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Improved risk-based TTC evaluation with system case partitioning
Electrical Power and Energy Systems 44 (2013) 530–539
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Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Improved risk-based TTC evaluation with system case partitioning Nattawut Paensuwan a,⇑, Akihiko Yokoyama b, Yoshiki Nakachi c, S.C. Verma c a
Department of Electrical Engineering, The University of Tokyo, Japan Department of Advanced Energy, The University of Tokyo, Japan c Research and Development Division, Chubu Electric Power Co. Inc., Japan b
a r t i c l e
i n f o
Article history: Received 5 November 2010 Received in revised form 1 July 2012 Accepted 25 July 2012 Available online 26 September 2012 Keywords: Decision tree Performance indices Total transfer capability Renewable energy
a b s t r a c t Managing the security of a system has always been one of the challenging tasks for a system operator, especially within a competitive environment where a system is commonly operated close to its limits. Therefore, the Total Transfer Capability (TTC) index was introduced by NERC in 1996 mainly to provide the system operator with information necessary for the system security management. In addition, the presence of variable generation from renewable energy further complicates the system operation and planning. When evaluating the TTC, it is important to consider the uncertainty associated with the system parameters and the system security related to the transient stability. This paper presents a riskbased TTC evaluation by means of Monte Carlo simulation together with transient stability consideration. However, this method requires excessive computing time in view of a large sample size; as a result, the TTC may not be available to the system operators in time. Since the heavy computational burden is a major impediment to the widespread use of this method; hence, a system case partitioning using two filters, i.e. performance indices and decision tree, is proposed as a means to reduce the sample size and enhance the computational speed. The validation of the proposed method is conducted through a numerical simulation on the 90-bus system integrated with wind power and photovoltaic (PV). Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction One of the system operator’s challenging tasks is to manage the security of a system. It is even more difficult particularly for a competitive power system where the system is usually operated close to its limits. As a means to assist in managing the security of a system, in 1996, North American Electricity Reliability Council (NERC) introduced a Total Transfer Capability (TTC) concept to be used as an index measuring the ability of a transmission network to reliably carry electric power from one location to another across the system [1]. The TTC evaluation is usually conducted in advance to provide the system operator with essential information related to the system operational limits necessary for the operation of a system. In the TTC evaluation, it is important to consider the uncertainty associated with the system parameters, and the security constraint related to transient stability. The TTC can be affected by the variation in load and renewable energy power output. Also, the TTC is sometimes limited by the transient stability constraint due to severe disturbances. Up to present, various methods have been proposed to evaluate the TTC. The TTC can be simply computed using Repeated Power Flow (RPF) [2] where the power flow problem is repeatedly solved ⇑ Corresponding author. E-mail address: [email protected] (N. Paensuwan). 0142-0615/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2012.07.041
as the specified generation and load vary until any security limit is reached. The TTC can also be found by means of an optimization based approach, e.g. security constrained optimal power flow (SCOPF) presented in [3]. Ref. [4], uses the hybrid mutation particle swarm optimization to solve the optimization problem for the transfer capability of a system with FACTs devices. Due to the uncertainty associated with the system conditions, a number of studies apply the probabilistic approaches to integrate the uncertainty into the TTC evaluation. In [5], the parametric bootstrap technique is used to generate bootstrap samples representing the uncertainty. Another commonly applied technique for the probabilistic TTC evaluation is Monte Carlo simulation [6–9]. Khaburi and Haghifam [6] apply Monte Carlo simulation to represent the load uncertainty. In addition to load uncertainty [7], also includes the uncertainty of the availability of the transmission lines. The impacts of uncertainty are now more pronounced as a result of a rapid growth and development of renewable energy due to its intermittent and uncertain power output characteristic. In [8], the TTC is evaluated by Monte Carlo simulation considering wind power output uncertainty. The clustering technique is applied to speed up the computation. Ref. [9] employs Monte Carlo simulation and risk-based TTC selection to examine the impact of renewable energy penetration on the TTC. Several efforts have been made to speed up the TTC evaluation. In [10], a two-step method is presented. The security constraint to
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which the system is prone is predicted in the prediction step, and the TTC is then found in the correction step. A radial basis function neural network based approach (RBFNN) is proposed in [11] in which the transfer capability is found from the mapping function as the RBFNN learns during the training. Nonetheless, none of these speed-improving techniques take into account transient stability constraint. When neglecting transient stability, the result is prone to be optimistic and may not be practical. Some studies include the transient stability constraint into the TTC evaluation using Transient Stability Constrained Optimal Power Flow (TSCOPF) [12], or energy function methods [13,14]. Nonetheless, most of these studies are simply limited to a system with only synchronous generators, and there is currently no investigation on a system with renewable energy. To address the aforementioned issues, this paper proposes a risk-based TTC evaluation using Monte Carlo simulation where the uncertainty associated with system parameters and renewable energy power output, and security constraint related to transient stability are properly taken into account. However, this method requires excessive computing time in view of a large sample size; as a result, the TTC may not be available to the system operators in time. Since the heavy computational burden is a major impediment to the widespread use of this method; hence, a system case partitioning method using two filters, i.e. performance indices and decision tree, is proposed as a means to enhance the computational speed. This paper is organized in the following sequences. Section 2 provides the definition of the TTC and describes the algorithm of the risk-based TTC evaluation by means of Monte Carlo simulation. Section 3 describes the modeling of wind power and PV system. Section 4 presents the proposed system case partitioning. Numerical examples, results, and discussion are presented in Section 5. Finally, the conclusion is given in Section 6.
2.1. System case generation First, a set of system cases is generated by the sampling process. In general, a system case depends on several uncertain parameters, e.g. short-term forecast demand, forecast renewable energy power output, equipment availability, and fault location. These parameters are sampled basically based on their probability distributions as described in the following. The forecast demand and PV power output are assumed here to have a normal probability distribution which is given by:
1 ðx lÞ2 f ðxÞ ¼ pffiffiffiffiffiffiffi exp 2r2 r 2p
! ð1Þ
where l is the forecast value, and r is the standard deviation representing the accuracy of the forecast value. To sample the parameter, first, a normally distributed random number, z (with zero mean and unity standard deviation) is generated either by the approximate inverse transform method or Box– Müller method. The sampled value is then converted from
x ¼ xl þ rz
ð2Þ
where x and xl are the sampled and forecast values. On the contrary, the probability distribution of the short-term wind power forecast changes with the amount of the power output. Based on the statistical analysis of wind power forecast error described in [17], a Beta distribution rather than a normal distribution can more accurately approximate the short-term wind power forecast. The Beta distribution is given by:
f ðx; a; bÞ ¼ Bða; bÞ ¼
Z
1 xa1 ð1 xÞb1 Bða; bÞ 1
xa1 ð1 xÞb1 dx
ð3Þ ð4Þ
0
2. Risk-based TTC evaluation This section starts with the TTC definition and moves onto the detailed description of the risk-based TTC evaluation with transient stability consideration. According to NERC, TTC was defined as the maximum of power that can be transferred in a reliable manner between a pair of defined source and sink locations in the interconnected system while meeting all of a specific set of defined pre- and post- contingency system conditions [1]. It is important to note from NERC’s document that only a general guideline not a standard method for determining, modeling the transfer capability is provided. Therefore, the TTC evaluation method may be slightly different from application to application. TTC can be simply evaluated by a deterministic method in which the TTC is determined from the minimum transfer capability within a predefined set of system conditions, e.g. all N 1 contingencies [15,16]. However, the result is commonly conservative due to a lack of the ability to properly reflect the uncertainty and stochastic nature of a system. Emphasized by the presence of renewable energy, the uncertainty is now realized essential to be taken into account in the TTC evaluation. This paper focuses on a point-to-point transfer of the interested path. The algorithm of the risk-based TTC evaluation is composed of four main steps, each of which will be described in detail in the following subsections. 1. 2. 3. 4.
System case generation. MTC calculation. Transient stability assessment. Risk-based TTC selection.
Here B(a, b) is the Beta function, and a, b are parameters related to the mean, l, and variance, r2 as follows:
l¼
a aþb
r2 ¼
ð5Þ
ab ða þ bÞ2 ða þ b þ 1Þ
ð6Þ
A Beta distributed random number, b is generated from
bða; bÞ ¼
gðaÞ gðbÞ þ gðaÞ
ð7Þ
where g is a Gamma distributed random number which is generated from
gðcÞ ¼ ln
c Y
Ui
ð8Þ
i¼1
where c is the shape parameter, and Ui is a uniform random number within [0, 1]. The uncertainty degree of the forecast parameters is specified here by a standard deviation expressed in percent of the forecast value. Equipment availability is determined from the operating status of each component in the system, i.e. a transmission line. This can be done by a state sampling technique [18] with the use of historical reliability data. In case of an outage due to a disturbance, the exact fault location on the faulted element is also of concern. The fault on a transmission line may occur in any location along the line, and its probability distribution generally does not follow a uniform distribution. Therefore, in this paper, the fault location is sampled from a discrete probability distribution. A transmission line is divided into
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three segments, i.e. the near-ended, middle, and far-ended, each of which is associated with the probability of the fault occurrence. 2.2. MTC calculation For each given system case, the maximum transferable power, i.e. Maximum Transfer Capability (MTC) is computed. Different operating conditions result in different amounts of the transferable power. For a point-to-point transfer, the amount of the additional electric power delivered from the source bus to the sink bus is specified by a parameter k. As a result, maximizing the transfer is equivalent to maximizing the k. With this additional electric power transferred to the sink bus, the active and reactive demands at the sink bus are modified as follows:
PD ¼ P0D þ k cosðw0 Þ QD ¼
Q 0D
0
þ k sinðw Þ
ð9Þ
s:t:
9 > > > =
Gðx; kÞ ¼ 0 > > > ; Hðx; kÞ 6 0
ð11Þ
where x is a vector of system control and state variables, G(x, k) are a set of equality constraints representing the power balance equations, and H(x, k) are a set of inequality constraints representing the system security limits including voltage magnitude limit, generation capacity limit, and equipment thermal limit. Once the maximum k is solved, the MTC is computed from
MTC ¼ P0D þ kmax cosðw0 Þ
3. Renewable energy modeling
ð10Þ
where PD, QD are active and reactive demands at the sink bus; P0D ; Q 0D are base-case active and reactive demands at the sink bus; k is a transfer parameter; and w0 is the base-case power factor angle of the sink bus. The optimization problem can be formulated in a general form as follows:
max k
MTC and select the TTC. In the past, the TTC was simply selected from the minimum MTC within a set of N 1 contingencies. This selection scheme is conservative and does not reflect stochastic nature of the system. In addition, the TTC is often masked by the severe system condition which usually has a small probability of occurrence; hence, it may not occur within the specified lead-time. Therefore, this paper instead selects the TTC based on the risk concept in which the stochastic nature of a system is well reflected through the PDF. The risk refers to the chance that the system fails to conduct the transfer at the specified value, i.e. the accumulated probability of the cases having the MTC smaller than the specified transfer, i.e. the shaded area shown in Fig. 1. The criterion for specifying the risk basically depends on the system security requirement. The optimal risk can also be determined from a benefitrisk analysis.
ð12Þ
In this paper, the OPF is solved by a well-known sequential quadratic programming (SQP). 2.3. Transient stability assessment As mentioned previously, the ability of a system to transfer electric power may be restricted by the transient stability constraint following a disturbance. In some studies, the transient stability constraint is directly added into the OPF (TSCOPF) [12], or assessed by energy function methods [13,14]. Nonetheless, these methods have the convergence difficulty and modeling limitations; hence, including renewable energy into the analysis would be relatively difficult. As a result, their applications are limited to a system with only synchronous generators, which may not be applicable for the future system as more renewable energy penetration is expected. Therefore, this paper employs an iterative process. Transient stability is assessed by a conventional time-domain simulation where the models of renewable energy based generation are included. Note that the MTC obtained in the previous step is computed from the post-disturbance condition. Therefore, this step checks whether the system can stably move from pre- to post-disturbance conditions. If unstable, the transfer is adjusted. The adjustment is done in a bi-section manner with a specified number of steps.
This paper focuses on wind power and photovoltaic (PV) generation. They have been successful and developed with a large penetration expected. Japanese government already set out a target of 6610 MW wind power and 53 GW PV by 2030 [19]. Their modeling is described here by an aggregated model in order to reduce the complexity and problem dimension. 3.1. Wind power generation This paper focuses on wind power generation with a constant speed wind turbine technology because this type of power plant has a more severe impact on the transient and voltage stabilities than the other types. The configuration is simply a wind turbine coupled with an induction generator and directly connected to the grid with a series of capacitor banks installed at the terminal for reactive power compensation. Usually, only the forecast wind power output and induction machine’s parameters are available. The other related parameters, i.e. terminal voltage, consumed reactive power, and rotor slip are computed from the modified power flow problem. The power balance equations of the bus i connected to the wind power generation are modified to include the complex power terms of the wind power generation as written in (13) and (14). Due to the addition of a new variable, i.e. a rotor slip, another equation, i.e. (15), is required for solving the power flow problem. Eq. (15) enforces the wind power output to its specified or forecast value.
fPi ¼ F Pi ðd; VÞ Pspec Wi þ P Di ¼ 0
ð13Þ
fQi ¼ F Qi ðd; VÞ þ Q Di þ Q Wi ðV i ; si Þ ¼ 0
ð14Þ
fPW i ¼ PWi ðV i ; si Þ Pspec Wi ¼ 0
ð15Þ
where
2.4. Risk-based TTC selection Steps 2–3 are repeated for a given number of system cases. The final step is to build the probability density function (PDF) of the
Fig. 1. Risk-based TTC selection.
N. Paensuwan et al. / Electrical Power and Energy Systems 44 (2013) 530–539
PWi ðV i ; si Þ ¼ Q Wi ðV i ; si Þ ¼ F Pi ðd; VÞ ¼
V 2i R2eq ðsi Þ
þ X 2eq ðsi Þ
which converts DC power fed from the PV arrays to AC power. In this paper, the steady-state model of a PV system is considered as a PQ bus operating at a unity power factor. The PV dynamic model used here is highly simplified. The power conditioning unit of a PV system is modeled as a current source [21]. The current is controlled to achieve the active and reactive power reference set-points, i.e. Pref and Qref. The power regulator is simply a PI controller as shown in Fig. 3. It is assumed that the inverter can adjust its current with the time delay T of 0.01 s fast enough to follow the current reference signals from the power regulator.
Req ðsi Þ
V 2i R2eq ðsi Þ
þ X 2eq ðsi Þ
X eq ðsi Þ
N X V i V j ðGij cos dij þ Bij sin dij Þ j¼1
N X F Qi ðd; VÞ ¼ V i V j ðGij sin dij Bij cos dij Þ: j¼1
Here PWi and QWi are the active and reactive power terms of the wind power generation expressed using load convention; P spec is Wi the specified or forecast wind power output; Gij and Bij are the real and imaginary parts of the ijth element in the admittance matrix; Vi and Vj are the voltage magnitudes at buses i and j respectively; dij is the voltage angle difference between buses i and j; and N is the number of buses. In (16) and (17), Req, Xeq are the equivalent resistance and reactance derived from a steady-state circuit of an induction generator as shown in Fig. 2.
Req ¼ 2 R r
s
R 2 r
X eq ¼
s
Rr X 2m s
þ ðX m þ X r Þ2
X m þ X m X r ðX m þ X r Þ Rr 2 þ ðX m þ X r Þ2 s
533
ð16Þ
ð17Þ
The dynamic model of wind power generation is based on an aggregated two-mass model. There are three differential equations describing the motion of the two rotating parts, i.e. wind turbine and induction generator rotors, and two differential equations associated with the electrical part of an induction generator. For more detailed descriptions, please refer to [20]. 3.2. Photovoltaic (PV) generation The photovoltaic (PV) power technology uses semiconductor cells to convert solar energy to electrical energy. A PV system is connected to the grid via a power conditioning unit, i.e. inverter,
4. System case partitioning Monte Carlo simulation usually requires a sufficiently large sample size to ensure the convergence; as a result, it is time-consuming. Referring to the risk-based TTC selection described in Section 2, it can be noted that not all the system cases are related to the risk, i.e. having small MTC. Therefore, to obtain the TTC, it is not necessary to compute the MTC of all the system cases, only those related to the risk are sufficient. Based on this concept, the system cases will be partitioned into two main groups based on their severity; risk-related and non-risk-related groups respectively. Considering only the above partitioned cases, some of the OPF calculation and time-domain simulation in the Monte Carlo simulation can be avoided; as a result, the run time can be saved. Most of the risk-related cases are severe cases whose transfer is restricted by either steady-state or transient stability constraint. Thus, two filters are used here to screen out such risk-related cases. Filter 1 screens out the risk-related cases due to the steady-state constraints using the performance indices (PIs) [22]. Filter 2 screens out the risk-related cases due to the transient stability constraint. Filter 2 employs a decision tree to predict the system transient stability. The partitioned set is then built by the partitioning process described in Section 4.3. It should be emphasized here that the proposed risk-based TTC evaluation with system case partitioning differs from a conventional TTC evaluation with contingency screening in that the proposed method starts off with much wider sample space where the uncertainty is fully taken into account. The process is based on a probabilistic method and focuses only on the sample set required to obtain the risk-based TTC value. The principles of performance indices and decision tree classification are described in the following subsections. 4.1. Performance indices
Fig. 2. Steady-state induction machine circuit.
The performance indices are commonly used in contingency ranking to measure the severity of the system conditions. They are also applied here to rank the system cases within the sample set according to their severity degree. First, the bus voltage performance index, JV is given by
JV ¼
2n N PQ X V i V i;mean cVi V i;tol i¼1
ð18Þ
max min ; Vi,tol is where Vi,mean is the mean bus voltage, i.e. 0:5 Vi þ Vi ; Vi is the voltage at the deviation tolerance, i.e. 0:5 V max V min i i bus i; cVi is a coefficient; n is a positive integer defining the exponent; and NPQ is the total number of PQ buses. Similarly, the generation reactive power performance index, JQ is given by
JQ ¼ Fig. 3. Power regulator of a PV system.
NG X Q i Q i;mean 2n cQi Q i;tol i¼1
ð19Þ
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where Qi,mean is the mean generated reactive power, i.e. ; Qi,tol is the deviation tolerance, i.e. 0:5 Q max þ Q min i i max min ; Qi is the generated reactive power by a generator 0:5 Q i Q i i; cQi is a coefficient; n is a positive integer defining the exponent; and NG is the total number of generators. The branch flow performance index, JS is given by
JS ¼
N br X i¼1
cSi
Si
2n
Smax i
ð20Þ
Fig. 4. Example of a decision tree.
where Si is the apparent power flow of the branch i; Smax is the theri mal rating; cSi is a coefficient; n is a positive integer defining the exponent; and Nbr is the total number of branches. Finally, the system performance index, Jsys can be obtained from
J sys ¼
wV J V þ wQ J Q þ wS J S wV þ wQ þ wS
ð21Þ
where wV, wQ, wS are the weighting factors. They are equally set to 1 in this paper. Note that the performance index, J is a function of the state variables, i.e. J(x). The state variables change as the system condition changes due to the uncertainty, outages, and addition of the transfer. Therefore, the severity of the system cases can be ranked according to the change of the system performance index value as follows:
DJsys ¼ Jsys ðxÞ Jsys ðx0 Þ
ð22Þ
where x0 and x are the vectors of state variables prior to and after the system condition change respectively. The coefficients in (18)–(20), cVi, cQi, cSi are 1 if the system condition after the system condition change is more severe than the base-case condition; and 0 otherwise. Here, the exponent n is set to 4 which is found large enough to avoid the masking problem. The algorithm for the system case ranking is summarized as follows: 1. Run power flow at the base-case condition, and find the basecase MTC. 2. Apply the system condition change, i.e. change of demand, renewable energy power output, system configuration, and add the transfer, here set to 75% of the base-case MTC. This addition of the transfer helps specify the direction where the system will move from the base-case condition; hence, the constraint which limits the transfer can be predicted. 3. Run power flow for the system condition after the system condition change, and compute the performance indices using (18)–(21). 4. Compute DJsys using (22), and rank the system cases. 4.2. Decision tree classification A decision tree is one of the classification methods in a form of a tree graph. Several applications of decision tree classification to the power system have been reported [23,24]. It is mostly used as a prediction tool for a fast system security assessment, e.g. an online transient stability assessment. A typical decision tree consists of three types of nodes: root node, decision node, and leaf node as shown in Fig. 4. A root node is at the top of the tree where the classification starts. A decision node conducts a test or question, where the cases are separated, i.e. splitting a node. A leaf node provides the output classification. A decision tree classifies the case as the case is moving down the tree passing through a series of tests until it falls to one of the leaf nodes from which the class is determined. In Fig. 4, X = {X1, X2, -
Fig. 5. Partitioning process.
. . . , Xn} is a set of n input attributes and c = {c0, c1} is a set of two output classes. A decision tree is grown offline from the training data. Growing a decision tree is simply splitting a node based on a splitting rule. The best split among all possible splitting rules can be achieved by comparing their scores representing how well they can separate the cases. In this paper, the score is calculated from the impurity measure GINI diversity index [25]. Therefore, the split with the best score will be selected. The nodes are split until they are either pure i.e. all cases in the node belong to the same class, or with the number of cases in the node smaller than the specified value, i.e. a minimum split criterion. This criterion is applied so as to prevent the tree from having too many leaf nodes, which is likely to over-fit the training data. The final step is to validate the obtained decision tree by testing with the test set to check its generalization. The input attributes are commonly selected from the system parameters or available measurements. In this paper, the input attribute of the decision tree for the system case partitioning filter is the shortest distance in per unit from the fault location to each generator determined from the Dijkstra’s algorithm [26]. This input attribute is chosen based on the fact that the disturbance is severe when the fault occurs near the generators. The decision tree helps manage this information and uses it for the classification in an effective way. Note that using only one input attribute may not give an accurate prediction but sufficient in roughly identifying the risk-related cases due to the transient stability constraint. When applied to the system case partitioning as Filter 2, for each outage case in the Monte Carlo sample set, the vector of the value of the input attribute (i.e. the shortest fault distance to the generators) is passed through a series of questions (i.e. decision nodes) in the decision tree until falling into one of the leaf nodes which indicates whether this case is stable or not. The system cases predicted by the decision tree as unstable are regarded as severe; hence, risk-related.
N. Paensuwan et al. / Electrical Power and Energy Systems 44 (2013) 530–539
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Fig. 6. Flowchart of the proposed algorithm.
4.3. Partitioning process The algorithm of the partitioning process is given in Fig. 5. The system cases from the system case generation are applied to two filters. Filter 1 ranks the cases based on the DJsys value in a descending order, i.e. from the most severe (largest DJsys) to the least severe cases (smallest DJsys). At the same time, the stability of the outage cases within the sample set is predicted using Filter 2, i.e. decision tree. The partitioned set is then built by selecting the cases from the results of these two filters. Here the results of Filter 2 (i.e. the outage cases classified by Filter 2 as unstable) are included first in the partitioned set and followed by those of Filter 1 (i.e. ranked system cases) until the specified number of cases for the partitioned set is attained. The flowchart of the algorithm for the improved TTC evaluation with system case partitioning is depicted in Fig. 6.
5. Numerical examples 5.1. Study system The proposed method is applied to the 90-bus system with a single-line diagram shown in Fig. 7. The system is obtained by connecting three IEEE 30-bus test systems [28] and with some modifications. The system has 14 thermal power plants with the total capacity of 890 MW, and total system demand of 624.36 MW. Nine
10 MW wind farms and nine 10 MW PV systems are added. The point-to-point transfer from bus#1 to bus#42 is evaluated. 5.2. Parameter setting and study cases The total number of generated cases is 5000 which is found to be sufficient for the convergence of the simulation. As a result of the sampling, out of the 5000 cases, there are 3490 non-outage and 1510 outage cases respectively. The forecast demand error is 10% of the forecast value. The forecast power output of each wind farm and PV system are 7.5 and 8 MW with the forecast error of 20%. The probability of the fault location is adopted from Ref. [27] as tabulated in Table 1. The simulation time for the time-domain simulation is 6 s with the time step of 0.01 s. The fault is cleared within 0.35 s. Note that the fault clearing time is relatively long so that the impact of the transient stability can be manifested. The decision tree is grown from the training data of 3000 cases and tested with the testing data of 1000 cases. These data sets are obtained from the system case generation used for the training data preparation. They can also be retrieved from actual historical operating records if available. The minimum split criterion is 100. The simulation is run on a Pentium Core2Duo 2.4 GHz PC with 4 GB RAM. To validate and examine the performance of the proposed method, four comparative study cases with different numbers of cases in the partitioned set are conducted as summarized in Table 2. The performance in terms of the accuracy and computational speed enhancement will be presented and discussed.
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Fig. 7. 90-Bus test system.
Table 1 Probability distribution of the fault location. Probability of the fault location
Near-ended Middle Far-ended
0.1307 0.7021 0.1672
Table 2 Summary of the partitioned sets. Cases
Number of cases
1 2 3 4
1000 2000 3000 5000
5% 10% Specified risks 15% 20%
0.025
Probability density
Line segment
0.03
0.02
0.015
0.01
0.005
0
0
10
20
30
40
50
60
70
80
MTC [MW] 5.3. Results and discussion The PDF of the MTC and the illustration of the risk-based TTC selection are depicted in Fig. 8. The decision tree for Filter 2 is shown in Fig. 9. The testing error is found to be 6.9%. The number
Fig. 8. PDF of the MTC.
of cases classified by this decision tree as unstable is 244 which is larger than the actual number of 146. However, the overall
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Fig. 9. Decision tree for Filter 2.
Table 4 Summary of the DTTC (Cases 1–3).
MTC [MW]
80 60
Risk (%)
DTTC (%)
40 20 0
0
1000
2000
3000
4000
5000
5 10 15 20
Case 1
Case 2
Case 3
0.09 2.02 6.15 34.13
0.00 0.27 0.61 1.11
0.00 0.03 0.13 0.24
Sample (without ranking) MTC [MW]
80 60 40 20 0
0
1000
2000
3000
4000
5000
Sample (with ranking) Fig. 10. Comparison of the MTC distribution.
Table 3 Summary of the TTC values (Cases 1–4). Risk (%)
5 10 15 20
TTC (MW) Case 1
Case 2
Case 3
Case 4
43.15 47.89 51.58 66.79
43.10 47.07 48.89 50.35
43.10 46.96 48.65 49.92
43.10 46.94 48.59 49.80
accuracy is not much a concern here as long as most of the actually unstable cases are classified correctly. The comparison of the MTC distribution between without and with the ranking is shown in Fig. 10. It can be seen from Fig. 10 that the proposed system case partitioning method can successfully screen out the risk-related cases. The TTC values selected at four different risk levels are summarized in Table 3. It is obvious from the simulation results that the more cases used, the more accurate result. The accuracy here is
determined by comparing with the result of Case 4 where all 5000 cases are used. It can be observed from Table 3 that the TTC obtained by using the partitioned set is slightly larger than the actual TTC. This implies that the partitioned set does not cover all the risk-related cases. The TTC errors from Case 4 in percent, DTTC, are also tabulated in Table 4. The results show that the TTC errors generally reduce as more cases are included in the partitioned set. Interestingly, when using a small number of cases, i.e. Case 1, the TTC error increases with the specified risk. This is because first obtaining the TTC at high risk requires a sufficient number of cases to build the probability distribution covering the range from which the TTC is selected. Thus, the partitioned set with a small number of cases may not cover all the risk-related cases. Second, the cases required for building the distribution at high risk are composed of both severe and fairly severe cases (i.e. having fairly large MTC). Unlike the severe cases, it is relatively difficult to accurately rank the fairly severe cases, i.e. the values of DJsys are not much different. To achieve a given level of accuracy for TTC, the number of cases should be more than that required for building the distribution. For example, given the total number of cases of 5000, the TTC at 20% Table 5 Summary of the run time (Cases 1–4). Cases
Run time (s)
1 2 3 4
4746 7727 10,962 16,793
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risk requires at least 1000 cases for building the distribution for the TTC selection. It can be seen from the result that using only 1000 cases cannot get the accurate TTC (i.e. with the error as much as 34.13%). In this example, it is better to use 3000 cases (Case 3) which is sufficient to obtain the TTC with the error less than 1%. The run time of Cases 1–4 is listed in Table 5. The run time for the system case partitioning process is about 50 s which is relatively small compared to the total run time of 16,793 s (Case 4). The decision tree growing takes less than 1 s. It is obvious from the results that the run time strongly depends on the number of cases, i.e. the more cases used, the more run time required. The proposed method can save a significant amount of the run time. For example, the TTC can be obtained within 7727 s (Case 2) accounting for only 46% of the total run time (i.e. 16,793 s) with the error less than 2%. The computational speed can be further improved if the simulation task is distributed to several PCs. Nonetheless, there is always a tradeoff between speed and accuracy. Therefore, to select the right partitioned set basically depends on the given accuracy level and security requirement. If the accuracy and security requirement are the primary concern, a sufficient number of cases should be included in the partitioned set.
Table A.6 Generator dynamic data. Gen#
MVAbase Ra
1 125 2, 6, 7, 10, 11 100 3, 4, 5, 9, 12, 13, 14 75
Xd
X 0d
T 0do
Xq
X 0q
T 0qo H
0.078 1.22 0.174 8.97 1.16 0.174 0.5 4.29 0.075 1.18 0.220 5.90 1.05 0.220 0.3 4.49 0.070 1.05 0.185 6.10 0.98 0.185 0.3 5.56
Fig. A.11. Exciter model.
6. Conclusion This paper proposes the system case partitioning for the efficient risk-based TTC evaluation. The simulation results show that the system case partitioning can capture the risk-related cases; hence, helps reduce the number of cases to be evaluated. As a result, it can save a significant amount of the run time while still obtaining an acceptable accuracy. The proposed method allows the TTC evaluation to take into account the uncertainty with less computational burden. However, when applying the proposed method, it is important to ensure that the partitioned set covers a sufficient number of cases for a given risk level. Our future research aims at further speeding up the TTC evaluation to a level that it can be included in the real energy management system. This makes possible to provide the system operator with information necessary for the decision making and efficient operation of a power system.
The values are in per unit on machine base. Please also refer to [20] for detailed descriptions on wind power model. PV system data: Kp = 0.05, Ki = 0.5 for both active and reactive power control, and T = 0.01.
Appendix A. Data for transient stability analysis
References
This section gives data of the generators, wind power, and PV systems for the transient stability analysis. Since the dynamic data of the IEEE-30 system is not available, this paper adapted the data from Ref. [29]. The generators are modeled by a 4th order model with the data given in Table A.6. The models for the exciter and turbine governor system are depicted in Figs. A.11 and A.12 respectively. The parameters for the exciter and turbine governor system are assumed to be identical for all generators. Due to the limit of the space, please refer to [30] for the detailed descriptions of the models and parameters. Exciter data:
[1] Transmission transfer capability task force. Available transfer capability definitions and determination. Princeton (NJ): North American Reliability Council; 1996. [2] Sun Rong-fu, Fan Yue, Song Yong-hua, Sun Yuan-zhang. Development and application of software for ATC calculation. In: PowerCon 2006, international conference on power system technology; 2006. p. 1–5. [3] Ou Y, Singh C. Assessment of available transfer capability and margins. IEEE Trans Power Syst 2002;22(5):463–8. [4] Farahmand H, Rashidinejad M, Mousavi A, Gharaveisi AA, Irving MR, Taylor GA. Hybrid mutation particle swarm optimization method for available transfer capability enhancement. Int J Electr Power Energy Syst 2012;42:240–9. [5] Othman MM, Musirin I. A novel approach to determine transmission reliability margin using parametric bootstrap technique. Int J Electr Power Energy Syst 2011;33:1666–74. [6] Khaburi MA, Haghifam MR. A probabilistic modeling based approach for total transfer capability enhancement using FACTs devices. Int J Electr Power Energy Syst 2010;32:12–6. [7] Audomvongseree Kulyos, Yokoyama Akihiko. Consideration of an appropriate TTC by probabilistic approach. IEEE Trans Power Syst 2004;19(1):375–83. [8] Ramezani M, Singh C, Haghifam MR. Role of clustering in the probabilistic evaluation of TTC in power systems including wind power generation. IEEE Trans Power Syst 2009;24(2):849–58. [9] Paensuwan Nattawut, Yokoyama Akihiko, Verma SC, Yoshiki Nakachi. Investigation of impact of renewable energy penetration on system total transfer capability at risk. IEEJ Trans Power Energy 2009;129(12):1523–31. [10] Audomvongseree Kulyos, Yokoyama Akihiko. Application of AC equivalent to total transfer capability evaluation using two-step method. In: Proc PowerCon 2002, international conference on power system technology, vol. 1; 2002. p. 383–7. [11] Jain T, Singh SN, Srivastava SC. Fast static available transfer capability using radial basis function neural network. Int J Appl Soft Comput 2011;11:2756–64.
T r ¼ 0:06;
K a ¼ 25; T a ¼ 0:2; V Rmax ¼ 3:438; V Rmin
¼ 3:438; K f ¼ 0:05; T f ¼ 0:35 Turbine governor system data:
R ¼ 20;
T min ¼ 0; T max ¼ 1; T s ¼ 0:09; T c ¼ 0:2; T 3 ¼ 0; T 4
¼ 1; T 5 ¼ 5 Wind power generation data (for each turbine):
MVAbase ¼ 2:25;
Rs ¼ 0:048; X s ¼ 0:075; X m ¼ 3:8; Rr
¼ 0:018; X r ¼ 0:12; Hm ¼ 3; Hxr ¼ 1; K s ¼ 0:3
Fig. A.12. Turbine governor system model.
N. Paensuwan et al. / Electrical Power and Energy Systems 44 (2013) 530–539 [12] Kubokawa J, Yuan Yue, Yorino N, Zoka Y, Sasaki H, Hakim L. A solution of total transfer capability using transient stability constrained optimal power flow. In: Proc PowerTech; 2007. p. 2018–22. [13] Rosales RA, Ruiz-Vega D, Ernst D. Online transient stability constrained ATC calculations. In: Proc IEEE power eng society summer meeting; 2000. p. 1291–6. [14] Yamada Y, Nagata M, Tanaka K. An energy function based on contingency screening method for TTC assessment with transient stability constraints. In: Proc IEEE/PES T&D Conf and Exhibition; 2002. p. 886–90. [15] Ejebe GC, Waight JG, Sanots-Nieto M, Tinney WF. Fast calculation of linear available transfer capability. IEEE Trans Power Syst 2000;15(3):1112–6. [16] Grijalva S, Saucer PW, Weber JD. Enhancement of linear ATC calculations by the incorporation of reactive power flow. IEEE Trans Power Syst 2003;18(2): 1112–6. [17] Bludszuweit H, Dominguez-Navarro JA. Llombart a statistical analysis of wind power forecast error. IEEE Trans Power Syst 2009;23(3):983–91. [18] Roy Billinton, Wenyuan Li. Reliability assessment of electric power systems using monte carlo methods. New York: Plenum; 1994. [19] http://www.meti.go.jp/committe/materials2/downloadfiles/g80908a04j.pdf. [20] Li H, Chen Z, Han L. Comparison and evaluation of induction generator models in wind turbine systems for transient stability of power system. In: Proc PowerCon; 2006. p. 1–6.
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[21] Fernandez-Bernal F, Rouco L, Centeno P, Gonzalez M, Alonso M. Modeling of photovoltaic plants for power system dynamic studies. In: Proc fifth int conf power syst management control; 2002. p. 341–6. [22] Stefopoulos George K, Yang Fang, Cokkinides George J, Sakis Meliopoulos AP. Advanced contingency selection methodology. In: Proc 37th annual north American power symp; 2005. p. 67–73. [23] Rovnyak S, Kretsinger S, Thorp J, Brown D. Decision trees for real-time transient stability prediction. IEEE Trans Power Syst 1994;9(3):1417–26. [24] Kai Sun, Likhate S, Vittal V, Kolluri VS, Mandal S. An online dynamic security assessment scheme using phasor measurements and decision trees. IEEE Trans Power Syst 2007;22(4):1935–43. [25] Breiman L, Friedman J, Olshen RA, Stone CJ. Classification and regression trees. Belmont, CA: Wadsworth; 1984. [26] Cormen Thomas H, Leiserson Charles E, Rivest Ronald L, Stein Clifford. Introduction to algorithms. 2nd ed. MIT Press, McGraw-Hill; 2001. [27] Wenyuan Li. Risk assessment of power systems. John Wiley & Sons, Inc.; 2005. [28] IEEE 30-Node System. Power system case achieve University of Washington.. [29] Anderson Paul M, Fouad AA. Power system control and stability. Revised printing, IEEE Press Power System Engineering Series; 2004. [30] Chow JH, Rogers Graham. Power system toolbox (PST) manual. .
Contents lists available at SciVerse ScienceDirect
Electrical Power and Energy Systems journal homepage: www.elsevier.com/locate/ijepes
Improved risk-based TTC evaluation with system case partitioning Nattawut Paensuwan a,⇑, Akihiko Yokoyama b, Yoshiki Nakachi c, S.C. Verma c a
Department of Electrical Engineering, The University of Tokyo, Japan Department of Advanced Energy, The University of Tokyo, Japan c Research and Development Division, Chubu Electric Power Co. Inc., Japan b
a r t i c l e
i n f o
Article history: Received 5 November 2010 Received in revised form 1 July 2012 Accepted 25 July 2012 Available online 26 September 2012 Keywords: Decision tree Performance indices Total transfer capability Renewable energy
a b s t r a c t Managing the security of a system has always been one of the challenging tasks for a system operator, especially within a competitive environment where a system is commonly operated close to its limits. Therefore, the Total Transfer Capability (TTC) index was introduced by NERC in 1996 mainly to provide the system operator with information necessary for the system security management. In addition, the presence of variable generation from renewable energy further complicates the system operation and planning. When evaluating the TTC, it is important to consider the uncertainty associated with the system parameters and the system security related to the transient stability. This paper presents a riskbased TTC evaluation by means of Monte Carlo simulation together with transient stability consideration. However, this method requires excessive computing time in view of a large sample size; as a result, the TTC may not be available to the system operators in time. Since the heavy computational burden is a major impediment to the widespread use of this method; hence, a system case partitioning using two filters, i.e. performance indices and decision tree, is proposed as a means to reduce the sample size and enhance the computational speed. The validation of the proposed method is conducted through a numerical simulation on the 90-bus system integrated with wind power and photovoltaic (PV). Ó 2012 Elsevier Ltd. All rights reserved.
1. Introduction One of the system operator’s challenging tasks is to manage the security of a system. It is even more difficult particularly for a competitive power system where the system is usually operated close to its limits. As a means to assist in managing the security of a system, in 1996, North American Electricity Reliability Council (NERC) introduced a Total Transfer Capability (TTC) concept to be used as an index measuring the ability of a transmission network to reliably carry electric power from one location to another across the system [1]. The TTC evaluation is usually conducted in advance to provide the system operator with essential information related to the system operational limits necessary for the operation of a system. In the TTC evaluation, it is important to consider the uncertainty associated with the system parameters, and the security constraint related to transient stability. The TTC can be affected by the variation in load and renewable energy power output. Also, the TTC is sometimes limited by the transient stability constraint due to severe disturbances. Up to present, various methods have been proposed to evaluate the TTC. The TTC can be simply computed using Repeated Power Flow (RPF) [2] where the power flow problem is repeatedly solved ⇑ Corresponding author. E-mail address: [email protected] (N. Paensuwan). 0142-0615/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.ijepes.2012.07.041
as the specified generation and load vary until any security limit is reached. The TTC can also be found by means of an optimization based approach, e.g. security constrained optimal power flow (SCOPF) presented in [3]. Ref. [4], uses the hybrid mutation particle swarm optimization to solve the optimization problem for the transfer capability of a system with FACTs devices. Due to the uncertainty associated with the system conditions, a number of studies apply the probabilistic approaches to integrate the uncertainty into the TTC evaluation. In [5], the parametric bootstrap technique is used to generate bootstrap samples representing the uncertainty. Another commonly applied technique for the probabilistic TTC evaluation is Monte Carlo simulation [6–9]. Khaburi and Haghifam [6] apply Monte Carlo simulation to represent the load uncertainty. In addition to load uncertainty [7], also includes the uncertainty of the availability of the transmission lines. The impacts of uncertainty are now more pronounced as a result of a rapid growth and development of renewable energy due to its intermittent and uncertain power output characteristic. In [8], the TTC is evaluated by Monte Carlo simulation considering wind power output uncertainty. The clustering technique is applied to speed up the computation. Ref. [9] employs Monte Carlo simulation and risk-based TTC selection to examine the impact of renewable energy penetration on the TTC. Several efforts have been made to speed up the TTC evaluation. In [10], a two-step method is presented. The security constraint to
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which the system is prone is predicted in the prediction step, and the TTC is then found in the correction step. A radial basis function neural network based approach (RBFNN) is proposed in [11] in which the transfer capability is found from the mapping function as the RBFNN learns during the training. Nonetheless, none of these speed-improving techniques take into account transient stability constraint. When neglecting transient stability, the result is prone to be optimistic and may not be practical. Some studies include the transient stability constraint into the TTC evaluation using Transient Stability Constrained Optimal Power Flow (TSCOPF) [12], or energy function methods [13,14]. Nonetheless, most of these studies are simply limited to a system with only synchronous generators, and there is currently no investigation on a system with renewable energy. To address the aforementioned issues, this paper proposes a risk-based TTC evaluation using Monte Carlo simulation where the uncertainty associated with system parameters and renewable energy power output, and security constraint related to transient stability are properly taken into account. However, this method requires excessive computing time in view of a large sample size; as a result, the TTC may not be available to the system operators in time. Since the heavy computational burden is a major impediment to the widespread use of this method; hence, a system case partitioning method using two filters, i.e. performance indices and decision tree, is proposed as a means to enhance the computational speed. This paper is organized in the following sequences. Section 2 provides the definition of the TTC and describes the algorithm of the risk-based TTC evaluation by means of Monte Carlo simulation. Section 3 describes the modeling of wind power and PV system. Section 4 presents the proposed system case partitioning. Numerical examples, results, and discussion are presented in Section 5. Finally, the conclusion is given in Section 6.
2.1. System case generation First, a set of system cases is generated by the sampling process. In general, a system case depends on several uncertain parameters, e.g. short-term forecast demand, forecast renewable energy power output, equipment availability, and fault location. These parameters are sampled basically based on their probability distributions as described in the following. The forecast demand and PV power output are assumed here to have a normal probability distribution which is given by:
1 ðx lÞ2 f ðxÞ ¼ pffiffiffiffiffiffiffi exp 2r2 r 2p
! ð1Þ
where l is the forecast value, and r is the standard deviation representing the accuracy of the forecast value. To sample the parameter, first, a normally distributed random number, z (with zero mean and unity standard deviation) is generated either by the approximate inverse transform method or Box– Müller method. The sampled value is then converted from
x ¼ xl þ rz
ð2Þ
where x and xl are the sampled and forecast values. On the contrary, the probability distribution of the short-term wind power forecast changes with the amount of the power output. Based on the statistical analysis of wind power forecast error described in [17], a Beta distribution rather than a normal distribution can more accurately approximate the short-term wind power forecast. The Beta distribution is given by:
f ðx; a; bÞ ¼ Bða; bÞ ¼
Z
1 xa1 ð1 xÞb1 Bða; bÞ 1
xa1 ð1 xÞb1 dx
ð3Þ ð4Þ
0
2. Risk-based TTC evaluation This section starts with the TTC definition and moves onto the detailed description of the risk-based TTC evaluation with transient stability consideration. According to NERC, TTC was defined as the maximum of power that can be transferred in a reliable manner between a pair of defined source and sink locations in the interconnected system while meeting all of a specific set of defined pre- and post- contingency system conditions [1]. It is important to note from NERC’s document that only a general guideline not a standard method for determining, modeling the transfer capability is provided. Therefore, the TTC evaluation method may be slightly different from application to application. TTC can be simply evaluated by a deterministic method in which the TTC is determined from the minimum transfer capability within a predefined set of system conditions, e.g. all N 1 contingencies [15,16]. However, the result is commonly conservative due to a lack of the ability to properly reflect the uncertainty and stochastic nature of a system. Emphasized by the presence of renewable energy, the uncertainty is now realized essential to be taken into account in the TTC evaluation. This paper focuses on a point-to-point transfer of the interested path. The algorithm of the risk-based TTC evaluation is composed of four main steps, each of which will be described in detail in the following subsections. 1. 2. 3. 4.
System case generation. MTC calculation. Transient stability assessment. Risk-based TTC selection.
Here B(a, b) is the Beta function, and a, b are parameters related to the mean, l, and variance, r2 as follows:
l¼
a aþb
r2 ¼
ð5Þ
ab ða þ bÞ2 ða þ b þ 1Þ
ð6Þ
A Beta distributed random number, b is generated from
bða; bÞ ¼
gðaÞ gðbÞ þ gðaÞ
ð7Þ
where g is a Gamma distributed random number which is generated from
gðcÞ ¼ ln
c Y
Ui
ð8Þ
i¼1
where c is the shape parameter, and Ui is a uniform random number within [0, 1]. The uncertainty degree of the forecast parameters is specified here by a standard deviation expressed in percent of the forecast value. Equipment availability is determined from the operating status of each component in the system, i.e. a transmission line. This can be done by a state sampling technique [18] with the use of historical reliability data. In case of an outage due to a disturbance, the exact fault location on the faulted element is also of concern. The fault on a transmission line may occur in any location along the line, and its probability distribution generally does not follow a uniform distribution. Therefore, in this paper, the fault location is sampled from a discrete probability distribution. A transmission line is divided into
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three segments, i.e. the near-ended, middle, and far-ended, each of which is associated with the probability of the fault occurrence. 2.2. MTC calculation For each given system case, the maximum transferable power, i.e. Maximum Transfer Capability (MTC) is computed. Different operating conditions result in different amounts of the transferable power. For a point-to-point transfer, the amount of the additional electric power delivered from the source bus to the sink bus is specified by a parameter k. As a result, maximizing the transfer is equivalent to maximizing the k. With this additional electric power transferred to the sink bus, the active and reactive demands at the sink bus are modified as follows:
PD ¼ P0D þ k cosðw0 Þ QD ¼
Q 0D
0
þ k sinðw Þ
ð9Þ
s:t:
9 > > > =
Gðx; kÞ ¼ 0 > > > ; Hðx; kÞ 6 0
ð11Þ
where x is a vector of system control and state variables, G(x, k) are a set of equality constraints representing the power balance equations, and H(x, k) are a set of inequality constraints representing the system security limits including voltage magnitude limit, generation capacity limit, and equipment thermal limit. Once the maximum k is solved, the MTC is computed from
MTC ¼ P0D þ kmax cosðw0 Þ
3. Renewable energy modeling
ð10Þ
where PD, QD are active and reactive demands at the sink bus; P0D ; Q 0D are base-case active and reactive demands at the sink bus; k is a transfer parameter; and w0 is the base-case power factor angle of the sink bus. The optimization problem can be formulated in a general form as follows:
max k
MTC and select the TTC. In the past, the TTC was simply selected from the minimum MTC within a set of N 1 contingencies. This selection scheme is conservative and does not reflect stochastic nature of the system. In addition, the TTC is often masked by the severe system condition which usually has a small probability of occurrence; hence, it may not occur within the specified lead-time. Therefore, this paper instead selects the TTC based on the risk concept in which the stochastic nature of a system is well reflected through the PDF. The risk refers to the chance that the system fails to conduct the transfer at the specified value, i.e. the accumulated probability of the cases having the MTC smaller than the specified transfer, i.e. the shaded area shown in Fig. 1. The criterion for specifying the risk basically depends on the system security requirement. The optimal risk can also be determined from a benefitrisk analysis.
ð12Þ
In this paper, the OPF is solved by a well-known sequential quadratic programming (SQP). 2.3. Transient stability assessment As mentioned previously, the ability of a system to transfer electric power may be restricted by the transient stability constraint following a disturbance. In some studies, the transient stability constraint is directly added into the OPF (TSCOPF) [12], or assessed by energy function methods [13,14]. Nonetheless, these methods have the convergence difficulty and modeling limitations; hence, including renewable energy into the analysis would be relatively difficult. As a result, their applications are limited to a system with only synchronous generators, which may not be applicable for the future system as more renewable energy penetration is expected. Therefore, this paper employs an iterative process. Transient stability is assessed by a conventional time-domain simulation where the models of renewable energy based generation are included. Note that the MTC obtained in the previous step is computed from the post-disturbance condition. Therefore, this step checks whether the system can stably move from pre- to post-disturbance conditions. If unstable, the transfer is adjusted. The adjustment is done in a bi-section manner with a specified number of steps.
This paper focuses on wind power and photovoltaic (PV) generation. They have been successful and developed with a large penetration expected. Japanese government already set out a target of 6610 MW wind power and 53 GW PV by 2030 [19]. Their modeling is described here by an aggregated model in order to reduce the complexity and problem dimension. 3.1. Wind power generation This paper focuses on wind power generation with a constant speed wind turbine technology because this type of power plant has a more severe impact on the transient and voltage stabilities than the other types. The configuration is simply a wind turbine coupled with an induction generator and directly connected to the grid with a series of capacitor banks installed at the terminal for reactive power compensation. Usually, only the forecast wind power output and induction machine’s parameters are available. The other related parameters, i.e. terminal voltage, consumed reactive power, and rotor slip are computed from the modified power flow problem. The power balance equations of the bus i connected to the wind power generation are modified to include the complex power terms of the wind power generation as written in (13) and (14). Due to the addition of a new variable, i.e. a rotor slip, another equation, i.e. (15), is required for solving the power flow problem. Eq. (15) enforces the wind power output to its specified or forecast value.
fPi ¼ F Pi ðd; VÞ Pspec Wi þ P Di ¼ 0
ð13Þ
fQi ¼ F Qi ðd; VÞ þ Q Di þ Q Wi ðV i ; si Þ ¼ 0
ð14Þ
fPW i ¼ PWi ðV i ; si Þ Pspec Wi ¼ 0
ð15Þ
where
2.4. Risk-based TTC selection Steps 2–3 are repeated for a given number of system cases. The final step is to build the probability density function (PDF) of the
Fig. 1. Risk-based TTC selection.
N. Paensuwan et al. / Electrical Power and Energy Systems 44 (2013) 530–539
PWi ðV i ; si Þ ¼ Q Wi ðV i ; si Þ ¼ F Pi ðd; VÞ ¼
V 2i R2eq ðsi Þ
þ X 2eq ðsi Þ
which converts DC power fed from the PV arrays to AC power. In this paper, the steady-state model of a PV system is considered as a PQ bus operating at a unity power factor. The PV dynamic model used here is highly simplified. The power conditioning unit of a PV system is modeled as a current source [21]. The current is controlled to achieve the active and reactive power reference set-points, i.e. Pref and Qref. The power regulator is simply a PI controller as shown in Fig. 3. It is assumed that the inverter can adjust its current with the time delay T of 0.01 s fast enough to follow the current reference signals from the power regulator.
Req ðsi Þ
V 2i R2eq ðsi Þ
þ X 2eq ðsi Þ
X eq ðsi Þ
N X V i V j ðGij cos dij þ Bij sin dij Þ j¼1
N X F Qi ðd; VÞ ¼ V i V j ðGij sin dij Bij cos dij Þ: j¼1
Here PWi and QWi are the active and reactive power terms of the wind power generation expressed using load convention; P spec is Wi the specified or forecast wind power output; Gij and Bij are the real and imaginary parts of the ijth element in the admittance matrix; Vi and Vj are the voltage magnitudes at buses i and j respectively; dij is the voltage angle difference between buses i and j; and N is the number of buses. In (16) and (17), Req, Xeq are the equivalent resistance and reactance derived from a steady-state circuit of an induction generator as shown in Fig. 2.
Req ¼ 2 R r
s
R 2 r
X eq ¼
s
Rr X 2m s
þ ðX m þ X r Þ2
X m þ X m X r ðX m þ X r Þ Rr 2 þ ðX m þ X r Þ2 s
533
ð16Þ
ð17Þ
The dynamic model of wind power generation is based on an aggregated two-mass model. There are three differential equations describing the motion of the two rotating parts, i.e. wind turbine and induction generator rotors, and two differential equations associated with the electrical part of an induction generator. For more detailed descriptions, please refer to [20]. 3.2. Photovoltaic (PV) generation The photovoltaic (PV) power technology uses semiconductor cells to convert solar energy to electrical energy. A PV system is connected to the grid via a power conditioning unit, i.e. inverter,
4. System case partitioning Monte Carlo simulation usually requires a sufficiently large sample size to ensure the convergence; as a result, it is time-consuming. Referring to the risk-based TTC selection described in Section 2, it can be noted that not all the system cases are related to the risk, i.e. having small MTC. Therefore, to obtain the TTC, it is not necessary to compute the MTC of all the system cases, only those related to the risk are sufficient. Based on this concept, the system cases will be partitioned into two main groups based on their severity; risk-related and non-risk-related groups respectively. Considering only the above partitioned cases, some of the OPF calculation and time-domain simulation in the Monte Carlo simulation can be avoided; as a result, the run time can be saved. Most of the risk-related cases are severe cases whose transfer is restricted by either steady-state or transient stability constraint. Thus, two filters are used here to screen out such risk-related cases. Filter 1 screens out the risk-related cases due to the steady-state constraints using the performance indices (PIs) [22]. Filter 2 screens out the risk-related cases due to the transient stability constraint. Filter 2 employs a decision tree to predict the system transient stability. The partitioned set is then built by the partitioning process described in Section 4.3. It should be emphasized here that the proposed risk-based TTC evaluation with system case partitioning differs from a conventional TTC evaluation with contingency screening in that the proposed method starts off with much wider sample space where the uncertainty is fully taken into account. The process is based on a probabilistic method and focuses only on the sample set required to obtain the risk-based TTC value. The principles of performance indices and decision tree classification are described in the following subsections. 4.1. Performance indices
Fig. 2. Steady-state induction machine circuit.
The performance indices are commonly used in contingency ranking to measure the severity of the system conditions. They are also applied here to rank the system cases within the sample set according to their severity degree. First, the bus voltage performance index, JV is given by
JV ¼
2n N PQ X V i V i;mean cVi V i;tol i¼1
ð18Þ
max min ; Vi,tol is where Vi,mean is the mean bus voltage, i.e. 0:5 Vi þ Vi ; Vi is the voltage at the deviation tolerance, i.e. 0:5 V max V min i i bus i; cVi is a coefficient; n is a positive integer defining the exponent; and NPQ is the total number of PQ buses. Similarly, the generation reactive power performance index, JQ is given by
JQ ¼ Fig. 3. Power regulator of a PV system.
NG X Q i Q i;mean 2n cQi Q i;tol i¼1
ð19Þ
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N. Paensuwan et al. / Electrical Power and Energy Systems 44 (2013) 530–539
where Qi,mean is the mean generated reactive power, i.e. ; Qi,tol is the deviation tolerance, i.e. 0:5 Q max þ Q min i i max min ; Qi is the generated reactive power by a generator 0:5 Q i Q i i; cQi is a coefficient; n is a positive integer defining the exponent; and NG is the total number of generators. The branch flow performance index, JS is given by
JS ¼
N br X i¼1
cSi
Si
2n
Smax i
ð20Þ
Fig. 4. Example of a decision tree.
where Si is the apparent power flow of the branch i; Smax is the theri mal rating; cSi is a coefficient; n is a positive integer defining the exponent; and Nbr is the total number of branches. Finally, the system performance index, Jsys can be obtained from
J sys ¼
wV J V þ wQ J Q þ wS J S wV þ wQ þ wS
ð21Þ
where wV, wQ, wS are the weighting factors. They are equally set to 1 in this paper. Note that the performance index, J is a function of the state variables, i.e. J(x). The state variables change as the system condition changes due to the uncertainty, outages, and addition of the transfer. Therefore, the severity of the system cases can be ranked according to the change of the system performance index value as follows:
DJsys ¼ Jsys ðxÞ Jsys ðx0 Þ
ð22Þ
where x0 and x are the vectors of state variables prior to and after the system condition change respectively. The coefficients in (18)–(20), cVi, cQi, cSi are 1 if the system condition after the system condition change is more severe than the base-case condition; and 0 otherwise. Here, the exponent n is set to 4 which is found large enough to avoid the masking problem. The algorithm for the system case ranking is summarized as follows: 1. Run power flow at the base-case condition, and find the basecase MTC. 2. Apply the system condition change, i.e. change of demand, renewable energy power output, system configuration, and add the transfer, here set to 75% of the base-case MTC. This addition of the transfer helps specify the direction where the system will move from the base-case condition; hence, the constraint which limits the transfer can be predicted. 3. Run power flow for the system condition after the system condition change, and compute the performance indices using (18)–(21). 4. Compute DJsys using (22), and rank the system cases. 4.2. Decision tree classification A decision tree is one of the classification methods in a form of a tree graph. Several applications of decision tree classification to the power system have been reported [23,24]. It is mostly used as a prediction tool for a fast system security assessment, e.g. an online transient stability assessment. A typical decision tree consists of three types of nodes: root node, decision node, and leaf node as shown in Fig. 4. A root node is at the top of the tree where the classification starts. A decision node conducts a test or question, where the cases are separated, i.e. splitting a node. A leaf node provides the output classification. A decision tree classifies the case as the case is moving down the tree passing through a series of tests until it falls to one of the leaf nodes from which the class is determined. In Fig. 4, X = {X1, X2, -
Fig. 5. Partitioning process.
. . . , Xn} is a set of n input attributes and c = {c0, c1} is a set of two output classes. A decision tree is grown offline from the training data. Growing a decision tree is simply splitting a node based on a splitting rule. The best split among all possible splitting rules can be achieved by comparing their scores representing how well they can separate the cases. In this paper, the score is calculated from the impurity measure GINI diversity index [25]. Therefore, the split with the best score will be selected. The nodes are split until they are either pure i.e. all cases in the node belong to the same class, or with the number of cases in the node smaller than the specified value, i.e. a minimum split criterion. This criterion is applied so as to prevent the tree from having too many leaf nodes, which is likely to over-fit the training data. The final step is to validate the obtained decision tree by testing with the test set to check its generalization. The input attributes are commonly selected from the system parameters or available measurements. In this paper, the input attribute of the decision tree for the system case partitioning filter is the shortest distance in per unit from the fault location to each generator determined from the Dijkstra’s algorithm [26]. This input attribute is chosen based on the fact that the disturbance is severe when the fault occurs near the generators. The decision tree helps manage this information and uses it for the classification in an effective way. Note that using only one input attribute may not give an accurate prediction but sufficient in roughly identifying the risk-related cases due to the transient stability constraint. When applied to the system case partitioning as Filter 2, for each outage case in the Monte Carlo sample set, the vector of the value of the input attribute (i.e. the shortest fault distance to the generators) is passed through a series of questions (i.e. decision nodes) in the decision tree until falling into one of the leaf nodes which indicates whether this case is stable or not. The system cases predicted by the decision tree as unstable are regarded as severe; hence, risk-related.
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Fig. 6. Flowchart of the proposed algorithm.
4.3. Partitioning process The algorithm of the partitioning process is given in Fig. 5. The system cases from the system case generation are applied to two filters. Filter 1 ranks the cases based on the DJsys value in a descending order, i.e. from the most severe (largest DJsys) to the least severe cases (smallest DJsys). At the same time, the stability of the outage cases within the sample set is predicted using Filter 2, i.e. decision tree. The partitioned set is then built by selecting the cases from the results of these two filters. Here the results of Filter 2 (i.e. the outage cases classified by Filter 2 as unstable) are included first in the partitioned set and followed by those of Filter 1 (i.e. ranked system cases) until the specified number of cases for the partitioned set is attained. The flowchart of the algorithm for the improved TTC evaluation with system case partitioning is depicted in Fig. 6.
5. Numerical examples 5.1. Study system The proposed method is applied to the 90-bus system with a single-line diagram shown in Fig. 7. The system is obtained by connecting three IEEE 30-bus test systems [28] and with some modifications. The system has 14 thermal power plants with the total capacity of 890 MW, and total system demand of 624.36 MW. Nine
10 MW wind farms and nine 10 MW PV systems are added. The point-to-point transfer from bus#1 to bus#42 is evaluated. 5.2. Parameter setting and study cases The total number of generated cases is 5000 which is found to be sufficient for the convergence of the simulation. As a result of the sampling, out of the 5000 cases, there are 3490 non-outage and 1510 outage cases respectively. The forecast demand error is 10% of the forecast value. The forecast power output of each wind farm and PV system are 7.5 and 8 MW with the forecast error of 20%. The probability of the fault location is adopted from Ref. [27] as tabulated in Table 1. The simulation time for the time-domain simulation is 6 s with the time step of 0.01 s. The fault is cleared within 0.35 s. Note that the fault clearing time is relatively long so that the impact of the transient stability can be manifested. The decision tree is grown from the training data of 3000 cases and tested with the testing data of 1000 cases. These data sets are obtained from the system case generation used for the training data preparation. They can also be retrieved from actual historical operating records if available. The minimum split criterion is 100. The simulation is run on a Pentium Core2Duo 2.4 GHz PC with 4 GB RAM. To validate and examine the performance of the proposed method, four comparative study cases with different numbers of cases in the partitioned set are conducted as summarized in Table 2. The performance in terms of the accuracy and computational speed enhancement will be presented and discussed.
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Fig. 7. 90-Bus test system.
Table 1 Probability distribution of the fault location. Probability of the fault location
Near-ended Middle Far-ended
0.1307 0.7021 0.1672
Table 2 Summary of the partitioned sets. Cases
Number of cases
1 2 3 4
1000 2000 3000 5000
5% 10% Specified risks 15% 20%
0.025
Probability density
Line segment
0.03
0.02
0.015
0.01
0.005
0
0
10
20
30
40
50
60
70
80
MTC [MW] 5.3. Results and discussion The PDF of the MTC and the illustration of the risk-based TTC selection are depicted in Fig. 8. The decision tree for Filter 2 is shown in Fig. 9. The testing error is found to be 6.9%. The number
Fig. 8. PDF of the MTC.
of cases classified by this decision tree as unstable is 244 which is larger than the actual number of 146. However, the overall
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Fig. 9. Decision tree for Filter 2.
Table 4 Summary of the DTTC (Cases 1–3).
MTC [MW]
80 60
Risk (%)
DTTC (%)
40 20 0
0
1000
2000
3000
4000
5000
5 10 15 20
Case 1
Case 2
Case 3
0.09 2.02 6.15 34.13
0.00 0.27 0.61 1.11
0.00 0.03 0.13 0.24
Sample (without ranking) MTC [MW]
80 60 40 20 0
0
1000
2000
3000
4000
5000
Sample (with ranking) Fig. 10. Comparison of the MTC distribution.
Table 3 Summary of the TTC values (Cases 1–4). Risk (%)
5 10 15 20
TTC (MW) Case 1
Case 2
Case 3
Case 4
43.15 47.89 51.58 66.79
43.10 47.07 48.89 50.35
43.10 46.96 48.65 49.92
43.10 46.94 48.59 49.80
accuracy is not much a concern here as long as most of the actually unstable cases are classified correctly. The comparison of the MTC distribution between without and with the ranking is shown in Fig. 10. It can be seen from Fig. 10 that the proposed system case partitioning method can successfully screen out the risk-related cases. The TTC values selected at four different risk levels are summarized in Table 3. It is obvious from the simulation results that the more cases used, the more accurate result. The accuracy here is
determined by comparing with the result of Case 4 where all 5000 cases are used. It can be observed from Table 3 that the TTC obtained by using the partitioned set is slightly larger than the actual TTC. This implies that the partitioned set does not cover all the risk-related cases. The TTC errors from Case 4 in percent, DTTC, are also tabulated in Table 4. The results show that the TTC errors generally reduce as more cases are included in the partitioned set. Interestingly, when using a small number of cases, i.e. Case 1, the TTC error increases with the specified risk. This is because first obtaining the TTC at high risk requires a sufficient number of cases to build the probability distribution covering the range from which the TTC is selected. Thus, the partitioned set with a small number of cases may not cover all the risk-related cases. Second, the cases required for building the distribution at high risk are composed of both severe and fairly severe cases (i.e. having fairly large MTC). Unlike the severe cases, it is relatively difficult to accurately rank the fairly severe cases, i.e. the values of DJsys are not much different. To achieve a given level of accuracy for TTC, the number of cases should be more than that required for building the distribution. For example, given the total number of cases of 5000, the TTC at 20% Table 5 Summary of the run time (Cases 1–4). Cases
Run time (s)
1 2 3 4
4746 7727 10,962 16,793
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risk requires at least 1000 cases for building the distribution for the TTC selection. It can be seen from the result that using only 1000 cases cannot get the accurate TTC (i.e. with the error as much as 34.13%). In this example, it is better to use 3000 cases (Case 3) which is sufficient to obtain the TTC with the error less than 1%. The run time of Cases 1–4 is listed in Table 5. The run time for the system case partitioning process is about 50 s which is relatively small compared to the total run time of 16,793 s (Case 4). The decision tree growing takes less than 1 s. It is obvious from the results that the run time strongly depends on the number of cases, i.e. the more cases used, the more run time required. The proposed method can save a significant amount of the run time. For example, the TTC can be obtained within 7727 s (Case 2) accounting for only 46% of the total run time (i.e. 16,793 s) with the error less than 2%. The computational speed can be further improved if the simulation task is distributed to several PCs. Nonetheless, there is always a tradeoff between speed and accuracy. Therefore, to select the right partitioned set basically depends on the given accuracy level and security requirement. If the accuracy and security requirement are the primary concern, a sufficient number of cases should be included in the partitioned set.
Table A.6 Generator dynamic data. Gen#
MVAbase Ra
1 125 2, 6, 7, 10, 11 100 3, 4, 5, 9, 12, 13, 14 75
Xd
X 0d
T 0do
Xq
X 0q
T 0qo H
0.078 1.22 0.174 8.97 1.16 0.174 0.5 4.29 0.075 1.18 0.220 5.90 1.05 0.220 0.3 4.49 0.070 1.05 0.185 6.10 0.98 0.185 0.3 5.56
Fig. A.11. Exciter model.
6. Conclusion This paper proposes the system case partitioning for the efficient risk-based TTC evaluation. The simulation results show that the system case partitioning can capture the risk-related cases; hence, helps reduce the number of cases to be evaluated. As a result, it can save a significant amount of the run time while still obtaining an acceptable accuracy. The proposed method allows the TTC evaluation to take into account the uncertainty with less computational burden. However, when applying the proposed method, it is important to ensure that the partitioned set covers a sufficient number of cases for a given risk level. Our future research aims at further speeding up the TTC evaluation to a level that it can be included in the real energy management system. This makes possible to provide the system operator with information necessary for the decision making and efficient operation of a power system.
The values are in per unit on machine base. Please also refer to [20] for detailed descriptions on wind power model. PV system data: Kp = 0.05, Ki = 0.5 for both active and reactive power control, and T = 0.01.
Appendix A. Data for transient stability analysis
References
This section gives data of the generators, wind power, and PV systems for the transient stability analysis. Since the dynamic data of the IEEE-30 system is not available, this paper adapted the data from Ref. [29]. The generators are modeled by a 4th order model with the data given in Table A.6. The models for the exciter and turbine governor system are depicted in Figs. A.11 and A.12 respectively. The parameters for the exciter and turbine governor system are assumed to be identical for all generators. Due to the limit of the space, please refer to [30] for the detailed descriptions of the models and parameters. Exciter data:
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T r ¼ 0:06;
K a ¼ 25; T a ¼ 0:2; V Rmax ¼ 3:438; V Rmin
¼ 3:438; K f ¼ 0:05; T f ¼ 0:35 Turbine governor system data:
R ¼ 20;
T min ¼ 0; T max ¼ 1; T s ¼ 0:09; T c ¼ 0:2; T 3 ¼ 0; T 4
¼ 1; T 5 ¼ 5 Wind power generation data (for each turbine):
MVAbase ¼ 2:25;
Rs ¼ 0:048; X s ¼ 0:075; X m ¼ 3:8; Rr
¼ 0:018; X r ¼ 0:12; Hm ¼ 3; Hxr ¼ 1; K s ¼ 0:3
Fig. A.12. Turbine governor system model.
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