Aggregation of Buses for a Network Reduction - IEEE Xplore
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 27, NO. 2, MAY 2012
705
Aggregation of Buses for a Network Reduction HyungSeon Oh, Member, IEEE
Abstract—A simple but precise model would improve the computation efficiency for planning a large-scale power system. Recent development of a new network reduction algorithm yields such a model with a given grouping of buses. While the flow over the reduced network is estimated precisely, a proper value of the flow limit still remains unresolved. In this study, a method is proposed to group buses based on the congestion profile in the original network and to assign the flow limits of a reduced network. The method is tested on modified IEEE 30-bus and 118-bus systems at various load profiles, and the simulation results are compared. Index Terms—Bulk, congestion, demand-rich area (DRA), near boundary, optimal power flow (OPF), power transfer distribution factor (PTDF), supply-rich area (SRA).
NOMENCLATURE PTDF elements corresponding to the lines of which flow constraints are binding. PTDF matrix, the sensitivity matrix of flow with respect to the injection, with cardinality of -by- . Number of lines in the original network. Number of buses in the original network. Load vector with the cardinality of
-by-1.
Flow vector with the cardinality of -by-1. Generation vector with the cardinality of -by-1. Injection vector with the cardinality of
-by-1.
Electrical distance from a congested line. Thickness of near boundary. Lagrangian. Shadow price of the binding flow constraints. Scalar to identify the membership of bus according to a congested line. Locational marginal price (LMP). System marginal cost. Threshold value to evaluate the impact of congestion. Manuscript received December 27, 2010; revised April 06, 2011 and May 18, 2011; accepted June 30, 2011. Date of publication January 09, 2012; date of current version April 18, 2012. Paper no. TPWRS-00971-2010. The author is with the State University of New York at Buffalo, Buffalo, NY 14260 USA ( e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPWRS.2011.2176758
Standard deviation. Measure of the impact of a congestion. I. INTRODUCTION ECENT efforts for the integration of renewable electric technologies into the power system have been increasing over the past decade due to the growing concern regarding climate change. These efforts result in the development of efficient expansion-planning algorithms for optimizing both the transmission network and generation [1]–[5]. These algorithms have limited applicability in the power system planning, however, due to the high computational cost associated with a large-scale power system. A traditional network reduction method yields a simple model [6]. This method typically involves computing impedances and eliminating unnecessary elements [6]–[10]. As a result, the reduced model includes a highly dense impedance matrix, which makes it difficult to increase efficiency significantly when using the reduced network. Equivalent networks reproduce the same voltages and currents of the remaining buses as the original systems do. However, the power transfer between areas is not preserved because the flows of the eliminated branches cannot be approximated. A reduction method based power transfer distribution factor (PTDF) is proposed to preserve the same flow pattern as that in the original network [11]. The model yields a similar flow profile at the injection profile for which the reduction is performed. However, the model also depends on the operation setpoint [12]. The solutions from planning studies using the model may be infeasible because the studies should consider various injection portfolios. A new method to reduce network was recently proposed to construct the flow sensitivity matrix to the reduced injection [12]. The reduced injections and the reduced flows are defined as the aggregated injections and the aggregated flows at a predefined group of buses. From the procedure, the method yields a simple but precise injection profile independent model. However, it is not clear how a group of buses should be defined. As a result, the injection and the flow profiles might be significantly different if the original network is used for an optimal power flow (OPF) study. In OPF studies, congestion often increases the system cost significantly because it may keep cheap generation from being dispatched. Congestion plays a critical role in OPF and planning studies. Because congestion arises where flow over a line equals the limit of the line, it is important to assign the flow limits for the studies. Therefore, it is also important to assign adequate flow limits for the aggregated lines (corridor). Even though the reduction algorithm yields a simple but precise model, its applicability can be limited without the proper assignment of the flow limit. In this paper, a new method is proposed to find a proper grouping of buses for the reduction and to assign the flow limits
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so that the congestion profile is preserved. The algorithms for the bus grouping and for the assigning flow limits are key features for preserving the congestion profiles. The method will be applied for OPF studies, and the OPF results are compared with those from the original networks in terms of the congestion profiles, the locational marginal prices (LMP), and generation profiles. II. CONGESTION MODELING A. Network Reduction Procedure Power system planning is, in general, computationally demanding due to its large scale. For reducing the computational cost, it is desirable to have a simple but precise scenario independent and reduced path independent model. A reduction method is described in [12]. Here is a brief description of the method: 1) First, define areas so that all the buses inside an area have similar properties. 2) Ignore all the lines connecting two buses that belong to the same area. 3) Lines connecting two buses that belong to different areas are aggregated. 4) PTDF of the reduced network is computed. 5) Reactance values for the reduced network are computed from the PTDF obtained in 4 and the node-branch incidence matrix for the reduced network. A simple but precise reduced model is obtained from the process if the areas are provided. [12] showed that power flow studies with the reduced model yield similar flow patterns. However, it is not clear how to determine the areas. For defining the areas, the buses inside an area should have similar properties. For optimal planning studies, significant importance is imposed on the economic impacts. Therefore, properties related to the economic impact need to be considered. Because LMP reflects the impact, the areas are defined based on the price. Therefore, area is a set of buses of which LMP falls into a similar value. OPF is an optimization process to find an optimal but feasible generation portfolio that will not violate the economic, operational, and engineering constraints. One can construct the Lagrangian, , and find the solution from a relaxation process. By definition, LMP is the increase of the value of the Lagrangian with respect to a one-unit increase of the load at the loca. There are three components in LMP: tion; i.e., the system marginal cost of electricity, ; the impact due to the congestion; and the cost associated with losses. Typically, the last component is marginal in comparison to other components. The first term affects LMP uniformly, and thus, it does not provide useful information for defining areas. Therefore, only the impact of the congestion on LMP will be discussed in this paper. B. Impact of the Congestion on LMP Congestion is the subset of the flow constraint that is binding; i.e., the flow over the congested line, , equals the flow limit of the line. Therefore, the second component is the product between the sensitivity of the binding constraints and the corresponding shadow prices; i.e., (1)
Fig. 1. Modified IEEE 30-bus system with a congested line (red line) between Bus 23 and 24. The solid back line shows the separation of a system into the supply-rich area and the demand-rich area.
Because the system cost decreases if the binding flow limits increase marginally, the values of are non-negative. As a result, the second component in LMP is proportional to the values of PTDF elements. Depending on the electric distance of buses from the congested line , the impact of the congestions varies. The impact can be negative or positive; i.e., LMP becomes higher or lower with congestion depending on . As a result, congestion divides the system into two areas: one with positively impacted LMPs, and the other with negatively impacted LMPs. When a line is congested, the power injected from one end of the line equals the one ejected from the other end if losses are ignored. At the ejection side, the power is deficient; i.e., the sum of loads is greater than that of generations. The area is termed demand-rich area (DRA). The congestion results in higher LMP than the system marginal cost. The injection side is termed supply-rich area (SRA) because the sum of loads is less than that of generation. The congestion results in lower LMP than the system marginal cost. Fig. 1 shows the separation of the modified IEEE 30-bus system [13] due to a congested line. If a system is separated into DRA and SRA, LMPs in DRA would be higher than those in SRA. C. Separation of Network Due to Congestion In a DC power flow model, the power flow over the transmission network equals the product between the PTDF matrix and the injection vector
(2) and are the elementary charges that generaNote that MW, respectively. tion and load carries, equal to 1 MW and Equation (2) implies that some electric power generations from SRA migrate to DRA through the congested line and that some
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junction and LMP separation due to congestion. Therefore, the classification to the four sub areas can be made in the power system due to a congested line. From the analogy between the electric potential in the p-n junction and LMP of a system with congestion, one can evaluate the impact congestion has on LMP as a function of the electric distance of a bus from the congested line using (3) with (1) as a case with a congested line: for for for for Fig. 2. Variation of LMP as a function of the electric distance from the congested line as in the case shown in Fig. 1. The broken line illustrates the variation only graphically.
loads from DRA also migrate to SRA. Both directional migrations occur until a steady state at the line is established. At the steady state, it is difficult to bring generation from SRA closer to the line and to bring load from DRA closer to the line. The difficulty can be evaluated using LMP because it is, by definition, the increase of Lagrangian by adding an additional load at a location of interest. In DRA, LMP becomes higher as a bus is located electrically closer to the congested line. In SRA, the opposite behavior is observed. Fig. 2 illustrates how LMP changes with respect to the electric distance from the interface. The broken line only represents the graphical change of LMP. Based on this price separation, it is possible to group buses into four categories if a proper criterion is provided for dissecting areas between near boundary and bulk areas: SRA near boundary, SRA bulk, DRA bulk, and DRA near boundary. There exists a similar behavior in p-n junction. p-type and n-type semiconductors have uniform electric potential. When they are attached, the charge carriers migrate across the boundary until a uniform Fermi level is established. At the steady state condition, the area near the interface has a low charge carrier density, i.e., depletion region. On energy point of view, it is difficult to locate charge carriers in the depletion region. An electric potential indicates the difficulty to bring an additional charge carrier. As a result, electric potential inside a p-n junction varies with the distance from the interface: for for
(4)
where is a proportionality constant, and is the electric distance that the impact of congestion on LMP is significant. Using (4), it is possible to enumerate the electric distance from the congested line from the congestion cost or PTDF matrix as shown in (1). is the number of The number of the PTDF elements in buses in the system; i.e., only elements are sampled. The distribution of selected PTDF elements is assumed to be independent and identically distributed, i.e., random distribution. In evaluating a PTDF matrix, typically the change in flow is observed when one unit is injected at a bus and ejected at a reference bus. Therefore, the values of a PTDF matrix depend on the choice of reference bus. One can evaluate a PTDF matrix for a different choice of reference bus because the evaluation process is identical with a different bus for ejection. The PTDF elements are evaluated as follows: (5) where is the PTDF element corresponding to the reference bus. depend on the Therefore, the values of the elements in is randomly distributed and choice of the reference bus. If is the mean value of . Then can be (4) holds, shifted by so that the mean value of the shifted becomes zero. Because the maximum value of may not be observed, 99% of covers approximately the entire range of . is a term to evaluate a threshold, which indicates how far the impact of congestion vanishes from -distribution [13]:
(3)
(6)
where is a scalar constant; represents the distance from is the width that the interface in the semiconductor; and the boundary effect exists. The semiconductor with p-n junction is often separated into four areas: p-type bulk, p-type near boundary, n-type near boundary, and n-type bulk. Equation (3) shows the electric potential as a function of the distance from the interface when the charge injected to one side equals that ejected from the other side. As the potential increases, it becomes more difficult to add an n-type charge carrier. On the other hand, the lower the electric potential is, the more difficult the addition of a p-type carrier. It is concluded that there exists analogy between the electric potential at p-n
where is the standard deviation of , ; is the threshold value of ; and is the distance from the congested . line corresponding to The overall impact on LMP due to congestion is the area under the curve in (4). Therefore, the fraction of area under the curve impacts how far congestion affects the system. quantito the impact at fies the relative impact of congestion at where is an electric distance from the congested line, and is the maximum distance that the impact is not negligible: (7)
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 27, NO. 2, MAY 2012
TABLE I TYPICAL VALUES OF THE FRACTION OF AREA AND THE THRESHOLD
For example, the choice of means that one wants to select the buses under any influence of congestion. Therefore, the choice yields all the buses in DRA or in SRA. On the other yields the set of buses that has a full influence. Behand, cause such influence is observed when a bus is infinitely close to the congested line, should be zero. Any choice between 0 and would yield a set of buses so that the influence of congestion is neither none nor full. From -distribution, all the area under the distribution curve . Let the area under the distribuis 99% covered when tion curve be . As shown in (7), is the fraction of area, and it with respect to . From is evaluated when equals ; i.e., the fraction of area , one can evaluate the electric distance from the congested line. Note that the electric distance is expressed in terms of the PTDF matrix shown in (4). Therefore, the PTDF matrix maps the area under the -distribution curve. Suppose corresponds to in -distrithe electric distance from bution. is a scalar value in -distribution that is corresponding to an area fraction of . The ratio of the PTDF element at the bus where electric disand at the bus infinitely close to tance from the boundary is the boundary is
Fig. 3. Bus grouping process to preserve the congestion profile.
(8)
Combining (7) and (8) leads to (9) Table I lists the value of the fraction of area and the threshold value. As decreases, increases; i.e., a large number of affected buses will be covered when a large threshold value of the PTDF element is considered. In this way, there are four sub-areas for a congested line: significantly affected area with positive impact, (DRA boundary); weakly affected area with positive impact, (DRA bulk); significantly affected area with neg(SRA boundary); and weakly afative impact, fected area with negative impact, (SRA bulk). When multiple lines are congested, a bus may belong to multiple areas. A set is defined so that all the buses in the set belong to the same multiple areas. D. Grouping Algorithm and Flow Limit Assignment With given values of and , four areas for each congested line are assigned. It is possible to define sets from a network provided the congestion profile is given. Suppose there exist sets. For each set , the buses that belong to the set are given
where . For the buses that belong to the same sets, the sum of the number ( ) should be equal. After such assignment, select all the buses to which the sum of the numbers assigned is equal. Note that the value for is selected so that the sum of a bus unequivocally yields the membership to which the bus belongs. For example, suppose two possible congested lines are considered. For the first congested line, the is 1.1. Then the buses in the system can be SRA value of near boundary, SRA bulk, DRA bulk, and DRA near boundary. Note that the order is according to the increasing order of the PTDF elements for the congested line. Then the possible values for the membership of the buses for the first congested line are for the second 1.1, 1.21, 1.331, and 1.4641. The choice of congested line can be 1.5 so that the sum can be a signature value. The grouping procedure is illustrated in Fig. 3. In this process, the congested lines belong to the intergroup lines [12]. Suppose a congested line connects two groups, and there exist other lines connecting the same groups. The method in [12] yields a way to aggregate all the lines connecting the groups. However, as mentioned in the Introduction, the flow limits are difficult to assign. Instead, the lines connecting two groups except the congested one are aggregated into a single line parallel to the congested line. Suppose the reduced PTDFs for and , the parallel and the congested lines are based on the method in [12].
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Kirchhoff’s laws dictate the flows at the given injections and, therefore, the PTDFs. Parallel lines that connect the same locations have the parallel PTDF elements regardless of the line properties: (10) and are the row vectors from corresponding to where the th and th parallel lines, and is a constant. Therefore, and need to follow (10). It is reasonable to because the line is physically existent and preserve congested while the other line is aggregated and not congested. should be adjusted parallel to . Therefore, -factorization is performed on :
(11) where and are the orthonormal basis sets spanning the , respectively; is a onereal and the null spaces of dimensional constant. Even though and are not necessarily identical, they should be similar because they connect two common areas. can be expressed by two components shown in (11):
(12) is a vector with a small magnitude because spans the null spaces of , which is similar to . The PTDF element for the parallel is adjusted to , and the second term in (12) is ignored. Because all the congested lines in the original network are preserved in the reduced network, the flow limits of the preserved lines are inherited from the original network. Other non-congested and aggregated inter-group lines are not congested, and therefore, there may be no need to assign any limits for them. E. Limitation of the Proposed Method The proposed method yields a way to group the buses to preserve the congestion profiles that the original network has. A purpose of the network reduction is for planning studies. Therefore, it is possible for unanticipated sets of congestion to occur. If the congested lines yield a completely different congestion profile, then the result from this algorithm may not reflect the proper congestion profiles. Therefore, it is important to include possibly congested lines as well as existing congested lines. III. SIMULATIONS AND DISCUSSION A. Modified IEEE 30-Bus System In the modified IEEE 30-bus system illustrated in Fig. 1 [13], there are two tie lines connecting between Area 2 and the rest of
Fig. 4. Modified IEEE 30-bus system. Tie lines connect Area 2 to the rest of the system. Groups of buses illustrated are listed in Table II.
the system. Due to high demand at Area 2 and tight flow limits of the tie lines, Area 2 is termed a load pocket. For the grouping algorithm proposed in this paper, only the list of the congested lines is required—not the shadow price or LMP. Provided that the tie lines become congested, the values for and are chosen to be 76% and 1, respectively. With the choice, four groups are classified for each congested line: SRA bulk, SRA near boundary, DRA bulk, and DRA near boundary. The values for are 1, 2, 3, and 4 for SRA bulk, SRA near boundary, DRA bulk, and DRA near boundary, respectively. The values for are 1.1 and 0.9 for the lines connecting 4–12 and 23–24, respectively. The second and third columns in Table II list the values. The two vectors listed in the second and third columns are added to uniquely identify the groups. The sum is listed in the forth column. The sum provides signature information to show the membership of the bus. For example, the sum for Bus 10 is 2.02. Because the value is a sum from the following formula, , the only feasible choice for integer is and , which implies that Bus 10 belongs to SRA bulk ( ) for the congestion between 4–12 and SRA ) for the congestion between 23–24. near boundary ( Based on the sum, six groups are formed and listed in the fifth column in Table II. The bus grouping is illustrated in Fig. 4. With the groups, a reduced network is created based on the reduction algorithm in [12]. Fig. 5 illustrates the reduced network. Note that there is a pair of parallel lines connecting Group II and Group III. One of the lines is a congested line connecting Bus 4 and Bus 12 in the original network shown in Fig. 1, while the other line represents the line connecting Bus 4 and Bus 16. DC OPF studies for various load profiles are performed on the original and the reduced network. Various loads are simulated to find different congestion profiles on the original network. Typical simulation results
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TABLE II BUS GROUPING PROCEDURE,
= 1:1 AND
= 0:9
TABLE III CASE WITH NO LINE CONGESTED
TABLE IV CASE WITH TIE LINES 4–12 CONGESTED
TABLE V CASE WITH TIE LINES 23–24 CONGESTED
TABLE VI CASE WITH TWO TIE LINES, 4–12 AND 23–24, CONGESTED
Fig. 5. Reduced 30-bus system based on the bus grouping listed in Table II. There are two parallel lines connecting Groups 2 and 3.
are listed in Tables III–VI. The case listed in Table III is a non-congested case; the case in Table IV includes a congested line between Area 1 and Area 2; the case in Table V contains a congested line between Area 2 and Area 3; and the two tie lines are congested in the case in Table VI. In all the cases listed in Tables III–VI, the congested profiles in the original network
are preserved in the reduced network. As shown in the tables, LMP and the generation portfolio using the reduced networks are approximately those using the original network. Therefore, OPF results with the reduced network are useful to estimate those with the original network. It is interesting to check how much error in LMP increases when unanticipated congestion occurs. When lines connecting 27–28, 9–10, and 6–10 are
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Fig. 6. Modified IEEE 118-bus system. Tie lines (Lines 37 and 38) can be congested. Groups of buses illustrated are listed in Table VII.
TABLE VII LIST OF BUSES FOR THE NINE GROUPS
Fig. 7. Reduced 118-bus system based on the bus grouping listed in Table VII.
congested, the error in LMP increases to 8%. It is clear that if the congestions were considered, Bus 10, 21, 27, 29, and 30 should belong to different areas than Group I. B. Modified IEEE 118-Bus System An IEEE 118-bus system in [13] is illustrated in Fig. 6, and the flow limits of some lines are modified to create congestion profiles. At peak periods, two lines (Lines 37 and 38) can be congested. A similar procedure described in Section IV-A yields the nine groups listed in Table VII, and a reduced network is formed based on the reduction algorithm in [12]. Fig. 7 illustrates the
reduced network resulting from the procedure. DC OPF studies for various load profiles are performed on the original and the reduced networks to compare the results. Various loads are simulated to find different congestion profiles. Typical simulation results are listed in Tables VIII–XI. The case listed in Table VIII is a non-congested case; the case in Table IX includes a congested line (Line 37); the case in Table X contains a congested line (Line 38); and the two tie lines are congested in the case in Table XI. In all the cases listed in Tables VIII–XI, the congested profiles in the original network are preserved in the reduced network. Similar to the results in Section IV-A, LMP and the generation portfolio using the reduced networks are approximately those using the original network. Therefore, OPF results with the reduced network are useful to estimate those with the original network.
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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 27, NO. 2, MAY 2012
TABLE VIII CASE WITH NO LINE CONGESTED
TABLE XI CASE WITH TWO TIE LINES, 37 AND 38, CONGESTED
TABLE IX CASE WITH TIE LINE 37 CONGESTED
the flow limits of the aggregated lines that are not congested do not need to be assigned precisely. As a result, along with the network reduction algorithm, it provides a concise and precise representation of the transmission network for a power flow study. Therefore, this method can be used in a large system OPF, national corridor, and renewable portfolio studies. REFERENCES
TABLE X CASE WITH TIE LINE 38 CONGESTED
IV. CONCLUSION For a large-scale power system, it is practically infeasible to find an optimal solution for power system planning. Recent development of network reduction allows precise modeling with a manageable computation expense. However, a proper assignment for the flow limits is missing, which is necessary for an OPF study. In this paper, an algorithm to find a proper group is proposed to assign for network reduction. The reduced network based on this group shows the same congestion profile as the original network in the OPF studies. Another advantage is that
[1] Northeast Power Coordinating Council, Guideline for Inter-Area Voltage Control, Nov. 1997. [2] ISO-NE, ISO New England Operating Procedure no. 19: Transmission Operation, Apr. 2007, pp. 7–8. [3] E. Fisher, R. O’Neill, and M. Ferris, “Optimal transmission switching,” IEEE Trans. Power Syst., vol. 23, no. 3, pp. 1346–1355, Aug. 2008. [4] K. Hedman, R. O’Neill, E. Fisher, and S. Oren, “Optimal transmission switching with contingency analysis,” IEEE Trans. Power Syst., vol. 24, no. 3, pp. 1577–1586, Aug. 2009. [5] K. Hedman, M. Ferris, R. O’Neill, E. Fisher, and S. Oren, “Co-optimization of generation unit commitment and transmission switching with N 1 reliability,” IEEE Trans. Power Syst., vol. 25, no. 2, pp. 1052–1063, May 2010. [6] J. B. Ward, “Equivalent circuits for power flow studies,” AIEE Trans. Power App. Syst., vol. 68, pp. 373–382, 1949. [7] S. Deckmann, A. Pizzolante, A. Monticelli, B. Stott, and O. Alsac, “Numerical testing of power system load flow equivalents,” IEEE Trans. Power App. Syst., vol. PAS-99, no. 6, pp. 2292–2300, Nov./Dec. 1980. [8] S. Deckmann, A. Pizzolante, A. Monticelli, B. Stott, and O. Alsac, “Studies on power system load flow equivalencing,” IEEE Trans. Power App. Syst., vol. PAS-99, no. 6, pp. 2301–2310, Nov./Dec. 1980. [9] M. K. Enns and J. J. Quada, “Sparsity-enhanced network reduction for fault studies,” IEEE Trans. Power Syst., vol. 6, no. 2, pp. 613–621, May 1991. [10] W. F. Tinney and J. M. Bright, “Adaptive reductions for power flow equivalents,” IEEE Trans. Power Syst., vol. 6, no. 2, pp. 613–621, May 1991. [11] X. Cheng and T. J. Overbye, “PTDF-based power system equivalents,” IEEE Trans. Power Syst., vol. 20, no. 4, pp. 1868–1876, Nov. 2005. [12] H. Oh, “A new network reduction methodology for power system planning studies,” IEEE Trans. Power Syst., vol. 25, no. 2, pp. 677–684, May 2010. [13] R. Zimmerman, C. E. Murillo-Sanchez, and D. Gan, MATPOWER: A MATLAB Power System Simulation Package, 2008. [Online]. Available: http://www.pserc.cornell.edu/matpower/.
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HyungSeon Oh received the Ph.D. degree in electrical and computer engineering from Cornell University, Ithaca, NY, in 2005. He is an Assistant Professor at the State University of New York at Buffalo. Before the appointment, he was a staff engineer at the National Renewable Energy Laboratory. His research interests include: energy systems economics, power system planning, smart grid, storage, computer simulation, and nonlinear dynamics.
705
Aggregation of Buses for a Network Reduction HyungSeon Oh, Member, IEEE
Abstract—A simple but precise model would improve the computation efficiency for planning a large-scale power system. Recent development of a new network reduction algorithm yields such a model with a given grouping of buses. While the flow over the reduced network is estimated precisely, a proper value of the flow limit still remains unresolved. In this study, a method is proposed to group buses based on the congestion profile in the original network and to assign the flow limits of a reduced network. The method is tested on modified IEEE 30-bus and 118-bus systems at various load profiles, and the simulation results are compared. Index Terms—Bulk, congestion, demand-rich area (DRA), near boundary, optimal power flow (OPF), power transfer distribution factor (PTDF), supply-rich area (SRA).
NOMENCLATURE PTDF elements corresponding to the lines of which flow constraints are binding. PTDF matrix, the sensitivity matrix of flow with respect to the injection, with cardinality of -by- . Number of lines in the original network. Number of buses in the original network. Load vector with the cardinality of
-by-1.
Flow vector with the cardinality of -by-1. Generation vector with the cardinality of -by-1. Injection vector with the cardinality of
-by-1.
Electrical distance from a congested line. Thickness of near boundary. Lagrangian. Shadow price of the binding flow constraints. Scalar to identify the membership of bus according to a congested line. Locational marginal price (LMP). System marginal cost. Threshold value to evaluate the impact of congestion. Manuscript received December 27, 2010; revised April 06, 2011 and May 18, 2011; accepted June 30, 2011. Date of publication January 09, 2012; date of current version April 18, 2012. Paper no. TPWRS-00971-2010. The author is with the State University of New York at Buffalo, Buffalo, NY 14260 USA ( e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TPWRS.2011.2176758
Standard deviation. Measure of the impact of a congestion. I. INTRODUCTION ECENT efforts for the integration of renewable electric technologies into the power system have been increasing over the past decade due to the growing concern regarding climate change. These efforts result in the development of efficient expansion-planning algorithms for optimizing both the transmission network and generation [1]–[5]. These algorithms have limited applicability in the power system planning, however, due to the high computational cost associated with a large-scale power system. A traditional network reduction method yields a simple model [6]. This method typically involves computing impedances and eliminating unnecessary elements [6]–[10]. As a result, the reduced model includes a highly dense impedance matrix, which makes it difficult to increase efficiency significantly when using the reduced network. Equivalent networks reproduce the same voltages and currents of the remaining buses as the original systems do. However, the power transfer between areas is not preserved because the flows of the eliminated branches cannot be approximated. A reduction method based power transfer distribution factor (PTDF) is proposed to preserve the same flow pattern as that in the original network [11]. The model yields a similar flow profile at the injection profile for which the reduction is performed. However, the model also depends on the operation setpoint [12]. The solutions from planning studies using the model may be infeasible because the studies should consider various injection portfolios. A new method to reduce network was recently proposed to construct the flow sensitivity matrix to the reduced injection [12]. The reduced injections and the reduced flows are defined as the aggregated injections and the aggregated flows at a predefined group of buses. From the procedure, the method yields a simple but precise injection profile independent model. However, it is not clear how a group of buses should be defined. As a result, the injection and the flow profiles might be significantly different if the original network is used for an optimal power flow (OPF) study. In OPF studies, congestion often increases the system cost significantly because it may keep cheap generation from being dispatched. Congestion plays a critical role in OPF and planning studies. Because congestion arises where flow over a line equals the limit of the line, it is important to assign the flow limits for the studies. Therefore, it is also important to assign adequate flow limits for the aggregated lines (corridor). Even though the reduction algorithm yields a simple but precise model, its applicability can be limited without the proper assignment of the flow limit. In this paper, a new method is proposed to find a proper grouping of buses for the reduction and to assign the flow limits
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0885-8950/$31.00 © 2012 IEEE
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so that the congestion profile is preserved. The algorithms for the bus grouping and for the assigning flow limits are key features for preserving the congestion profiles. The method will be applied for OPF studies, and the OPF results are compared with those from the original networks in terms of the congestion profiles, the locational marginal prices (LMP), and generation profiles. II. CONGESTION MODELING A. Network Reduction Procedure Power system planning is, in general, computationally demanding due to its large scale. For reducing the computational cost, it is desirable to have a simple but precise scenario independent and reduced path independent model. A reduction method is described in [12]. Here is a brief description of the method: 1) First, define areas so that all the buses inside an area have similar properties. 2) Ignore all the lines connecting two buses that belong to the same area. 3) Lines connecting two buses that belong to different areas are aggregated. 4) PTDF of the reduced network is computed. 5) Reactance values for the reduced network are computed from the PTDF obtained in 4 and the node-branch incidence matrix for the reduced network. A simple but precise reduced model is obtained from the process if the areas are provided. [12] showed that power flow studies with the reduced model yield similar flow patterns. However, it is not clear how to determine the areas. For defining the areas, the buses inside an area should have similar properties. For optimal planning studies, significant importance is imposed on the economic impacts. Therefore, properties related to the economic impact need to be considered. Because LMP reflects the impact, the areas are defined based on the price. Therefore, area is a set of buses of which LMP falls into a similar value. OPF is an optimization process to find an optimal but feasible generation portfolio that will not violate the economic, operational, and engineering constraints. One can construct the Lagrangian, , and find the solution from a relaxation process. By definition, LMP is the increase of the value of the Lagrangian with respect to a one-unit increase of the load at the loca. There are three components in LMP: tion; i.e., the system marginal cost of electricity, ; the impact due to the congestion; and the cost associated with losses. Typically, the last component is marginal in comparison to other components. The first term affects LMP uniformly, and thus, it does not provide useful information for defining areas. Therefore, only the impact of the congestion on LMP will be discussed in this paper. B. Impact of the Congestion on LMP Congestion is the subset of the flow constraint that is binding; i.e., the flow over the congested line, , equals the flow limit of the line. Therefore, the second component is the product between the sensitivity of the binding constraints and the corresponding shadow prices; i.e., (1)
Fig. 1. Modified IEEE 30-bus system with a congested line (red line) between Bus 23 and 24. The solid back line shows the separation of a system into the supply-rich area and the demand-rich area.
Because the system cost decreases if the binding flow limits increase marginally, the values of are non-negative. As a result, the second component in LMP is proportional to the values of PTDF elements. Depending on the electric distance of buses from the congested line , the impact of the congestions varies. The impact can be negative or positive; i.e., LMP becomes higher or lower with congestion depending on . As a result, congestion divides the system into two areas: one with positively impacted LMPs, and the other with negatively impacted LMPs. When a line is congested, the power injected from one end of the line equals the one ejected from the other end if losses are ignored. At the ejection side, the power is deficient; i.e., the sum of loads is greater than that of generations. The area is termed demand-rich area (DRA). The congestion results in higher LMP than the system marginal cost. The injection side is termed supply-rich area (SRA) because the sum of loads is less than that of generation. The congestion results in lower LMP than the system marginal cost. Fig. 1 shows the separation of the modified IEEE 30-bus system [13] due to a congested line. If a system is separated into DRA and SRA, LMPs in DRA would be higher than those in SRA. C. Separation of Network Due to Congestion In a DC power flow model, the power flow over the transmission network equals the product between the PTDF matrix and the injection vector
(2) and are the elementary charges that generaNote that MW, respectively. tion and load carries, equal to 1 MW and Equation (2) implies that some electric power generations from SRA migrate to DRA through the congested line and that some
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junction and LMP separation due to congestion. Therefore, the classification to the four sub areas can be made in the power system due to a congested line. From the analogy between the electric potential in the p-n junction and LMP of a system with congestion, one can evaluate the impact congestion has on LMP as a function of the electric distance of a bus from the congested line using (3) with (1) as a case with a congested line: for for for for Fig. 2. Variation of LMP as a function of the electric distance from the congested line as in the case shown in Fig. 1. The broken line illustrates the variation only graphically.
loads from DRA also migrate to SRA. Both directional migrations occur until a steady state at the line is established. At the steady state, it is difficult to bring generation from SRA closer to the line and to bring load from DRA closer to the line. The difficulty can be evaluated using LMP because it is, by definition, the increase of Lagrangian by adding an additional load at a location of interest. In DRA, LMP becomes higher as a bus is located electrically closer to the congested line. In SRA, the opposite behavior is observed. Fig. 2 illustrates how LMP changes with respect to the electric distance from the interface. The broken line only represents the graphical change of LMP. Based on this price separation, it is possible to group buses into four categories if a proper criterion is provided for dissecting areas between near boundary and bulk areas: SRA near boundary, SRA bulk, DRA bulk, and DRA near boundary. There exists a similar behavior in p-n junction. p-type and n-type semiconductors have uniform electric potential. When they are attached, the charge carriers migrate across the boundary until a uniform Fermi level is established. At the steady state condition, the area near the interface has a low charge carrier density, i.e., depletion region. On energy point of view, it is difficult to locate charge carriers in the depletion region. An electric potential indicates the difficulty to bring an additional charge carrier. As a result, electric potential inside a p-n junction varies with the distance from the interface: for for
(4)
where is a proportionality constant, and is the electric distance that the impact of congestion on LMP is significant. Using (4), it is possible to enumerate the electric distance from the congested line from the congestion cost or PTDF matrix as shown in (1). is the number of The number of the PTDF elements in buses in the system; i.e., only elements are sampled. The distribution of selected PTDF elements is assumed to be independent and identically distributed, i.e., random distribution. In evaluating a PTDF matrix, typically the change in flow is observed when one unit is injected at a bus and ejected at a reference bus. Therefore, the values of a PTDF matrix depend on the choice of reference bus. One can evaluate a PTDF matrix for a different choice of reference bus because the evaluation process is identical with a different bus for ejection. The PTDF elements are evaluated as follows: (5) where is the PTDF element corresponding to the reference bus. depend on the Therefore, the values of the elements in is randomly distributed and choice of the reference bus. If is the mean value of . Then can be (4) holds, shifted by so that the mean value of the shifted becomes zero. Because the maximum value of may not be observed, 99% of covers approximately the entire range of . is a term to evaluate a threshold, which indicates how far the impact of congestion vanishes from -distribution [13]:
(3)
(6)
where is a scalar constant; represents the distance from is the width that the interface in the semiconductor; and the boundary effect exists. The semiconductor with p-n junction is often separated into four areas: p-type bulk, p-type near boundary, n-type near boundary, and n-type bulk. Equation (3) shows the electric potential as a function of the distance from the interface when the charge injected to one side equals that ejected from the other side. As the potential increases, it becomes more difficult to add an n-type charge carrier. On the other hand, the lower the electric potential is, the more difficult the addition of a p-type carrier. It is concluded that there exists analogy between the electric potential at p-n
where is the standard deviation of , ; is the threshold value of ; and is the distance from the congested . line corresponding to The overall impact on LMP due to congestion is the area under the curve in (4). Therefore, the fraction of area under the curve impacts how far congestion affects the system. quantito the impact at fies the relative impact of congestion at where is an electric distance from the congested line, and is the maximum distance that the impact is not negligible: (7)
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TABLE I TYPICAL VALUES OF THE FRACTION OF AREA AND THE THRESHOLD
For example, the choice of means that one wants to select the buses under any influence of congestion. Therefore, the choice yields all the buses in DRA or in SRA. On the other yields the set of buses that has a full influence. Behand, cause such influence is observed when a bus is infinitely close to the congested line, should be zero. Any choice between 0 and would yield a set of buses so that the influence of congestion is neither none nor full. From -distribution, all the area under the distribution curve . Let the area under the distribuis 99% covered when tion curve be . As shown in (7), is the fraction of area, and it with respect to . From is evaluated when equals ; i.e., the fraction of area , one can evaluate the electric distance from the congested line. Note that the electric distance is expressed in terms of the PTDF matrix shown in (4). Therefore, the PTDF matrix maps the area under the -distribution curve. Suppose corresponds to in -distrithe electric distance from bution. is a scalar value in -distribution that is corresponding to an area fraction of . The ratio of the PTDF element at the bus where electric disand at the bus infinitely close to tance from the boundary is the boundary is
Fig. 3. Bus grouping process to preserve the congestion profile.
(8)
Combining (7) and (8) leads to (9) Table I lists the value of the fraction of area and the threshold value. As decreases, increases; i.e., a large number of affected buses will be covered when a large threshold value of the PTDF element is considered. In this way, there are four sub-areas for a congested line: significantly affected area with positive impact, (DRA boundary); weakly affected area with positive impact, (DRA bulk); significantly affected area with neg(SRA boundary); and weakly afative impact, fected area with negative impact, (SRA bulk). When multiple lines are congested, a bus may belong to multiple areas. A set is defined so that all the buses in the set belong to the same multiple areas. D. Grouping Algorithm and Flow Limit Assignment With given values of and , four areas for each congested line are assigned. It is possible to define sets from a network provided the congestion profile is given. Suppose there exist sets. For each set , the buses that belong to the set are given
where . For the buses that belong to the same sets, the sum of the number ( ) should be equal. After such assignment, select all the buses to which the sum of the numbers assigned is equal. Note that the value for is selected so that the sum of a bus unequivocally yields the membership to which the bus belongs. For example, suppose two possible congested lines are considered. For the first congested line, the is 1.1. Then the buses in the system can be SRA value of near boundary, SRA bulk, DRA bulk, and DRA near boundary. Note that the order is according to the increasing order of the PTDF elements for the congested line. Then the possible values for the membership of the buses for the first congested line are for the second 1.1, 1.21, 1.331, and 1.4641. The choice of congested line can be 1.5 so that the sum can be a signature value. The grouping procedure is illustrated in Fig. 3. In this process, the congested lines belong to the intergroup lines [12]. Suppose a congested line connects two groups, and there exist other lines connecting the same groups. The method in [12] yields a way to aggregate all the lines connecting the groups. However, as mentioned in the Introduction, the flow limits are difficult to assign. Instead, the lines connecting two groups except the congested one are aggregated into a single line parallel to the congested line. Suppose the reduced PTDFs for and , the parallel and the congested lines are based on the method in [12].
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Kirchhoff’s laws dictate the flows at the given injections and, therefore, the PTDFs. Parallel lines that connect the same locations have the parallel PTDF elements regardless of the line properties: (10) and are the row vectors from corresponding to where the th and th parallel lines, and is a constant. Therefore, and need to follow (10). It is reasonable to because the line is physically existent and preserve congested while the other line is aggregated and not congested. should be adjusted parallel to . Therefore, -factorization is performed on :
(11) where and are the orthonormal basis sets spanning the , respectively; is a onereal and the null spaces of dimensional constant. Even though and are not necessarily identical, they should be similar because they connect two common areas. can be expressed by two components shown in (11):
(12) is a vector with a small magnitude because spans the null spaces of , which is similar to . The PTDF element for the parallel is adjusted to , and the second term in (12) is ignored. Because all the congested lines in the original network are preserved in the reduced network, the flow limits of the preserved lines are inherited from the original network. Other non-congested and aggregated inter-group lines are not congested, and therefore, there may be no need to assign any limits for them. E. Limitation of the Proposed Method The proposed method yields a way to group the buses to preserve the congestion profiles that the original network has. A purpose of the network reduction is for planning studies. Therefore, it is possible for unanticipated sets of congestion to occur. If the congested lines yield a completely different congestion profile, then the result from this algorithm may not reflect the proper congestion profiles. Therefore, it is important to include possibly congested lines as well as existing congested lines. III. SIMULATIONS AND DISCUSSION A. Modified IEEE 30-Bus System In the modified IEEE 30-bus system illustrated in Fig. 1 [13], there are two tie lines connecting between Area 2 and the rest of
Fig. 4. Modified IEEE 30-bus system. Tie lines connect Area 2 to the rest of the system. Groups of buses illustrated are listed in Table II.
the system. Due to high demand at Area 2 and tight flow limits of the tie lines, Area 2 is termed a load pocket. For the grouping algorithm proposed in this paper, only the list of the congested lines is required—not the shadow price or LMP. Provided that the tie lines become congested, the values for and are chosen to be 76% and 1, respectively. With the choice, four groups are classified for each congested line: SRA bulk, SRA near boundary, DRA bulk, and DRA near boundary. The values for are 1, 2, 3, and 4 for SRA bulk, SRA near boundary, DRA bulk, and DRA near boundary, respectively. The values for are 1.1 and 0.9 for the lines connecting 4–12 and 23–24, respectively. The second and third columns in Table II list the values. The two vectors listed in the second and third columns are added to uniquely identify the groups. The sum is listed in the forth column. The sum provides signature information to show the membership of the bus. For example, the sum for Bus 10 is 2.02. Because the value is a sum from the following formula, , the only feasible choice for integer is and , which implies that Bus 10 belongs to SRA bulk ( ) for the congestion between 4–12 and SRA ) for the congestion between 23–24. near boundary ( Based on the sum, six groups are formed and listed in the fifth column in Table II. The bus grouping is illustrated in Fig. 4. With the groups, a reduced network is created based on the reduction algorithm in [12]. Fig. 5 illustrates the reduced network. Note that there is a pair of parallel lines connecting Group II and Group III. One of the lines is a congested line connecting Bus 4 and Bus 12 in the original network shown in Fig. 1, while the other line represents the line connecting Bus 4 and Bus 16. DC OPF studies for various load profiles are performed on the original and the reduced network. Various loads are simulated to find different congestion profiles on the original network. Typical simulation results
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TABLE II BUS GROUPING PROCEDURE,
= 1:1 AND
= 0:9
TABLE III CASE WITH NO LINE CONGESTED
TABLE IV CASE WITH TIE LINES 4–12 CONGESTED
TABLE V CASE WITH TIE LINES 23–24 CONGESTED
TABLE VI CASE WITH TWO TIE LINES, 4–12 AND 23–24, CONGESTED
Fig. 5. Reduced 30-bus system based on the bus grouping listed in Table II. There are two parallel lines connecting Groups 2 and 3.
are listed in Tables III–VI. The case listed in Table III is a non-congested case; the case in Table IV includes a congested line between Area 1 and Area 2; the case in Table V contains a congested line between Area 2 and Area 3; and the two tie lines are congested in the case in Table VI. In all the cases listed in Tables III–VI, the congested profiles in the original network
are preserved in the reduced network. As shown in the tables, LMP and the generation portfolio using the reduced networks are approximately those using the original network. Therefore, OPF results with the reduced network are useful to estimate those with the original network. It is interesting to check how much error in LMP increases when unanticipated congestion occurs. When lines connecting 27–28, 9–10, and 6–10 are
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Fig. 6. Modified IEEE 118-bus system. Tie lines (Lines 37 and 38) can be congested. Groups of buses illustrated are listed in Table VII.
TABLE VII LIST OF BUSES FOR THE NINE GROUPS
Fig. 7. Reduced 118-bus system based on the bus grouping listed in Table VII.
congested, the error in LMP increases to 8%. It is clear that if the congestions were considered, Bus 10, 21, 27, 29, and 30 should belong to different areas than Group I. B. Modified IEEE 118-Bus System An IEEE 118-bus system in [13] is illustrated in Fig. 6, and the flow limits of some lines are modified to create congestion profiles. At peak periods, two lines (Lines 37 and 38) can be congested. A similar procedure described in Section IV-A yields the nine groups listed in Table VII, and a reduced network is formed based on the reduction algorithm in [12]. Fig. 7 illustrates the
reduced network resulting from the procedure. DC OPF studies for various load profiles are performed on the original and the reduced networks to compare the results. Various loads are simulated to find different congestion profiles. Typical simulation results are listed in Tables VIII–XI. The case listed in Table VIII is a non-congested case; the case in Table IX includes a congested line (Line 37); the case in Table X contains a congested line (Line 38); and the two tie lines are congested in the case in Table XI. In all the cases listed in Tables VIII–XI, the congested profiles in the original network are preserved in the reduced network. Similar to the results in Section IV-A, LMP and the generation portfolio using the reduced networks are approximately those using the original network. Therefore, OPF results with the reduced network are useful to estimate those with the original network.
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TABLE VIII CASE WITH NO LINE CONGESTED
TABLE XI CASE WITH TWO TIE LINES, 37 AND 38, CONGESTED
TABLE IX CASE WITH TIE LINE 37 CONGESTED
the flow limits of the aggregated lines that are not congested do not need to be assigned precisely. As a result, along with the network reduction algorithm, it provides a concise and precise representation of the transmission network for a power flow study. Therefore, this method can be used in a large system OPF, national corridor, and renewable portfolio studies. REFERENCES
TABLE X CASE WITH TIE LINE 38 CONGESTED
IV. CONCLUSION For a large-scale power system, it is practically infeasible to find an optimal solution for power system planning. Recent development of network reduction allows precise modeling with a manageable computation expense. However, a proper assignment for the flow limits is missing, which is necessary for an OPF study. In this paper, an algorithm to find a proper group is proposed to assign for network reduction. The reduced network based on this group shows the same congestion profile as the original network in the OPF studies. Another advantage is that
[1] Northeast Power Coordinating Council, Guideline for Inter-Area Voltage Control, Nov. 1997. [2] ISO-NE, ISO New England Operating Procedure no. 19: Transmission Operation, Apr. 2007, pp. 7–8. [3] E. Fisher, R. O’Neill, and M. Ferris, “Optimal transmission switching,” IEEE Trans. Power Syst., vol. 23, no. 3, pp. 1346–1355, Aug. 2008. [4] K. Hedman, R. O’Neill, E. Fisher, and S. Oren, “Optimal transmission switching with contingency analysis,” IEEE Trans. Power Syst., vol. 24, no. 3, pp. 1577–1586, Aug. 2009. [5] K. Hedman, M. Ferris, R. O’Neill, E. Fisher, and S. Oren, “Co-optimization of generation unit commitment and transmission switching with N 1 reliability,” IEEE Trans. Power Syst., vol. 25, no. 2, pp. 1052–1063, May 2010. [6] J. B. Ward, “Equivalent circuits for power flow studies,” AIEE Trans. Power App. Syst., vol. 68, pp. 373–382, 1949. [7] S. Deckmann, A. Pizzolante, A. Monticelli, B. Stott, and O. Alsac, “Numerical testing of power system load flow equivalents,” IEEE Trans. Power App. Syst., vol. PAS-99, no. 6, pp. 2292–2300, Nov./Dec. 1980. [8] S. Deckmann, A. Pizzolante, A. Monticelli, B. Stott, and O. Alsac, “Studies on power system load flow equivalencing,” IEEE Trans. Power App. Syst., vol. PAS-99, no. 6, pp. 2301–2310, Nov./Dec. 1980. [9] M. K. Enns and J. J. Quada, “Sparsity-enhanced network reduction for fault studies,” IEEE Trans. Power Syst., vol. 6, no. 2, pp. 613–621, May 1991. [10] W. F. Tinney and J. M. Bright, “Adaptive reductions for power flow equivalents,” IEEE Trans. Power Syst., vol. 6, no. 2, pp. 613–621, May 1991. [11] X. Cheng and T. J. Overbye, “PTDF-based power system equivalents,” IEEE Trans. Power Syst., vol. 20, no. 4, pp. 1868–1876, Nov. 2005. [12] H. Oh, “A new network reduction methodology for power system planning studies,” IEEE Trans. Power Syst., vol. 25, no. 2, pp. 677–684, May 2010. [13] R. Zimmerman, C. E. Murillo-Sanchez, and D. Gan, MATPOWER: A MATLAB Power System Simulation Package, 2008. [Online]. Available: http://www.pserc.cornell.edu/matpower/.
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HyungSeon Oh received the Ph.D. degree in electrical and computer engineering from Cornell University, Ithaca, NY, in 2005. He is an Assistant Professor at the State University of New York at Buffalo. Before the appointment, he was a staff engineer at the National Renewable Energy Laboratory. His research interests include: energy systems economics, power system planning, smart grid, storage, computer simulation, and nonlinear dynamics.
