Estimation of Power System Variability due to
Wind Power
George Papaefthymiou, Member, IEEE, Jody Verboomen, Member, IEEE and Lou van der Sluis, Senior Member, IEEE
Abstract— The incorporation of wind power generation to the power system leads to an increase in the variability of the system power flows. The assessment of this variability is necessary for the planning of the necessary system reinforcements. For the assessment of this variability, the uncertainty in the system inputs should be modeled, comprising of the time-dependent stochastic-ity of the system loads and the correlated wind resources. In this contribution, a unified Monte-Carlo simulation methodology is presented that addresses both issues. The application of the method for the analysis of the wind power integration in the New England test system is presented.
I. INTRODUCTION
T
HE share of wind power in the power system energy mix is increasing in recent years. The large-scale wind power integration to the power system imposes radical system changes, namely the transition from the vertical to a hori-zontal power system structure [1]. The main characteristic of this horizontal power system structure is the connection of non-regulated power generation to different voltage levels of the system. This ubiquitous power generation leads to an increase in the variability of the system power flows, allowing reverse power flows from the distribution systems towards the transmission system, when the local generation exceeds the consumption. For system planning, it is necessary to evaluate the variability of the system power flows, in order to assess the risk of overloading the system lines and plan the necessary reinforcements [2].For this, the stochasticity in the system uncertain inputs (loads-wind generators) should be modeled. The modeling should comply to the following restrictions:
1) Time-conditioning: the system loads present a high time-dependence. This is due to the dependence of the human activities on a cyclic-deterministic phenomenon (time of day, day of week, season). This time-dependence is removed by performing the analysis separately for time-periods with similar statistical characteristics (Time-Frames Analysis [3]). This procedure corresponds to conditioning the calculation in time. The load in each time-frame (TF) can be modeled by superimposing a random noise variable to the conditional mean. Thus, given the period of consumption, all system loads can be modeled as independent normal random variables G. Papaefthymiou, J. Verboomen and L. van der Sluis are with the Electrical Power System Laboratory (EPS), Faculty of Electrical Engi-neering, Mathematics & Computer Science, Delft University of Technol-ogy (TU Delft), P.O. Box 5031, 2600 GA Delft, the Netherlands (e-mail:g.papaefthymiou@tudelft.nl).
around the expected mean value. Further, an aggrega-tion procedure may be applied in order to obtain the distribution for the whole period of concern.
2) Non-normality: In the case of wind power generation, this time-conditioning cannot be applied. The wind speed in each generation site follows a Weibull distri-bution [4]. The wind power distridistri-bution is obtained by the propagation of this wind speed distribution through the wind speed/wind power characteristic of the energy converter (wind turbine generator). In Fig. 1, the wind speed and wind power distributions for a typical wind turbine generator are presented. As may be seen, the non-linear wind speed/power characteristic of the wind turbine generator leads to non-standardized wind power distributions which present accumulation of probability masses in the zero and nominal wind power values [5]. 3) Correlated resources: the wind resources are geograph-ically correlated. Hence, the wind power infeeds from different wind parks in the system are also correlated. This correlation is high for generation sites that are geographically close to each other and remains signifi-cant for remote areas. The dependence structure has a direct impact on the aggregate wind power distribution. The arbitrary assumption of independence produces a systematic tendency to disregard the effect of the de-pendence. This generally is an underestimation of the system variability and can lead to serious pitfalls, since positive dependence between the stochastic resources leads to to an increase in the variability of the system power flows [6], [7].
In this contribution, a unified Monte-Carlo simulation (MCS) methodology is presented that addresses all these issues. The core point is the presentation of appropriate algorithms that can provide the sampling of the input distributions based on the restrictions of time-dependence for the system loads and non-normality/dependence for the system wind power generation. II. MODELING OF TIME-AND STOCHASTIC-DEPENDENCE For the modeling of the system stochastic inputs, the mod-eling effort should be split in two separate tasks [8]:
• Marginals: model the one-dimensional marginal distrib-utions.
• Stochastic dependence: model the stochastic dependence structure between the inputs.
Wind Speed (m/s) WTG Power Output (p.u.) Fig. 1: WTG wind speed/power characteristic and distributions.
U
X
1( )
XX
F
U
-10 -5 0 5 10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 Random variable X Cu m u la ti v e di s tr ib ut io n f un c ti o n FX Fig. 2: Sampling of a r.v. in MCS. the interval[0, 1]1.This relationship forms the basis for the sampling of any r.v. in MCS studies. For the sampling of a single r.v. X with invertible cdf FX, first a random realization u from a
uniform r.v. U in[0, 1] is generated and then the transformation x= FX−1(u) is applied. In this case, the samples x follow the distribution FX. The above mentioned procedure is presented
in Fig. 2.
This transformation may be further used for the modeling of correlated r.v. In particular, for the modeling of a n-vector of r.v. X1, X2, . . . , Xn, the uniform r.v. Ui, i= 1, . . . , n should
1The proof of this statement is as follows:F or r ∈ [0, 1] : P (F X(X) ≤
r) = P (X ≤ FX−1(r)) = FX[FX−1(r)] = r. Thus, if U is the uniform
distribution,FX(X) = U ⇔ X = FX−1(U). Therefore, FX−1(U) follows the distribution ofX.
be used. In order to generate samples correlated according to a given correlation matrixR, we should generate the uniforms Ui, i = 1, . . . , n, correlated according to R and use these
correlated uniforms to generate the r.v.’s X1, X2, . . . , Xn. For
this, the theory of the multivariate normal distribution is used (Joint Normal Transform methodology). The algorithm for the generation of correlated samples is the following [9]:
1) Dependence structure:
• Simulate an n-vector of independent standard
nor-malsZ = [Z1, . . . , Zn].
• Calculate the matrix product y = T · z, where T is a lower triangular matrix such thatR = T · TT
(Cholesky decomposition). The n-vector y forms a sample drawn from a random vector Y that follows a standard multivariate normal distribution with product moment correlation matrixR: Y = T · Z.
• Transform the correlated standard normals Y to
correlated uniformsUC= Φ(Y ).
2) Marginals:
• Transform the correlated uniforms UC into the
given marginalsF = Fi−1(UC). A. Wind power distributions
For the modeling of wind speed distribution, FX equals
the Weibull cdf. After the generation of the wind speed distribution, the wind speed/power characteristic of the wind turbine generator is used for the generation of the wind turbine power output (Fig. 1). In Fig. 3, the scatter diagram for the wind speed distributions correlated with a correlation ρ= 0.7 is presented.
B. Load distributions (time-conditioning)
Fig. 3: Wind speed scatter diagram, ρ= 0.7 (10000-sample MCS). 1500 200 250 300 350 400 450 500 550 600 50 100 150 200 250 300 Load - bus 4 (MW) Num b er o f sam p le s
Fig. 4: Load distribution for 4-TF segmentation (10000-sample MCS).
an independent uniform r.v. UT F as TF-indicator. In particular,
based on the relative duration of each TF, each sample drawn from UT F is matched to a specific TF. According to the
indicator, a sample is drawn from the normal distribution belonging to the specific TF. In table I the TF settings for a 4-TF segmentation are presented. According to the time ratio of this table, for UT F >0.8, TF4 is chosen (high-load-TF),
thus a sample is drawn from a normal r.v. with mean value equal to the high-load-TF mean μT F 4and standard deviation
0.03 · μT F 4, for 0.5 < UT F < 0.8, TF3 is chosen, thus a
sample is drawn from a normal r.v. with mean value equal to the μT F 3= 0.85 × (high − load − T F mean) and standard
deviation0.1·μT F 3etc. In Fig. 4, the resulting distribution for
a 10000-sample MCS for a 4-TF segmentation is presented. As may be seen, the 4-TF segmentation offers a very good approximation to the system load distribution.
Since the TF analysis corresponds to time-conditioning,
Fig. 5: Scatter diagram between two system loads for 4-TF segmentation (10000-sample MCS).
TABLE I: TF settings for 4-TF load modeling for the New England test system
Time Ratio Mean Load St. Deviation
(% high-load-TF mean) (% mean load)
T F 1 0.2 0.5 0.06
T F 2 0.3 0.65 0.1
T F 3 0.3 0.85 0.1
T F 4 0.2 1 0.03
when a TF is chosen, all system loads are considered to be in the same TF. Therefore, the same as TF-indicator UT F is
used for the choice of TF for all system loads; further, the loads in each TF are modeled independently. This imposes a high dependence between the load distributions. In Fig. 5, the scatter diagram for a 10000-sample MCS for the modeling of two loads in the system is presented. As may be seen, the methodology captures the time-dependence between the system loads. Although in each TF the r.v. are independent, the resulting distributions are highly correlated. In particular, the rank correlation obtained at the output samples is 0.89.
III. STUDYCASE:WIND INTEGRATION IN BULK POWER SYSTEM
A. System Data
W2
W3 W15
Fig. 6: Single line diagram of the 39-bus New England test system [10].
TABLE II: Characteristics of the New England test system. System characteristic value
Busses 39
Central generators 10
Loads 19
Transmission lines 46
Voltage level 100kV
Total generation 6140.7MW / 1264.3MVar Total load 6097.1MW / 1408.7MVar
B. MCS data
In order to proceed to the MCS analysis of the system, the system uncertain inputs are sampled based on the theory presented in section II. Thus, all the system loads and wind power generators are sampled based on the given marginal distributions and dependence structure. For the specific study case, the system load is considered as independent from the wind activity at the system.
A 4-TF segmentation is chosen for the modeling of the system loads, according to the analysis presented in section II-B. Accordingly, a normal distribution is used for the modeling of the time-conditioned load in each TF and the resulting distribution is obtained by an aggregation procedure as a mixture of these normals. The settings for the 4-TF modeling are presented in Table I. In Fig. 4, the resulting distributions for a 10000-sample MCS for the loads in buses 4 and 20 are presented.
The wind resources are considered to be correlated through-out the system. The mutual correlation between the wind speed r.v. for the 15 wind parks is defined as 0.7. In Fig. 3, the scatter diagram for the wind speed distributions between two sites in the system is presented.
For the system planning, two basic study cases are consid-ered:
1) No penetration: this case corresponds to the system operation with no wind power (vertically-operated power system).
2) 200MW capacity for each wind park: the nominal wind power capacity in the system is in this case 15 × 200MW = 3000MW , corresponding to a penetration level of 50%.
For these study cases, the variability in the power flows in the system is investigated. In particular, 10000 MCS samples are generated for the system inputs and for each sample, the steady-state analysis of the system is performed. The MCS input data are generated in Matlab. These data are written to a text file which is read by a Python2 script. PSS/E is used as the load flow solver. The actual MCS is performed by a combination of PSS/E and Python.
The dispatch of the central generators in the system should be modeled for each system state, defined by the sampling of the system loads and wind power. In the specific study case, the CG units in the system are considered to be thermal units of the same type. For each sample, the power production of the CG units (except for the slack bus) is pro-rata. For each CG unit, the minimum power output is 10% of the unit capacity, due to restrictions with shutting down thermal units. The system dispatch is set to follow the system net load for each sample, which corresponds to perfect forecast. In this case, the system slack bus covers the system losses and also consumes the excess of wind power when the CG units are at
Fig. 7: Box-plot for the power flows in the system lines in case of no wind power penetration (10000-sample MCS). -800 -600 -400 -200 0 200 400 600 Power flow (MW) System lines 21-22 1-39 23-36 6-7 25-37 2-25 17-27 20-34 15-16 17-18 5-8 2-30 10-11 29-38 12-13 10-13 22-23 8-9 3-4 13-14 4-5 9-39 7-8 19-20 28-29 26-29 16-24 23-24 11-12 16-21 4-14 19-33 16-19 14-15 6-31 16-17 22-35 25-26 1-2 2-3 26-28 5-6 10-32 6-11 26-27 3-18
flow distributions extend to both the positive and negative axis).
IV. CONCLUSIONS
The integration of wind power generation in the power system leads to an increase in the variability of the power flows in the system lines. For the planning of the system with high penetration of such stochastic power generation, a stochastic methodology is necessary that can offer the modeling of the uncertainty in the system and comply with the time-dependent stochasticity of the system loads and the stochastic dependence of the system wind power. The presented Monte-Carlo simulation methodology offers a unified solution to this problem.
The methodology presented here offers the basis for plan-ning of power systems with high penetration of wind power generation. Based on measurements on the wind speed regime in the areas of interest and an assessment of the correlation between the wind resources, one can assess the risk of over-loading of the system lines for different penetration levels of wind power. The minimization of this risk forms the basis for the assessment of system planning decisions.
ACKNOWLEDGMENT
This research has been performed within the framework of the research program ’Intelligent Power Systems’, which is sponsored by the IOP/EMVT research program of Senter-Novem. SenterNovem is an agency of the Dutch ministry of Economic Affairs.
REFERENCES
[1] Delft & Eindhoven Universities of Technology, “Intelligent power sys-tems project website,” http://www.intelligentpowersyssys-tems.nl, 2003. [2] G. Papaefthymiou, P. H. Schavemaker, L. van der Sluis, W. L. Kling,
D. Kurowicka, and R. M. Cooke, “Integration of stochastic generation in power systems,” Electrical Power and Energy Systems, vol. 28, pp. 655–667, 2006.
3In a box-plot, the box has lines at the lower quartile, median, and upper quartile values. Lines extending from each end of the box to show the extent of the rest of the data (whiskers). Outliers are data with values beyond the ends of the whiskers. Each of these data is represented by the marker+ and corresponds to the tails of the distributions.
Statistics, 2006.
[9] G. Papaefthymiou, A. Tsanakas, D. Kurowicka, P. H. Schavemaker, and L. van der Sluis, “Probabilistic power flow methodology for the modeling of horizontally-operated power systems,” in International
Conference on Future Power Systems (FPS2005), November 16-18 2005.
[10] M. A. Pai, Energy Function Analysis for Power System Stability. Kluwer Academic Publishers, Boston, 1989.
George Papaefthymiou obtained his Dipl-Eng in
Electrical and Computer Engineering from the Uni-versity of Patras, Greece in 1999. He is currently pursuing Ph.D. at the Electrical Power Systems Lab-oratory of the Department of Electrical Engineering, Mathematics and Computer Science of the Delft University of Technology, in the main framework of Intelligent Power Systems. His current research interests include modeling of uncertainty in power systems and design of systems with large-scale pen-etration of distributed and stochastic generation.
Jody Verboomen obtained his Master in Industrial
Sciences (Electronics) from Groep T Hogeschool in Leuven, Belgium in 2001. He obtained his M.Sc. in Electrical Engineering from the Catholic University of Leuven (KUL), Belgium in 2004. He is currently working towards a Ph.D. on the application of FACTS and phase shifters in transmission systems in the Electrical Power System (EPS) laboratory of the Delft University of Technology, The Netherlands. His research is funded by SenterNovem, an agency of the Dutch ministry of Economical Affairs.
Lou van der Sluis obtained his M.Sc. in electrical
