include uncertainty analysis in cost/worth studies and security assessment. This chapter focuses on those two analysis tools and illustrates the differences between the Monte Carlo simulation method and the analytical probability method. There has been renewed interest in the subject of uncertainty in generation with the incorporation of renewable resources. Mainly these resources are wind energy, solar energy, and concentrated solar energy. In each of these cases, uncertainty in the supply power occurs because of the variability of the wind and the stochastic nature of wind forecasts, as well as the variability of insolation (solar intensity) due to weather and man-made degradation to atmospheric transmissivity. One approach in these areas is to “bracket” the variability of the generation (e.g., high wind case, high solar case versus low wind case, low solar case). Another is to model the probability density function of the wind or solar generation, and then use various algorithms that process the probability density functions of the sources to obtain the probability density functions of the line loads and bus voltages (i.e., a “stochastic load flow study”). The stochastic analysis described makes available such information as the probability of insufficient generation, the probability of high line loads (e.g., overload), and the probability of high bus voltage (this may occur during light load conditions with high penetration of solar generation at unity power factor common to residential solar photovoltaic generation systems).

20.2  Deterministic Power Flow Studies

Conventional, deterministic power flow studies (load flow studies) are one of the most widely used analysis tools for both planning and operations (Heydt, 1996). They can imprecisely be described through three features:

•  Power flow analysis is the computation of steady-state conditions for a given set of loads P (active power) and Q (reactive power) and a given system configuration (interconnecting lines/transformers).

•  Active-power generation and bus voltage magnitudes are specified at the generator buses.

•  Power flow analysis gives a solution of system states, mainly bus voltage phase angles (δ) and magnitudes (V).

Additional complexities such as phase-shifting and tap-changing-under-load (TCUL) transformers and other devices are often added to this basic description. And of course many other “solution” quantities such as generator reactive power output and line power flows are often added.

The main method used for power flow analysis is the Newton–Raphson (NR) method, a method to find the zeros of f(X) = 0 where f is a vector-valued function of P and Q summations at each bus (Kirchhoff’s current law in terms of power), and X is the system state vector (primarily δ and V). There is normally a natural weak coupling between real power and voltage magnitude (P and V) as well as reactive power and voltage angle (Q and δ). This weak coupling can be exploited in the NR method to speed up solutions. Figures 20.1 and 20.2 are pictorials of this decoupling.

FIGURE 20.1  Conventional NR power flow study.

FIGURE 20.2  Decoupled NR power flow study.

This decoupling can be utilized to increase the speed of computing exact solutions, and also can be used to create superfast approximate linear solutions. In both cases, the inputs to the power flow solution algorithms shown are deterministic. When there is uncertainty in the input data, this can be considered using the two methods introduced earlier—Monte Carlo and analytical probability. These two methods are discussed next in the context of power flow analysis.

20.3  Monte Carlo Power Flow Studies

In Monte Carlo power flow studies, multiple power flow studies are run using different sets of input data. The various sets of input data are typically obtained from random number generators to correspond to some desired statistical distribution. For example, the true future total load of a power system (and its allocation to system buses) might be unknown, but an expected value might be known together with a variance with specified statistical distribution (i.e., uniform or normal). This problem could be a problem with a scalar random variable (random total load at one time instant with known deterministic allocation to system buses), or it could be a problem with a vector of random variables (random total load at one time instant plus random allocation to system buses). In the scalar case, this would require numerous power flow studies using numerous random samples for the total load. In the vector case, this would require numerous power flow studies using numerous random samples for the total load (perhaps one statistical distribution) plus numerous random samples for the factors that allocate the total load to the system buses (perhaps another statistical distribution). This could perhaps be defined more easily by simply considering each system bus load to be a random variable. This would require a decision of which “multivariate” statistical distribution to use (i.e., all bus load values are independent and normally distributed, or all bus loads are fully correlated and normally distributed, or something in between). In most cases, the Monte Carlo power flow studies are done for “variations” from the expected value of the random variables. As such, the expected value of the variation in bus load from the expected value would be zero. The first Monte Carlo power flow study is usually the case where all variations are equal to their expected value (zero). This is also often called the “base case.” Figure 20.3 shows the Monte Carlo simulation method for considering uncertainty in the input data.

The statistical description of the distribution is typically a histogram. This is a plot of the value of the output variable on the horizontal axis and the number of times that value occurs on the vertical axis. The horizontal values are grouped into “ranges” of possible values (i.e., all values from 0.1 to 0.2, 0.2 to 0.3, etc.). These histograms are similar in concept to the probability density function of analytical methods. There are also numerical descriptions (moments) of the statistical distribution that can be computed—i.e., mean, variance, skewness, kurtosis, etc. One problem with the Monte Carlo method is in determining the number of samples that are sufficient to properly represent the variation in the uncertain parameters. For example, are 100 samples sufficient to represent all possible variations in the total system load? How about 1000? One way to look at this is—“The deterministic solution is just one sample, so anything more than one is better than the deterministic solution.” But, what is important in the choice of samples is the specification of what is most likely to represent the most likely possible values of the uncertain data. This normally requires a large number of samples. Another problem with Monte Carlo methods is the large computational burden required to study a large number of samples.

FIGURE 20.3  Monte Carlo power flow study.

Multiple power flow studies can also be used to compute the variation of system bus voltages (and all other output quantities) throughout the day, week, or year as the load varies. For example, if there was an interest in computing the line flows for every hour of the day for 1 year (assuming all load and generation data are known for each hour), this would require 8760 power flow studies (1/h). It is important to realize that although this is not really a Monte Carlo simulation as the variation of the load data is not from a random phenomenon (i.e., daily, weekly, and seasonal variations in load are somewhat predictable), it is computationally the same as running 8760 “samples” and observing the distribution of the output quantities. The “statistical distribution” would give an indication of the percentage of time throughout the year during which a line flow was in a certain range. If there was an interest in considering the uncertainty in the total load variation throughout the year, some mechanism for specifying the “variations” from the base case would need to be given for each hour of everyday. Presumably, these variations would take the form of random samples created from some known statistical distribution. A Monte Carlo simulation of this could easily result in the need for over one million power flow studies. And, if the uncertainty of the data was extended to random samples of load at each bus for each hour, the result could be the need for over one billion power flow studies.

20.4  Analytical Probabilistic Power Flow Studies

Because it is very difficult to perform analytical probability analysis for large nonlinear systems, the most probabilistic power flow studies exploit two properties of power systems that lead to a linear, smaller model. First, the decoupling that was discussed above allows the real power and voltage angle to be completely separated from the reactive power and voltage magnitude in computation. This not only reduces the size of the problem, but also reduces the accuracy and loses voltage information. Second, the fact that “small” changes in bus real power injections result in “small” changes in bus voltage angles leads to a linear approximation as follows:

$\Delta {\delta }_{\text{bus}}=T_1\Delta P_{\text{bus}}$

(20.1)

So, for an initial power flow solution giving the bus voltage angle vector ${\text{δ}}_{\text{bus}}^0$ for an initial bus real power injection vector, $P_{\text{bus}}^0$, an approximate power flow solution for a problem with $P_{\text{bus}}^1=P_{\text{bus}}^0+\Delta P_{\text{bus}}$ is given by ${\text{δ}}_{\text{bus}}^1={\text{δ}}_{\text{bus}}^0+T_1\Delta P_{\text{bus}}$.

It is important to point out that the vector of bus real power injection changes does not include the injection change at the swing (or slack) bus of the power flow study. And, the vector of bus voltage angles does not include an entry for the swing (or slack) bus of the power flow study. This is because the swing (or slack) bus is chosen as the fixed-angle reference for the power flow study and also provides the balance in real power for any specified condition. In this manner, the vector of bus real power injection changes could include a single entry and the result of this injection change would automatically be offset by a nearly equal and opposite injection change at the swing bus.

This linear analysis can be extended to approximate the resulting change in real power line flows as

$\Delta P_{\text{line}}=T_2\Delta {\delta }_{\text{bus}}$

(20.2)

This combination of linear approximations gives the traditional “generation shift distribution factor,” which is also the “power transfer distribution factor” for power transfers from a bus to or from the swing bus (Heydt, 1996):

$\Delta P_{\text{line}}=T_2{T}_1\Delta P_{\text{bus}}=T\Delta P_{\text{bus}}$

(20.3)

This relationship is the heart of the linear power flow, which enables analytical probabilistic methods. One reason a linear computation is important is that the linear sum of jointly normal random variables is also jointly normal (Papoulis, 1965). This makes the computation of the statistical distribution of the output variables very easy to compute from the specified statistical distribution of the input variables. For example, if there is some uncertainty in the value of the loads, an initial power flow study could be done using the expected value of the loads. This would be a deterministic power flow study. To account for the uncertainty in load, an analytical probabilistic power flow could be done as follows:

Step 1. Specify the statistical distribution of the variation of load (ΔP_bus)—for a scalar (total load) this might be a zero mean, normal distribution with some specified variance. For a vector of zero-mean jointly normal load variations, the statistical distribution would be specified by also giving the covariance between each load variation. A covariance of zero would mean that the two uncertain jointly normal variations are statistically “independent.” A covariance of ±1.0 would mean that the two jointly normal variations are fully correlated (linearly related). The covariance matrix C_load contains the complete statistical description of all the load variations through the variances (diagonal) and covariances (off diagonal).

Step 2. Compute the statistical distribution of the variation of line real power flows. When the variations in loads are assumed to be zero-mean jointly normal, the complete description of the statistical distribution of the variation of line real power flows is found easily as (Papoulis, 1965)

$C_{\text{line}}=T_3{C}_{\text{load}}{T}_3^{\text{t}}$

(20.4)

where “t” denotes matrix transposition. The mean value of the variation in line real power flows is taken as zero because the mean value of the bus real power injection variations was assumed zero and the two are assumed to be linearly related. Now, the fact that the variations in line real power flows are not really linearly related to the bus real power injection variations means that these statistical distributions will not be exact. Figure 20.4 shows how the analytic probabilistic power flow study is done.

Stochastic power flow is another term that has been used for analytical probabilistic power flow. Since stochastic processes are statistical processes involving a number of random variables depending on a variable parameter (usually time), this terminology has been adopted as equivalent. One of the first publications on this method was of Borkowska (1974). The subject continues to attract interest (Vorsic et al., 1991).

FIGURE 20.4  Analytic probabilistic power flow study.

20.5  Applications for Available Transfer Capability (ATC)

In order to realize open access to electric power transmission networks and promote generation competition and customer choice, the Federal Energy Regulatory Commission requires that ATC be made available on a publicly accessible open access same-time information system (OASIS). The ATC is defined as a measure of the transfer capability, or available “room” in the physical transmission network for transfers of power for further commercial activity, over and above already committed uses (NERC, 1996). The ATC is defined by NERC as the total transfer capability (TTC) minus the transmission reliability margin (TRM), capacity benefit margin (CBM), and existing power flows:

$\text{ATC}=\text{TTC}-\text{TRM}-\text{CBM}-\text{Existing}\text{power}\text{flows}$

(20.5)

where the TTC is the total amount of power that can be sent from bus A to bus B within a power network in a reliable manner, the TRM is the amount of transmission transfer capability necessary to ensure network security under uncertainties, and the CBM is the amount of TTC reserved by load serving entities to ensure access to generation from interconnected systems to meet generation reliability requirements. Neglecting TRM and CBM reduces the computation of ATC to

$\text{ATC}=\text{TTC}-\text{Existing}\text{power}\text{flows}\text{.}$

(20.6)

The ATC between two buses (or groups of buses) determines the maximum additional power that can be transmitted in an interchange schedule between the specified buses. ATC is clearly determined by load flow study results and transmission limits. As an illustration, consider Figure 20.5 in which 1000 MW is transmitted from bus A to bus B. If the line rating is 1500 MW and if no parallel paths exist from A to B, the ATC from A to B is 500 MW.

In a real power system, the network and the computation of the ATC are much more complicated as parallel flows make the relationship between transfers and flows less obvious. In addition, the ATC is uncertain due to the uncertainty of power system equipment availability and power system loads. An evaluation of the stochastic behavior of the ATC is important to reduce the likelihood of congestion. The ATC is determined by power flow studies subject to transmission system limits. These phenomena are nonlinear in behavior. As discussed above, linearization can be used to estimate line flows in a power flow study model. This section presents a method of finding the stochastic ATC using a linear transformation of line power flows into ATC.

FIGURE 20.5  An example of ATC.

Increasing the transfer power increases the loading in the network. At some point, operational or physical limits to various elements are reached that prevent further increase. The largest value of transfer that causes no limit violations is used to compute the TTC and ATC. Limits are affected by the power injections at both buses A and B. This effect can be found analytically by finding the distribution factors of the lines and other components. In this context, a “distribution factor” refers to the power transfer distribution factor for line i – j and bus k, where an equal and opposite injection is automatically made at the swing bus.

The complexity of the ATC calculation is drastically reduced by linearizing the power flow study problem and considering only thermal transmission limits. The linearization is most accurate when only small deviations from the point of linearization are encountered. The use of thermal limits is justified for short transmission circuits. The transmission system linearization in this case is done using power transfer distribution factors, which are discussed above. Starting with a base case power flow solution, the addition of a power transfer from bus A to bus B impacts the flow in the line between bus i and bus j as

$P_{ij}^1=P_{ij}^0+\left(T_{ij,\text{A}}-T_{ij,\text{B}}\right)P_{\text{AB}}$

(20.7)

where P_AB is the power transfer from bus A to bus B (assumed to be positive). When (T_ij,A – T_ij,B) is positive, the transfer that results in rated power flow on line i – j is

$P_{\mathrm{AB},ij}=\frac{\left(P_{ij}^{\text{Rated}}-P_{ij}^0\right)}{\left(T_{ij,\text{A}}-T_{ij,\text{B}}\right)}$

(20.8)

When (T_{ij, A} – T_ij,B) is negative, the transfer that results in rated power flow on line i – j is

$P_{\mathrm{AB},ij}=\frac{\left(-P_{ij}^{\text{Rated}}-P_{ij}^0\right)}{\left(T_{ij,\text{A}}-T_{ij,\text{B}}\right)}$

(20.9)

Each line will have a value of P_AB,ij that represents the maximum value of power transfer from bus A to bus B without overloading line ij. The minimum of all these values would be the ATC for the AB transfer.

Once this minimum is determined, the question is, “What if there is uncertainty in the initial bus and line loadings?” As discussed above, there are at least two ways to incorporate uncertainty in this computation. The first way is the Monte Carlo method that specifies a statistical distribution of the uncertain parameter (perhaps total load) and utilizes pseudorandom number samples to perform brute force repeated ATC solutions. This could involve full nonlinear power flow studies, or faster linear power flow studies. The collection of ATC solutions that come from a large sample of total load values gives the statistical distribution of the ATC, perhaps as a histogram. If the critical line that determines the ATC of the AB transfer does not change with the uncertainty of the initial bus and line loadings, then the possible variation in ATC can be easily computed from the statistical distribution of the initial line flow. For example, if the uncertainty of initial line flows is normally distributed with zero mean and some variance, then the ATC from A to B is also normally distributed with zero mean, as it is a linear function of the initial flow in the critical line. The variance is also directly available because of the linear transformation.

TABLE 20.1 Comparison of Monte Carlo and Analytical Methods for ATC

20.6  An Example of Stochastic ATC

The following example illustrates the basic principles of stochastic ATC. The IEEE 14-bus system was studied with the base case bus and line variables as given in Christie (1999) and Pai (1979). For the uncertainty analysis by Monte Carlo methods, the bus loads were generated on the computer using 100 samples from a random number generator. The bus real power loads were assumed to be statistically independent with a normal distribution that had a mean equal to the base case and a standard deviation equal to 10% of the base case. Bus load power factor was maintained at the base case value. A comparison of the Monte Carlo results with the analytical method is given in Table 20.1.

The Monte Carlo results required 100 full nonlinear power flow studies of ATC, whereas the analytical results required a simple direct multiplication. Additional results for this example are available in Stahlhut et al. (2005).

20.7  An Example of Expected Financial Income from Transmission Tariffs

A second illustration of the two methods for considering uncertainty deals with the estimation of transmission tariff revenue because of unknown future loading. This illustration was reported also in Westendorf (2005) and Stahlhut et al. (2005). The work is presented in three parts. The first includes a study of transmission tariff revenue based on an estimated hourly forecast for 1 year. This “base case” solution provides the expected revenue for the year for that forecasted loading level. The second part includes a Monte Carlo simulation of possible variations in revenue for a forecast error of known statistical distribution. This is a repeat of the base case solution for 100 different possible cases (per hour) sampled from a normal distribution function—using total load as the random variable. This error in total load was then “distributed” to the individual buses using the base case percentage allocation. The third part included an analytical solution to estimate the expected statistical distribution of the transmission tariff revenue using linear power flow techniques.

This analysis used the IEEE 14-bus test system (Pai, 1979; Christie, 1999) and computed revenue from transmission tariffs using an assumed tariff of $0.04 per MW-mile for each transmission line. An annual income was also computed by summing the hourly incomes for the year. Because the transmission tariff is a function of the length of each transmission line, an estimate of the transmission line lengths was necessary. The total impedance of a transmission line is dependent on the length of the line. The line length is directly proportional to the transmission line impedance. The method for calculating the lengths of the transmission lines in the IEEE 14-bus test system based on the total line impedance of each line was developed as follows. A power base for the system was assumed to be S_base = 100 MVA. Using the nominal voltages, V_base, of the transmission lines and the power base of the system, each transmission line reactance value, the length of the line was calculated by assuming a conversion factor of 0.7 Ω/mile. Secondly, lines containing transformers (i.e., lines 8–10) were considered to be zero-length lines.

A load forecast taking into account hourly variations was developed for this investigation by using publicly accessible approximate historical load data for each hour of everyday for the year 2004 from the PJM Web site (PJM, 2005). The integrated hourly load data for the PJM-E area were scaled to match the system load of 259 MW of the IEEE 14-bus test system. Using the hourly load data, power flows were conducted to determine the megawatt flows on the transmission lines allowing the calculation of the each line revenue for the hour. The total revenue generated for each hour was calculated by summing all of the line revenues calculated from the power flows. While standard nonlinear power flow methods were used to compute the Monte Carlo solutions, standard linear power flow methods were used to calculate the analytical solutions.

The Monte Carlo simulations and the analytical methods were compared. Both methods provided results for the distribution of the system revenue generated for each hour. Although an entire year was analyzed, Table 20.2 lists the results from both methods for just 1 day.

TABLE 20.2 Comparison of Monte Carlo and Analytical Methods for Expected Revenue for 1 Day

Time	Mean Total Revenue ($)	Base Case Total Revenue ($)	Simulated Standard Deviation σrevenue ($)	Analytical Standard Deviation σrevenue ($)
100	390.66	390.96	12.49	9.96
200	382.23	382.95	11.28	9.76
300	378.27	381.11	12.82	9.72
400	383.13	384.06	12.29	9.79
500	394.60	393.56	13.23	10.03
600	427.99	426.22	14.47	10.83
700	478.56	479.04	14.23	12.13
800	510.09	510.85	16.94	12.91
900	505.21	506.51	14.03	12.81
1000	501.30	501.86	13.83	12.69
1100	497.15	496.49	13.93	12.56
1200	490.09	491.41	12.79	12.43
1300	494.68	490.73	13.83	12.42
1400	488.98	488.53	15.66	12.36
1500	484.86	483.91	15.24	12.25
1600	484.97	484.78	14.77	12.27
1700	502.80	504.46	14.16	12.76
1800	537.99	538.33	17.39	13.59
1900	543.86	544.99	16.76	13.75
2000	536.59	538.02	19.71	13.58
2100	527.99	527.33	14.94	13.32
2200	500.62	502.73	15.19	12.71
2300	464.40	466.21	13.47	11.81
2400	427.75	430.47	14.41	10.93

Column 1 is the time of day studied. Column 2 is the expected revenue using the Monte Carlo simulation, which created deviations from the base case using a zero-mean random number generator (100 samples per hour). Column 3 shows the revenue from the base case. The fact that these two columns are quite close is due to the fact that the deviations were sampled from a zero-mean number generator and the system behavior is nearly linear for deviations of this size. The standard deviation of the sampled distribution was 3%. This was designed to produce a sample load forecast with expected variations between plus and minus 10% of the base case. Column 4 shows the Monte Carlo sample standard deviation, and Column 5 shows the analytical standard deviation found using the assumed linear nature of the calculation and the assumed normal distribution of the forecast error.

The error in standard deviations can be attributed to several things. The linear power flow method is an approximation that is used in the analytical method but not in the Monte Carlo simulation method. The Monte Carlo method only used 100 random samples to create the sample standard deviation. This is not a large number of samples and contributes to the discrepancy in the two methods.

20.8  Wind Energy Resources and Stochastic Power Flow Studies

Wind is caused by regions on Earth that have different air pressure. Perez (2007) and Pavia (1986) are brief samples of the extensive meteorological literature in this area. The wind speed is often characterized as a random process, and one particular model used frequently is the Weibull distribution (see Figure 20.6) because this distribution is one sided (i.e., the random variable is always non-negative), and actual wind data appear to behave in this fashion. The Weibull probability density f(x) is

$f_x\left(x\right)=\{\begin{array}{ll}\frac{k{x}^{k-1}{e}^{-\left(x/\lambda \right)k}}{{\lambda }^k}\hfill & forx\ge 0\hfill \\ 0\hfill & otherwise\hfill \end{array}$

where

λ is the scale of the density

k is termed the shape parameter

The actual power contained in wind is proportional to the cube of the wind speed under some conditions, and therefore the available wind power, if viewed as a random process, does not possess a Weibull probability density. The power output of a wind turbine, however, is not simply characterized because the cubic behavior of the wind power with wind speed only holds below a certain threshold of wind speed (typically in the <50 m/s range)—and there are also nonlinear confounding factors of air temperature, wind direction, and humidity. To illustrate the complexity of the problem, this discussion suggests that wind power is indeed a random process, but one cannot rely on a simple statistical model such as a Weibull density as an accurate representation of wind power.

Having modeled the probability density function of the wind resource (active power P), it is then possible to use algorithms to calculate the probability density function of the line flows and bus voltage magnitudes in the interconnected power system. While this has been done under laboratory and experimental conditions, a full-scale commercialized application software package is still lacking.

FIGURE 20.6  A representative Weibull probability density curve to model wind speed.

20.9  Solar Photovoltaic Energy Resources and Stochastic Power Flow Studies

Similar to the preceding discussion with regard to wind resources, solar photovoltaic resources also can be modeled probabilistically, and a probability density function of the solar generation can be estimated. This generation has two components, one being completely deterministic. The deterministic component comes from the “equation of time,” which gives the sun’s position for every day of the year for any place on Earth. This equation is known to a high precision and basically the equation is a trigonometric expression. The second component is probabilistic due to weather conditions, cloud cover, and man-made smoke and haze. There are estimates of the probability density function of the stochastic component; these vary in accuracy and, invariably, are obtained for specific locations. As in the case of wind resource calculations, it is possible to process the probability density functions of the solar insolation in order to obtain the statistics of system line power flows and bus voltages. Commercial software packages that encompass all elements of solar energy uncertainty are lacking.

20.10  Conclusions

Power system planning will always have a need to consider the uncertainty of future conditions and the impact that these uncertainties have on technical and financial issues. When uncertainties can be assumed small, linear power flow methods that are needed for analytical techniques can significantly reduce computational effort. A significant element in stochastic power flow analysis is the appearance of wind, solar photovoltaic, and concentrated solar energy resources in power systems. These resources have significant uncertainty, and stochastic power flow studies will be needed to analyze the impact of these renewable resources on system operating parameters.

References

Borkowska, B., Probabilistic load flow, IEEE Trans. Power Appar. Syst., PAS-93(3), 1974, 752–759.

Christie, R., Power system test archive, 1999, http://www.ee.washington.edu/research/pstca/pf14/pg_tca-14bus.htm

Heydt, G.T., Computer Analysis Methods for Power Systems, Macmillan, New York, 1996.

Pérez, I.A., Sánchez, M.L., and Ángeles García, M. Weibull wind speed distribution: Numerical considerations and use with solar data, J. Geophysical Research-Atmospherics, October 2007, V.112, record locator D20112, doi: 10.1029/2006JD008278, 2007.

NERC, Available transfer capability definitions and determination, North American Electric Reliability Council, Princeton, NJ, June 1996.

Pai, M.A., Computer Techniques in Power System Analysis, Tata McGraw Hill, New Delhi, India, 1979.

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Pavia, E.G. and J.J. O’Brien, Weibull statistics of wind speed over the ocean, J. Climate and Applied Meteorology, 25, 1324–1332, 1986.

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Rau, N., Grigg, C.H., and Silverstein, B., Living with uncertainty: R&D trends and needs in applying probability methods to power system planning and operation, IEEE PES Rev., November 1994, pp. 24–25.

Stahlhut, J., Feng, G., Hedman, K., Westendorf, B., Heydt, G., Sauer, P., and Sheble, G., Uncertain power flows and transmission expansion planning, Accepted for presentation at the 2005 North American Power Symposium, Ames, IA, October 23–25, 2005.

Vorsic, J., Muzek, V., and Skerbinek, G., Stochastic load flow analysis, in Proceedings of the Electrotechnical Conference, Ljubljana, Slovenia, 2, 1445–1448, May 22–24, 1991.

Westendorf, B., Stochastic transmission revenues, MSEE thesis, University of Illinois at Urbana-Champaign, Urbana, IL, May 2005.