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Dynamic transmission planning under uncertainty
Electric Power Systems Research, 8 (1984/85} 131 - 136
Dynamic T r a n s m i s s i o n
131
Planning Under Uncertainty
G. J. BERG
Electrical Engineering Department, The University of Calgary, Calgary, Alta. (Canada) T. A. M. SHARAF
Electrical Engineering Department, Cairo University (Egypt) (Received August 4, 1984)
SUMMARY
A new approximate method is presented to solve the dynamic optimization problem in long-range transmission capacity expansion planning. The method employs a static optimization technique along with an iterative procedure to simulate the dynamic solution. The proposed method can be simplified to provide a fast technique, also discussed in the paper. Comparison is made between the fast technique and the conventional horizon year planning approach.
1. INTRODUCTION
As stated in ref. 1, the objective of longrange transmission planning is to define a series of optimal expansion plans subject to specific technical and other constraints. The determination of such plans requires a dynamic optimization technique, i.e. time should be included in the optimization process as an independent variable. The main difficulty with available dynamic optimization techniques is in the complexity of the computational process, particularly when dealing with the high dimensionality encountered in transmission planning. Several procedures have been developed to overcome this difficulty. Henault et al. [2] used dynamic programming, assuming the load at one bus as a stochastic variable with all other loads deterministic. It was also assumed that at each stage the load at the next stage would be certain, but subsequent loads would be statistical variables. In order to reduce the number of variables, a single variable was used to identify system configuration, chosen from a set of predesigned configurations. 0378-7796/85/$3.30
Obviously, the solution could only be as good as the predesigned set. Dusonchet and E1-Abiad [3] combined the deterministic search procedure in dynamic programming with a probabilistic search and a heuristic stopping criterion. The main problem with this technique is in constructing the neighbourhood of a strategy. As a result the solution obtained may not be optimal, but rather sub-optimal. Mixed integer programming was used by Adams and Laughton [4] to solve the dynamic planning problem. In order to make the problem manageable, a fixed-cost transportation model was used to simulate the power flow in the network. Such a model tends to be oversimplified, and the results obtained very rough. In the most common approach, expansion plans are obtained by applying a static optimization technique sequentially. Berg et al. [5] have discussed a method in which alternative plans are developed for every stage. Then these plans are used to generate alternative overall plans by applying the k shortest paths technique. Recently, Meliopoulos et al. [6] have used a dynamic optimization procedure based on a nonlinear branch and bound algorithm to develop optimal overall plans. A screening algorithm curbs the number of alternatives obtained by enumeration. A survey of available methods of dynamic transmission planning under uncertainty suggests that improvement is desirable in both the system model and the optimization technique. The system model should be sufficiently detailed to include the uncertainties in load projections and network topology usually encountered in long-range planning. Given such a model, a suitable © Elsevier Sequoia/Printed in The Netherlands
132
optimization technique must be employed, so that the resulting m e t h o d is practical from a computational point of view. In this paper, a m e t h o d is proposed which simulates the dynamic (ideal) solution approximately [7]. The m e t h o d is simple and its computational requirements are relatively moderate. It is flexible enough to handle a wide range of system models. The system model developed in ref. 1 which accounts quantitatively for the uncertainties in both load projections and system topology has been used in the present study. The proposed m e t h o d m a y be simplified, resulting in an approximate plan that is referred to here as the 'dynamic horizon year plan'. The simplification reduces the computational requirements considerably. In §2, the proposed dynamic m e t h o d is discussed. Simplification of the m e t h o d leading to the dynamic horizon year plan is introduced. The application of the new optimization strategies to the IEEE 5-bus test system is presented in § 3 along with typical results.
2. THE PROPOSED DYNAMIC METHOD
2.1. General The installation of a new transmission element at a specific time during the planning period influences future system performance and the present worth of investment in a time d e p e n d e n t manner. Therefore, it is necessary to identify the timing of the installation of new elements so as to minimize the present worth of investment through the planning period subject to technical and other constraints. If the formulation of the transmission expansion planning problem is to be dynamic, a difference relation between the present independent and dependent variables and the dependent variables in the next time interval must be defined. Implementation of this formulation is n o t really practical or feasible, particularly in the presence of uncertainties in both load projections and system topology. As already mentioned, integer programming and dynamic programming have been applied to the dynamic planning problem. The models used with these methods tend to
be oversimplified, and the logic required to limit the number of configurations to be considered at each stage remains a significant problem. It is also worth mentioning that the computational effort associated with these methods increases rapidly with system size and the number of stages. In this paper a new model is presented to solve the dynamic planning problem. The m e t h o d is sufficiently flexible to acc o m m o d a t e different system models. The computational requirements increase almost linearly with the size of the network and the number of stages. On the other hand, the solution obtained is not exact because certain approximations are made in the analysis. 2.2. The dynamic method Instead of updating the physical system state in response to intermediate stage decisions during the optimization process, intermediate states are estimated before optimization starts. An iterative procedure is then followed which improves on the initial estimates. The main computational steps of the proposed m e t h o d are indicated in Fig. 1. To evaluate the initial estimates a static optimization technique is applied sequentially to every stage of the planning period through to the horizon year. The o u t p u t from the first stage is used to update the system before applying the optimization technique to the second stage. The o u t p u t of the second stage is used to update the system for the third stage, and so on. Given the initial estimates of system states, the static optimization technique is applied to determine the optimal expansion pattern including all stages. Here, optimality is considered relative to the initial systems generated in the sequential process. Therefore, the resulting expansion schedule may be used to produce a new estimate of system states which form a basis for re-evaluation of the expansion at all stages. The process is repeated until the difference in results of two consecutive iterations is less than a prescribed quantity, e. In other words, the proposed method combines a static procedure for simultaneous optimization over all the stages of the planning period, with an iterative procedure which takes into account the system changes due to intermediate stage decisions.
133
II EA0 INPUT0A A l -I
NO
I ,PP y STAT,CSTAGE ToOPT'M'ZAT' 1 ' 0N
~YES~NO
I RETURN i = lTO THE I INITIAl. SYSTEM l
UPDATE THE SYSTEM USING THE I I OUTPUTOF STAGE i. SAVE THE [,_NEW STATE.
J
NO APPLY THE STATIC OPTIMIZATION TECHNIQUETO THE UPOATED SYSTEMSTO EVALUATE THE OUTPUTOF THE K STAGES SIMULTANEOUSLY.
Fig. 1. The computational procedure for the proposed dynamic method.
2.3. Dynamic horizon year planning technique In power system planning, it is frequently desirable to have an approximate preliminary or rough solution that takes into consideration the projections for the last stage of the planning period. Since it is rough, and is frequently required, the technique producing this solution should be fast. The proposed dynamic technique described in § 2.2 may be simplified to provide a fast method suitable for obtaining a preliminary (approximate) solution. Appropriate simplifications, and the resulting algorithm, are described below. Instead of optimizing the expansions at all stages simultaneously while meeting the constraints associated with each stage, the optimization process is performed by considering expansion at all stages simultaneously subject to the constraints in the horizon year only. Although the accuracy of the results will suffer somewhat, the reduction in the computational requirements is significant. The simplified technique seems conceptually superior to the conventional procedure of simply evaluating a horizon year plan (i.e. determine the transmission capacity requirements to handle the final year projections of
load and generation). Also, the new method provides more useful information regarding the intermediate stages. The proposed simplified technique is referred to here as the 'dynamic horizon year planning technique'. In order to appreciate the reduction in computational requirements achieved by the simplifications, consider that in a linear programming formulation involving K stages and M constraints per stage, the number of constraints before simplification is K X M, while after simplification it is just M. For purposes of long-range planning, the factor K may have a value in the 10 - 20 range. Application of the proposed technique and its simplified version is discussed in the next section along with a comparison between the simplified version and the conventional horizon year planning approach.
3. EXAMPLES
3.1. General As explained in §2.2, a static optimization algorithm is required for the proposed dynamic method. The static optimization algorithm described in ref. 1 is used. Its application as
134
part of the proposed dynamic method is discussed in this section, and is illustrated b y a few examples. The emphasis is on m e t h o d o l o g y rather than particular numerical results.
3.2. Specific problem and assumptions The proposed dynamic planning method has been applied to the IEEE 5-bus test system as presented in ref. 1. The per unit resistances and reactances are given by Table 1 of ref. I along with maximum allowable power flows for the network branches at the beginning of the planning period. The projected mean values of peak loads at the beginning of the planning period, and the corresponding generation, are given by Table 2 of ref. 1. It is assumed that the system peak loads have independent normal distributions and their mean values are growing at an annual rate of 5% over a planning period of 12 years. The standard deviations are assumed to be 25%, 30% and 40% of the increase in the mean values for the first, second and last four-year periods respectively. The element outage rates are those shown in Table 3 of ref. 1. They are assumed unchanged through the planning period. It is also assumed that changing the element capacities does n o t affect these rates. These assumptions are made for purposes of computational convenience. Dependence of the outage rates upon time or capacity can be accommodated. The expected demand not served at the beginning of the planning period is estimated at 0.49 p.u. Three prescribed values of the expected demand n o t served (EDNSp), 0.55, 0.6 and 0.7 p.u., are assumed for the
first, second and third four-year period respectively. It is assumed that right-of-way requirements are unchanged and the weighting coefficients in the objective function (eqn. (17) of ref. 1) are equal for each stage. The limits on phase-angle differences, (~k) ma× and ( ~ J m + k ) max a r e assumed equal for k = 1, 2, . . . , m . They are again assumed unchanged through the planning period. The confidence levels of the chance constraints, u, are assumed equal at 85%, 80% and 75% for the three periods, respectively. No action has been taken in respect of interest and inflation rates since they vary widely from one projection to another. The effect of interest and inflation rates on the plan may be considered by modifying the weighting coefficients of the objective function of eqn. (17), ref. 1.
3.3. Results and discussion 3.3.1. Dynamic planning method The twelve-year planning period is divided into six stages, each stage of two-year duration. An initial solution is obtained by applying the static optimization technique of ref. 1 to the six stages sequentially. The results of the first stage are used to update the capacity sensitivity matrix of eqns. (22) and (23) [1], and the reliability sensitivity coefficients of eqn. (20). The static optimization technique is applied to the modified system to determine the capacity additions for the second stage, and so on. The initial solution thus obtained is shown in Table 1. The static optimization technique is then applied to the updated systems to assess overall performance of the six stages simultaneously. Results obtained are used to re-
TABLE 1 Transmission capacity requirements determined by sequential optimization Stage
1 2 3 4 5 6
C a p a c i t y a d d i t i o n s (p.u.) for b r a n c h e s 1 to 7 1
2
3
4
5
6
7
0.0 0.0 0.0 0.0 0.3917 1.4152
0.3361 0.6242 0.2651 0.2320 0.3051 0.5250
0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.3859 1.1501 1.2581 1.2808 1.1158
0.0 0.0 0.0 0.1620 0.3189 0.8871
0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0
T o t a l per stage (p.u.) 0.3361 1.0101 1.4152 1.6521 2.2965 3.9431
135 TABLE 2 Transmi~ioncapacityrequirementsdeterminedbytheproposeddynamicplanning
method
Total per stage (p.u.)
Capacity additions (p.u.) for branches 1 to 7
Stage
1 2 3 4 5 6
1
2
3
4
5
6
7
0.0 0.0 0.0 0.0 0.0 0.0
0.3361 1.8644 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0
0.0 4.5302 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0
m o d i f y the intermediate system states. This process continues until the accuracy target is achieved, as illustrated in Fig. 1. In the example discussed, only t w o iterations were required to reach a solution of 0.01 p.u. accuracy. The solution is given in Table 2. Comparing t he results of the dynamic m e t h o d with those of the c o m m o n l y used sequential m e t h o d , t he first-stage r e qui r em ent is the same. Additions of 10.3 p.u. were required f o r th e subsequent stages using the sequential m e t h o d , against 6.4 p.u. using the dynamic m e t h o d . In ot he r words a difference o f mo r e than 50% in the capacity requirements has been observed while using the same model and th e same constraints, b u t different m e t h o d s o f analysis.
3.3.2. Dynamic technique
horizon
year planning
The simplified t e c h n i q u e described in § 2.3 has also been applied to t he IEEE 5-bus test system with data and assumptions as given in §3.2. The initial solution o f Table 1 is obtained as explained in §3.3.1. Updated intermediate system states are t he n used in an optimization
0.3361 6.3946 0.0 0.0 0.0 0.0
process to determine the requirements for the six stages simultaneously. However, in this case only the constraints of the last stage are applied. Consequently, the loading is considered as t he difference between t he projections for the beginning and end o f t he planning period. The intermediate system states are then re-modified according to the new estimates of the capacity requirements, and the process is repeated until the results converge to the 0.01 p.u. accuracy margin. The results are shown in Table 3. F o u r iterations were required. A conventional horizon year plan has also been determined using the same system model. Results for t he conventional study are presented in Table 4. A comparison between these results and those of Table 3 n o t only reveals the difference in accuracy, it also displays the ability o f the new formulation, even in the simplified form, to provide t he intermediate stage requirements for optimal long-range planning. Of course the gain in accuracy is achieved at the expense of increased com put at i onal requirements. Instead o f applying the static optimization technique just once, as in the
TABLE 3 Transmission capacity requirements determined by the 'dynamic horizon year planning technique'
Stage
1 2 3 4 5 6
Capacity additions (p.u.) for branches 1 to 7 1
2
3
4
5
6
7
Total per stage (p.u.)
0.0 0.0 0.0 0.0 0.0 1.7892
0.0 2.5406 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0
0.0 4.8336 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 1.4068 0.0
0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0
0.0 7.3742 0.0 0.0 1.4068 1.7892
136 TABLE 4 Transmission capacity requirements determined by the 'conventional horizon year planning technique' Capacity additions (p.u.) for branches 1 to 7 1
2
3
4
5
6
7
0.0200
1.5148
0.0
2.4905
0.5083
0.0
0.0
case of the conventional horizon year planning approach, it is applied K (the number of stages) times for the initial solution, plus an additional 3 to 4 times (with the number of independent variables multiplied b y K) in the iterative process. This is done for purposes of updating the sensitivity matrices according to intermediate stage capacity modifications. In the present study with the computing facilities used (Multics System) the computing time required for the initial solution was a b o u t 4 seconds per stage. Estimated average computing time for the six-stage model is 8 seconds per iteration. The corresponding figure for the conventional horizon year plan is 4 seconds. The time required b y the dynamic planning method was 12 seconds for the same six-stage model. It has been found that the length of the time period of the stage affects the accuracy o f the results. This is to be expected, firstly because increasing the number of stages should move the solution closer to the ideal dynamic solution, or because of the iterative procedure followed in estimating the intermediate system states; secondly, the approximate linear model used to represent the system produces higher errors as the stage length increases. The length of the stage therefore reflects the trade-off between the accuracy of the solution and computing time. A significant advantage of the proposed technique is that any system representation m a y be incorporated with the desired degree of detail. As seen from the above example, statistical load projection can be used together with quantitative reliability constraints. 4. CONCLUSIONS A new formulation of the dynamic transmission capacity expansion planning problem has been introduced. A quantitative reliability
index has been observed and uncertainties in load projections have been accounted for. The new formulation is flexible. It can handle a wide range of reliability criteria and system models. The proposed iterative procedure to solve the planning problem appears to converge quickly to a specified accuracy. In the cases studied no more than four iterations were needed. Computational requirements are reasonable, increasing almost linearly with the dimension of the problem. The simplified technique, referred to as the 'dynamic horizon year planning technique' provides fast approximate plans. It is capable of providing a more accurate solution than the conventional horizon year planning technique because intermediate stages are considered. It may be regarded as a quasidynamic algorithm because the constraints on the intermediate stages are applied only for the initial solution and then relaxed. ACKNOWLEDGEMENT
The authors acknowledge with thanks the funds granted by the Natural Sciences and Engineering Research Council of Canada in support of this study. REFERENCES 1 T. A. M. Sharaf and G. J. Berg, Static transmission capacity planning under uncertainty, Electr. Power Syst. Res., 7 (1984) 289 - 296. 2 P. H. Henault, R. B. Eastvedt, J. Peschon and L. P. Hajdu, Power system long-term planning in the presence of uncertainty, IEEE Trans., PAS-89 (1970) 1 5 6 - 164. 3 Y. P. Dusonchet and A. E1-Abiad, Transmission • planning using discrete dynamic optimizing, IEEE Trans., PAS-92 (1973) 1358 - 1371. 4 R. N. Adams and M. A. Laughton, Optimal planning of power networks using mixed-integer programming, Part I -- Static and time-phased network synthesis, Proc. Inst. Electr. Eng., 121 (1974) 1 3 9 - 147. 5 G. J. Berg, K. R. C. Mamandur and P. Subramanian, A new technique for transmission system planning, Can. Electr. Eng. J., 3 (3) (1978) 35 40. 6 A. P. Meliopoulos, R. P. Webb, R. J. Bennon and J. A. Juves, Optimal long-range transmission planning with AC load flow, IEEE Trans., PAS101 (1982) 4156 - 4163. 7 T. A. M. Sharaf, Probabilistic measures and analytical techniques in power system transmission expansion planning, Ph.D. Thesis, University of Calgary, Canada, July 1982.
Dynamic T r a n s m i s s i o n
131
Planning Under Uncertainty
G. J. BERG
Electrical Engineering Department, The University of Calgary, Calgary, Alta. (Canada) T. A. M. SHARAF
Electrical Engineering Department, Cairo University (Egypt) (Received August 4, 1984)
SUMMARY
A new approximate method is presented to solve the dynamic optimization problem in long-range transmission capacity expansion planning. The method employs a static optimization technique along with an iterative procedure to simulate the dynamic solution. The proposed method can be simplified to provide a fast technique, also discussed in the paper. Comparison is made between the fast technique and the conventional horizon year planning approach.
1. INTRODUCTION
As stated in ref. 1, the objective of longrange transmission planning is to define a series of optimal expansion plans subject to specific technical and other constraints. The determination of such plans requires a dynamic optimization technique, i.e. time should be included in the optimization process as an independent variable. The main difficulty with available dynamic optimization techniques is in the complexity of the computational process, particularly when dealing with the high dimensionality encountered in transmission planning. Several procedures have been developed to overcome this difficulty. Henault et al. [2] used dynamic programming, assuming the load at one bus as a stochastic variable with all other loads deterministic. It was also assumed that at each stage the load at the next stage would be certain, but subsequent loads would be statistical variables. In order to reduce the number of variables, a single variable was used to identify system configuration, chosen from a set of predesigned configurations. 0378-7796/85/$3.30
Obviously, the solution could only be as good as the predesigned set. Dusonchet and E1-Abiad [3] combined the deterministic search procedure in dynamic programming with a probabilistic search and a heuristic stopping criterion. The main problem with this technique is in constructing the neighbourhood of a strategy. As a result the solution obtained may not be optimal, but rather sub-optimal. Mixed integer programming was used by Adams and Laughton [4] to solve the dynamic planning problem. In order to make the problem manageable, a fixed-cost transportation model was used to simulate the power flow in the network. Such a model tends to be oversimplified, and the results obtained very rough. In the most common approach, expansion plans are obtained by applying a static optimization technique sequentially. Berg et al. [5] have discussed a method in which alternative plans are developed for every stage. Then these plans are used to generate alternative overall plans by applying the k shortest paths technique. Recently, Meliopoulos et al. [6] have used a dynamic optimization procedure based on a nonlinear branch and bound algorithm to develop optimal overall plans. A screening algorithm curbs the number of alternatives obtained by enumeration. A survey of available methods of dynamic transmission planning under uncertainty suggests that improvement is desirable in both the system model and the optimization technique. The system model should be sufficiently detailed to include the uncertainties in load projections and network topology usually encountered in long-range planning. Given such a model, a suitable © Elsevier Sequoia/Printed in The Netherlands
132
optimization technique must be employed, so that the resulting m e t h o d is practical from a computational point of view. In this paper, a m e t h o d is proposed which simulates the dynamic (ideal) solution approximately [7]. The m e t h o d is simple and its computational requirements are relatively moderate. It is flexible enough to handle a wide range of system models. The system model developed in ref. 1 which accounts quantitatively for the uncertainties in both load projections and system topology has been used in the present study. The proposed m e t h o d m a y be simplified, resulting in an approximate plan that is referred to here as the 'dynamic horizon year plan'. The simplification reduces the computational requirements considerably. In §2, the proposed dynamic m e t h o d is discussed. Simplification of the m e t h o d leading to the dynamic horizon year plan is introduced. The application of the new optimization strategies to the IEEE 5-bus test system is presented in § 3 along with typical results.
2. THE PROPOSED DYNAMIC METHOD
2.1. General The installation of a new transmission element at a specific time during the planning period influences future system performance and the present worth of investment in a time d e p e n d e n t manner. Therefore, it is necessary to identify the timing of the installation of new elements so as to minimize the present worth of investment through the planning period subject to technical and other constraints. If the formulation of the transmission expansion planning problem is to be dynamic, a difference relation between the present independent and dependent variables and the dependent variables in the next time interval must be defined. Implementation of this formulation is n o t really practical or feasible, particularly in the presence of uncertainties in both load projections and system topology. As already mentioned, integer programming and dynamic programming have been applied to the dynamic planning problem. The models used with these methods tend to
be oversimplified, and the logic required to limit the number of configurations to be considered at each stage remains a significant problem. It is also worth mentioning that the computational effort associated with these methods increases rapidly with system size and the number of stages. In this paper a new model is presented to solve the dynamic planning problem. The m e t h o d is sufficiently flexible to acc o m m o d a t e different system models. The computational requirements increase almost linearly with the size of the network and the number of stages. On the other hand, the solution obtained is not exact because certain approximations are made in the analysis. 2.2. The dynamic method Instead of updating the physical system state in response to intermediate stage decisions during the optimization process, intermediate states are estimated before optimization starts. An iterative procedure is then followed which improves on the initial estimates. The main computational steps of the proposed m e t h o d are indicated in Fig. 1. To evaluate the initial estimates a static optimization technique is applied sequentially to every stage of the planning period through to the horizon year. The o u t p u t from the first stage is used to update the system before applying the optimization technique to the second stage. The o u t p u t of the second stage is used to update the system for the third stage, and so on. Given the initial estimates of system states, the static optimization technique is applied to determine the optimal expansion pattern including all stages. Here, optimality is considered relative to the initial systems generated in the sequential process. Therefore, the resulting expansion schedule may be used to produce a new estimate of system states which form a basis for re-evaluation of the expansion at all stages. The process is repeated until the difference in results of two consecutive iterations is less than a prescribed quantity, e. In other words, the proposed method combines a static procedure for simultaneous optimization over all the stages of the planning period, with an iterative procedure which takes into account the system changes due to intermediate stage decisions.
133
II EA0 INPUT0A A l -I
NO
I ,PP y STAT,CSTAGE ToOPT'M'ZAT' 1 ' 0N
~YES~NO
I RETURN i = lTO THE I INITIAl. SYSTEM l
UPDATE THE SYSTEM USING THE I I OUTPUTOF STAGE i. SAVE THE [,_NEW STATE.
J
NO APPLY THE STATIC OPTIMIZATION TECHNIQUETO THE UPOATED SYSTEMSTO EVALUATE THE OUTPUTOF THE K STAGES SIMULTANEOUSLY.
Fig. 1. The computational procedure for the proposed dynamic method.
2.3. Dynamic horizon year planning technique In power system planning, it is frequently desirable to have an approximate preliminary or rough solution that takes into consideration the projections for the last stage of the planning period. Since it is rough, and is frequently required, the technique producing this solution should be fast. The proposed dynamic technique described in § 2.2 may be simplified to provide a fast method suitable for obtaining a preliminary (approximate) solution. Appropriate simplifications, and the resulting algorithm, are described below. Instead of optimizing the expansions at all stages simultaneously while meeting the constraints associated with each stage, the optimization process is performed by considering expansion at all stages simultaneously subject to the constraints in the horizon year only. Although the accuracy of the results will suffer somewhat, the reduction in the computational requirements is significant. The simplified technique seems conceptually superior to the conventional procedure of simply evaluating a horizon year plan (i.e. determine the transmission capacity requirements to handle the final year projections of
load and generation). Also, the new method provides more useful information regarding the intermediate stages. The proposed simplified technique is referred to here as the 'dynamic horizon year planning technique'. In order to appreciate the reduction in computational requirements achieved by the simplifications, consider that in a linear programming formulation involving K stages and M constraints per stage, the number of constraints before simplification is K X M, while after simplification it is just M. For purposes of long-range planning, the factor K may have a value in the 10 - 20 range. Application of the proposed technique and its simplified version is discussed in the next section along with a comparison between the simplified version and the conventional horizon year planning approach.
3. EXAMPLES
3.1. General As explained in §2.2, a static optimization algorithm is required for the proposed dynamic method. The static optimization algorithm described in ref. 1 is used. Its application as
134
part of the proposed dynamic method is discussed in this section, and is illustrated b y a few examples. The emphasis is on m e t h o d o l o g y rather than particular numerical results.
3.2. Specific problem and assumptions The proposed dynamic planning method has been applied to the IEEE 5-bus test system as presented in ref. 1. The per unit resistances and reactances are given by Table 1 of ref. I along with maximum allowable power flows for the network branches at the beginning of the planning period. The projected mean values of peak loads at the beginning of the planning period, and the corresponding generation, are given by Table 2 of ref. 1. It is assumed that the system peak loads have independent normal distributions and their mean values are growing at an annual rate of 5% over a planning period of 12 years. The standard deviations are assumed to be 25%, 30% and 40% of the increase in the mean values for the first, second and last four-year periods respectively. The element outage rates are those shown in Table 3 of ref. 1. They are assumed unchanged through the planning period. It is also assumed that changing the element capacities does n o t affect these rates. These assumptions are made for purposes of computational convenience. Dependence of the outage rates upon time or capacity can be accommodated. The expected demand not served at the beginning of the planning period is estimated at 0.49 p.u. Three prescribed values of the expected demand n o t served (EDNSp), 0.55, 0.6 and 0.7 p.u., are assumed for the
first, second and third four-year period respectively. It is assumed that right-of-way requirements are unchanged and the weighting coefficients in the objective function (eqn. (17) of ref. 1) are equal for each stage. The limits on phase-angle differences, (~k) ma× and ( ~ J m + k ) max a r e assumed equal for k = 1, 2, . . . , m . They are again assumed unchanged through the planning period. The confidence levels of the chance constraints, u, are assumed equal at 85%, 80% and 75% for the three periods, respectively. No action has been taken in respect of interest and inflation rates since they vary widely from one projection to another. The effect of interest and inflation rates on the plan may be considered by modifying the weighting coefficients of the objective function of eqn. (17), ref. 1.
3.3. Results and discussion 3.3.1. Dynamic planning method The twelve-year planning period is divided into six stages, each stage of two-year duration. An initial solution is obtained by applying the static optimization technique of ref. 1 to the six stages sequentially. The results of the first stage are used to update the capacity sensitivity matrix of eqns. (22) and (23) [1], and the reliability sensitivity coefficients of eqn. (20). The static optimization technique is applied to the modified system to determine the capacity additions for the second stage, and so on. The initial solution thus obtained is shown in Table 1. The static optimization technique is then applied to the updated systems to assess overall performance of the six stages simultaneously. Results obtained are used to re-
TABLE 1 Transmission capacity requirements determined by sequential optimization Stage
1 2 3 4 5 6
C a p a c i t y a d d i t i o n s (p.u.) for b r a n c h e s 1 to 7 1
2
3
4
5
6
7
0.0 0.0 0.0 0.0 0.3917 1.4152
0.3361 0.6242 0.2651 0.2320 0.3051 0.5250
0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.3859 1.1501 1.2581 1.2808 1.1158
0.0 0.0 0.0 0.1620 0.3189 0.8871
0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0
T o t a l per stage (p.u.) 0.3361 1.0101 1.4152 1.6521 2.2965 3.9431
135 TABLE 2 Transmi~ioncapacityrequirementsdeterminedbytheproposeddynamicplanning
method
Total per stage (p.u.)
Capacity additions (p.u.) for branches 1 to 7
Stage
1 2 3 4 5 6
1
2
3
4
5
6
7
0.0 0.0 0.0 0.0 0.0 0.0
0.3361 1.8644 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0
0.0 4.5302 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0
m o d i f y the intermediate system states. This process continues until the accuracy target is achieved, as illustrated in Fig. 1. In the example discussed, only t w o iterations were required to reach a solution of 0.01 p.u. accuracy. The solution is given in Table 2. Comparing t he results of the dynamic m e t h o d with those of the c o m m o n l y used sequential m e t h o d , t he first-stage r e qui r em ent is the same. Additions of 10.3 p.u. were required f o r th e subsequent stages using the sequential m e t h o d , against 6.4 p.u. using the dynamic m e t h o d . In ot he r words a difference o f mo r e than 50% in the capacity requirements has been observed while using the same model and th e same constraints, b u t different m e t h o d s o f analysis.
3.3.2. Dynamic technique
horizon
year planning
The simplified t e c h n i q u e described in § 2.3 has also been applied to t he IEEE 5-bus test system with data and assumptions as given in §3.2. The initial solution o f Table 1 is obtained as explained in §3.3.1. Updated intermediate system states are t he n used in an optimization
0.3361 6.3946 0.0 0.0 0.0 0.0
process to determine the requirements for the six stages simultaneously. However, in this case only the constraints of the last stage are applied. Consequently, the loading is considered as t he difference between t he projections for the beginning and end o f t he planning period. The intermediate system states are then re-modified according to the new estimates of the capacity requirements, and the process is repeated until the results converge to the 0.01 p.u. accuracy margin. The results are shown in Table 3. F o u r iterations were required. A conventional horizon year plan has also been determined using the same system model. Results for t he conventional study are presented in Table 4. A comparison between these results and those of Table 3 n o t only reveals the difference in accuracy, it also displays the ability o f the new formulation, even in the simplified form, to provide t he intermediate stage requirements for optimal long-range planning. Of course the gain in accuracy is achieved at the expense of increased com put at i onal requirements. Instead o f applying the static optimization technique just once, as in the
TABLE 3 Transmission capacity requirements determined by the 'dynamic horizon year planning technique'
Stage
1 2 3 4 5 6
Capacity additions (p.u.) for branches 1 to 7 1
2
3
4
5
6
7
Total per stage (p.u.)
0.0 0.0 0.0 0.0 0.0 1.7892
0.0 2.5406 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0
0.0 4.8336 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 1.4068 0.0
0.0 0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0 0.0
0.0 7.3742 0.0 0.0 1.4068 1.7892
136 TABLE 4 Transmission capacity requirements determined by the 'conventional horizon year planning technique' Capacity additions (p.u.) for branches 1 to 7 1
2
3
4
5
6
7
0.0200
1.5148
0.0
2.4905
0.5083
0.0
0.0
case of the conventional horizon year planning approach, it is applied K (the number of stages) times for the initial solution, plus an additional 3 to 4 times (with the number of independent variables multiplied b y K) in the iterative process. This is done for purposes of updating the sensitivity matrices according to intermediate stage capacity modifications. In the present study with the computing facilities used (Multics System) the computing time required for the initial solution was a b o u t 4 seconds per stage. Estimated average computing time for the six-stage model is 8 seconds per iteration. The corresponding figure for the conventional horizon year plan is 4 seconds. The time required b y the dynamic planning method was 12 seconds for the same six-stage model. It has been found that the length of the time period of the stage affects the accuracy o f the results. This is to be expected, firstly because increasing the number of stages should move the solution closer to the ideal dynamic solution, or because of the iterative procedure followed in estimating the intermediate system states; secondly, the approximate linear model used to represent the system produces higher errors as the stage length increases. The length of the stage therefore reflects the trade-off between the accuracy of the solution and computing time. A significant advantage of the proposed technique is that any system representation m a y be incorporated with the desired degree of detail. As seen from the above example, statistical load projection can be used together with quantitative reliability constraints. 4. CONCLUSIONS A new formulation of the dynamic transmission capacity expansion planning problem has been introduced. A quantitative reliability
index has been observed and uncertainties in load projections have been accounted for. The new formulation is flexible. It can handle a wide range of reliability criteria and system models. The proposed iterative procedure to solve the planning problem appears to converge quickly to a specified accuracy. In the cases studied no more than four iterations were needed. Computational requirements are reasonable, increasing almost linearly with the dimension of the problem. The simplified technique, referred to as the 'dynamic horizon year planning technique' provides fast approximate plans. It is capable of providing a more accurate solution than the conventional horizon year planning technique because intermediate stages are considered. It may be regarded as a quasidynamic algorithm because the constraints on the intermediate stages are applied only for the initial solution and then relaxed. ACKNOWLEDGEMENT
The authors acknowledge with thanks the funds granted by the Natural Sciences and Engineering Research Council of Canada in support of this study. REFERENCES 1 T. A. M. Sharaf and G. J. Berg, Static transmission capacity planning under uncertainty, Electr. Power Syst. Res., 7 (1984) 289 - 296. 2 P. H. Henault, R. B. Eastvedt, J. Peschon and L. P. Hajdu, Power system long-term planning in the presence of uncertainty, IEEE Trans., PAS-89 (1970) 1 5 6 - 164. 3 Y. P. Dusonchet and A. E1-Abiad, Transmission • planning using discrete dynamic optimizing, IEEE Trans., PAS-92 (1973) 1358 - 1371. 4 R. N. Adams and M. A. Laughton, Optimal planning of power networks using mixed-integer programming, Part I -- Static and time-phased network synthesis, Proc. Inst. Electr. Eng., 121 (1974) 1 3 9 - 147. 5 G. J. Berg, K. R. C. Mamandur and P. Subramanian, A new technique for transmission system planning, Can. Electr. Eng. J., 3 (3) (1978) 35 40. 6 A. P. Meliopoulos, R. P. Webb, R. J. Bennon and J. A. Juves, Optimal long-range transmission planning with AC load flow, IEEE Trans., PAS101 (1982) 4156 - 4163. 7 T. A. M. Sharaf, Probabilistic measures and analytical techniques in power system transmission expansion planning, Ph.D. Thesis, University of Calgary, Canada, July 1982.