Assessing the Risk of Transmission Line Overloads in Power Grids ...
Assessing the Risk of Transmission Line Overloads in Power Grids with Stochastic Generation Markus Schl¨apfer ETH Zurich, Laboratory for Safety Analysis, Sonneggstrasse 3, CH-8092 Z¨ urich, Switzerland, [email protected]
1 Introduction The production of electrical energy from renewable sources has, in recent years, increased significantly with wind generation as the predominant technology. This development is expected to continue in the near future. However, the integration of such stochastic and intermittent power sources into the existing power system implies a higher ratio of non-dispatchable generation which, in turn, leads to less predictable and more volatile flows on the network. The intermittent nature poses a challenge for an adequate generation capacity planning, and the operational uncertainty of the time-varying load flow regime makes the anticipation of critical situations such as transmission line overloads highly complicated. This extended abstract presents a hybrid modeling and simulation approach with the objective of assessing the impact of intermittent generation on the probability of thermal line overloads and of analyzing the time sequences of subsequent cascading line outages. Thereby, the power flow dependent, dynamic line temperatures are explicitly modeled in time. In order to overcome the problem of the slow simulation speed coming along with the continuous integration of the heat balance equation, we apply a technique for the fast simulation of rare events, called RESTART (REpetitive Simulation Trials After Reaching Thresholds) on the present problem. The modeling framework is applied to the IEEE Reliability Test System 1996 whereas the effect of different stochastic power injection patterns on the line outage probability and on subsequent cascading failures are analyzed.
2 Modeling Framework 2.1 Thermal Line Model Each phase of a transmission line is heated by its temperature dependent resistive losses L` (T` (t)) = I` (t)2 R(T` (t)) and by the solar heat gain Qs` (t), while cooling is due to convection Qc` (T` (t)) and radiation Qr` (T` (t)) [1, 2, 3]. This heat balance yields the following first order nonlinear differential equation for the line temperature T` (t): ρ`
d T` (t) = L` (T` (t)) + Qs` (t) − Qc` (T` (t)) − Qr` (T` (t)) dt
(1)
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Markus Schl¨ apfer
where ρ` is the heat capacity of the line. During a simulation run Eq. (1) is integrated numerically using the RK4 algorithm. 2.2 Stochastic Power Generation Model The output states of the individual intermittent generating units are represented by a simple twostate model according to Fig. 1. The power output, Pg (t), is either at its maximum value, Pgmax , or zero. λg Up state
Down state μg
Fig. 1. Two-state model for an individual intermittent generating unit
The stochastic up-down-up cycle assumes constant transition rates λg and µg . Hence, this alternating renewal process is characterized by the cumulative distribution functions of the up 0 state times, τg , and down state times, τg , respectively [4]: Fg (tu ) = P r{τg ≤ tu } = 1 − e−λg tu 0
Gg (td ) = P r{τg ≤ td } = 1 − e−µg td
(2) (3)
where tu and td are the time spans measured from the moment of entering the up-state and downstate respectively. The average frequency of changing from the down state to the up state is called the renewal density, fr , and is calculated by [5]: fr =
λg µg λg + µg
(4)
By increasing the renewal density of the individual generators a higher volatility of the combined power output of several units is obtained. Therefore, both transition rates are multiplied by the same factor in order to keep the value for the mean power output constant, see Fig. 2, upper part. Furthermore, by grouping different numbers of intermittent generators into a cluster, for which the same random generator holds during a simulation, a broad spectrum of different stochastic power injection patterns can be reproduced. Given a large number of small units being connected to one or several nodes, we denote the clustering factor C as the number of aggregated generators which change their output state simultaneously. The mean power output is independent of C. As depicted in figure 2, lower part, a small clustering factor is leading to more smooth power output time-series, while a high clustering factor implies a strong fluctuation around the mean value. According to [8], such a comonotonic behavior of a large group of strongly positively correlated units has more adverse effects on the frequency of line overloads, as the extremes of the combined power outputs are more frequently reached.
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Assessing Transmission Line Overloads in Power Grids with Stochastic Generation
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Fig. 2. Different combined power injection patterns due to different renewal densities fr and clustering factors C for 18 generating units with a capacity of 2.5 MW each. a) fr =0.028h−1 , C=1; b) fr =0.28h−1 , C=1; c) fr =0.028h−1, C=6; d) fr =0.28h−1 , C=6.
3 Simulation Speedup Applying RESTART The continuous integration of the heat balance equation for the estimation of the transmission line overload frequencies by applying a crude Monte Carlo simulation would be highly time-consuming. In order to overcome this rare-event problem we make use of an accelerated simulation method, called RESTART [6]. Opposite to other variance reduction techniques such as importance sampling, the method has no influence on the time-sequence of the discrete events. Following, we introduce the informal basics of the method applied to our specific transitory state simulation problem. Suppose the objective of the study is to evaluate the occurrence probability of an event A within a predefined time period [t0 , te ) given a defined system state at t0 . In our case, the event A is the exceedance of the maximum tolerable temperature T max on a specific line. A crude simulation would repeatedly simulate the system in the given period, whereas the line temperature would be significantly below T max most of the time. The basic idea of RESTART is to repeatedly perform simulation runs in those regions of the state space, where the event of interest is more often provoked. Therefore, the method divides the temperature state space [T 0 , T max ] into m intermediate intervals [T 0 , T 1 ), [T 1 , T 2 ), . . . , (T m−1 , T m ] with thresholds T 0 < T 1
1 Introduction The production of electrical energy from renewable sources has, in recent years, increased significantly with wind generation as the predominant technology. This development is expected to continue in the near future. However, the integration of such stochastic and intermittent power sources into the existing power system implies a higher ratio of non-dispatchable generation which, in turn, leads to less predictable and more volatile flows on the network. The intermittent nature poses a challenge for an adequate generation capacity planning, and the operational uncertainty of the time-varying load flow regime makes the anticipation of critical situations such as transmission line overloads highly complicated. This extended abstract presents a hybrid modeling and simulation approach with the objective of assessing the impact of intermittent generation on the probability of thermal line overloads and of analyzing the time sequences of subsequent cascading line outages. Thereby, the power flow dependent, dynamic line temperatures are explicitly modeled in time. In order to overcome the problem of the slow simulation speed coming along with the continuous integration of the heat balance equation, we apply a technique for the fast simulation of rare events, called RESTART (REpetitive Simulation Trials After Reaching Thresholds) on the present problem. The modeling framework is applied to the IEEE Reliability Test System 1996 whereas the effect of different stochastic power injection patterns on the line outage probability and on subsequent cascading failures are analyzed.
2 Modeling Framework 2.1 Thermal Line Model Each phase of a transmission line is heated by its temperature dependent resistive losses L` (T` (t)) = I` (t)2 R(T` (t)) and by the solar heat gain Qs` (t), while cooling is due to convection Qc` (T` (t)) and radiation Qr` (T` (t)) [1, 2, 3]. This heat balance yields the following first order nonlinear differential equation for the line temperature T` (t): ρ`
d T` (t) = L` (T` (t)) + Qs` (t) − Qc` (T` (t)) − Qr` (T` (t)) dt
(1)
2
Markus Schl¨ apfer
where ρ` is the heat capacity of the line. During a simulation run Eq. (1) is integrated numerically using the RK4 algorithm. 2.2 Stochastic Power Generation Model The output states of the individual intermittent generating units are represented by a simple twostate model according to Fig. 1. The power output, Pg (t), is either at its maximum value, Pgmax , or zero. λg Up state
Down state μg
Fig. 1. Two-state model for an individual intermittent generating unit
The stochastic up-down-up cycle assumes constant transition rates λg and µg . Hence, this alternating renewal process is characterized by the cumulative distribution functions of the up 0 state times, τg , and down state times, τg , respectively [4]: Fg (tu ) = P r{τg ≤ tu } = 1 − e−λg tu 0
Gg (td ) = P r{τg ≤ td } = 1 − e−µg td
(2) (3)
where tu and td are the time spans measured from the moment of entering the up-state and downstate respectively. The average frequency of changing from the down state to the up state is called the renewal density, fr , and is calculated by [5]: fr =
λg µg λg + µg
(4)
By increasing the renewal density of the individual generators a higher volatility of the combined power output of several units is obtained. Therefore, both transition rates are multiplied by the same factor in order to keep the value for the mean power output constant, see Fig. 2, upper part. Furthermore, by grouping different numbers of intermittent generators into a cluster, for which the same random generator holds during a simulation, a broad spectrum of different stochastic power injection patterns can be reproduced. Given a large number of small units being connected to one or several nodes, we denote the clustering factor C as the number of aggregated generators which change their output state simultaneously. The mean power output is independent of C. As depicted in figure 2, lower part, a small clustering factor is leading to more smooth power output time-series, while a high clustering factor implies a strong fluctuation around the mean value. According to [8], such a comonotonic behavior of a large group of strongly positively correlated units has more adverse effects on the frequency of line overloads, as the extremes of the combined power outputs are more frequently reached.
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Assessing Transmission Line Overloads in Power Grids with Stochastic Generation
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Fig. 2. Different combined power injection patterns due to different renewal densities fr and clustering factors C for 18 generating units with a capacity of 2.5 MW each. a) fr =0.028h−1 , C=1; b) fr =0.28h−1 , C=1; c) fr =0.028h−1, C=6; d) fr =0.28h−1 , C=6.
3 Simulation Speedup Applying RESTART The continuous integration of the heat balance equation for the estimation of the transmission line overload frequencies by applying a crude Monte Carlo simulation would be highly time-consuming. In order to overcome this rare-event problem we make use of an accelerated simulation method, called RESTART [6]. Opposite to other variance reduction techniques such as importance sampling, the method has no influence on the time-sequence of the discrete events. Following, we introduce the informal basics of the method applied to our specific transitory state simulation problem. Suppose the objective of the study is to evaluate the occurrence probability of an event A within a predefined time period [t0 , te ) given a defined system state at t0 . In our case, the event A is the exceedance of the maximum tolerable temperature T max on a specific line. A crude simulation would repeatedly simulate the system in the given period, whereas the line temperature would be significantly below T max most of the time. The basic idea of RESTART is to repeatedly perform simulation runs in those regions of the state space, where the event of interest is more often provoked. Therefore, the method divides the temperature state space [T 0 , T max ] into m intermediate intervals [T 0 , T 1 ), [T 1 , T 2 ), . . . , (T m−1 , T m ] with thresholds T 0 < T 1
