Article Search
닫기

## Original Article

Split Viewer

International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(2): 128-134

Published online June 25, 2022

https://doi.org/10.5391/IJFIS.2022.22.2.128

© The Korean Institute of Intelligent Systems

## Security-constrained Economic Dispatch (SCED) Considering Load Level Uncertainty

BenJeMar-Hope Flores1 and Hwachang Song2

1Department of Electrical and Information Engineering, Seoul National University of Science and Technology, Seoul, Korea
2Department of Smart Energy System Engineering, Seoul National University of Science and Technology, Seoul, Korea

Correspondence to :
Hwachang Song (hcsong@seoultech.ac.kr)

Received: September 24, 2021; Revised: September 24, 2021; Accepted: January 10, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

In power system operation, security-constrained economic dispatch (SCED) is used to provide a reference point for the next time period for each dispatchable generator, and load forecasting needs to be involved in the SCED procedure. In real-time operation, forecasting errors and the nonlinearity of the load change might cause undesirable dispatch patterns in terms of system security. This study presents an SCED that considers the uncertainty of the system load level. For this purpose, a fuzzy model was employed to express the uncertainty in the load level change, and a practical two-stage SCED solution method was adopted. In addition, to resolve the difficulty in determining the reference generation pattern from solutions using this model, a procedure is required to determine the secure upper and lower limits for the dispatchable generators as the final SCED outcome. The numerical results were obtained using a 43-bus test system to test the feasibility of the proposed method.

Keywords: Economic dispatch, Load level, Generation pattern, Security constraints, Uncertainty model

The optimal generation dispatch must be determined to operate power systems securely and economically, so that the system can operate within the secure operation region while supplying the corresponding load, and electrical power is supplied based on a predefined pattern [13]. Security-constrained optimal power flow (SCOPF) [35] is a tool for the optimal adjustment of power generation to prevent the system from securing region violations under normal and contingency conditions. However, in real-time system operation, a security-constrained economic dispatch (SCED) [68], which is a simplified form of SCOPF, is employed instead. SCED considers several linearized security constraints, including branch flow limits and reserve power constraints, and needs to use real-time state estimation data and network analysis results. The optimal generation correction is given to generators that have adjustable ramp-up and ramp-down rates. The energy management system adopts an external module of automatic generation control (AGC) for transmitting the correction signal.

The system load level critically affects the optimal generation dispatch obtained by the SCED. The target time point of the SCED application is usually a few minutes after the current time, so another external module for short-term load forecasting is needed, whose prediction results are passed to the SCED. In addition, conventional SCED relies entirely on the forecasting results, and the solutions are determined to satisfy the change in the predicted load level within the target time snapshot. Thus, it is desirable to increase the accuracy of the load prediction. If the actual load level deviates from the forecasted load level, the determined generation pattern may cause violations of the system security limits in real-time operation. To avoid this, increasing the accuracy of short-term load forecasting can be considered as an applicable countermeasure. However, it is obvious that there is a limit to improving the accuracy. Therefore, for secure system operation, it is important to properly deal with the uncertainty in load prediction during the SCED solution process.

In this study, an SCED formulation that considers the uncertainty of the system load level and its solution method are proposed. To express the nonlinearity and forecasting error in the load change of a real system operation, a fuzzy model with a set of three break points is adopted herein. However, when the uncertainty model for load change is applied, it might be difficult to determine the final solution for one reference generation point because it is expressed in the same form as the SCED solution. To manage this difficulty, a procedure for determining the secure upper and lower limits for dispatchable generators as the final outcome of SCED is proposed in this study. Furthermore, a two-stage solution technique corresponding to a practical SCED solution is employed. To test the feasibility of the proposed method, this paper includes numerical results, which were obtained using a modified 43-bus test system.

An SCED formulation with uncertain system load level parameters can be expressed as follows:

$min∑i∈SGfi(PG,i),$$s.t.∑i∈SGPG,i=P˜D+PL,$$Presmin≤∑i∈SG(PG,imax-PG,i),$$PG,imin≤PG,i≤PG,imax, i∈SG,$$∣PF, k∣≤PF,kmax, k∈SBR,$$|PF,kc| ≤PF,kmaxc, k∈SBR, c∈SCA,$

where

• fi (•) cost function for the i-th generator,

• PG,i active power generation of the i-th generator,

• D uncertain active power system load level,

• PL active power system losses,

• PF,k active power flow at the k-th branch,

• $PF,kc$ active power flow at the k-th branch during contingency c,

• Presmin minimum limit of active power reserve,

• PG,imax maximum limit of active power generation,

• PG,imin minimum limit of active power generation,

• PFn,kmax maximum limit of active power flow at the k-th branch,

• PFc,kmax maximum limit of active power flow at the k-th branch during contingency c,

• SBR set of branches,

• SCA set of contingencies,

• SG set of generators.

From the power balance in Eq. (2), it can be seen that an uncertainty parameter is included for the system load level. The uncertainty in the deviation of the actual and forecasted load-level parameters affects the optimal generation dispatch and system security. The above formulation includes the branch flow constraints for the normal and contingent states that require power flow calculation; therefore, an external power flow calculation module is needed to obtain branch flows for the states.

### 3. Solution Technique for SCED with Load Level Uncertainty

Various models have been proposed to express uncertain parameters in power systems, as in [2,5,9,10], but in this study a fuzzy model is adopted for the change in the system load level from the previous value. The model is illustrated in Figure 1. In Figure 1, ΔPD and $ΔPDest$ represent the actual change in the system load level from the one time point to the next one and its predicted value, respectively; α and β represent the inferior and superior dispersion values for the uncertainty in ΔPD. The change in the system load level has a fuzzy membership function value μPD) in the range of [0, 1].

The conventional SCED problem considers only one forecasted load level. Therefore, a two-stage solution technique is employed for practical solutions to this problem. The simple procedure for the conventional technique is shown in Figure 2. The first stage in the solution procedure, which is called unconstrained economic dispatch (UED), is to obtain the generation change of each AGC participating generator and follow ΔPD by considering only the corresponding ramp-up and ramp-down rates. The second stage, called constrained economic dispatch (CED), corrects the generation dispatch from the first stage to remove possible violations of the security constraints. Because the dispatch pattern after the first stage is different from that at the current time point, there should be another process for the re-evaluation of the branch and interface line loading between the UED and CED.

When obtaining a solution that considers the uncertainty model of the system load level change, as described above, it is necessary to determine the modified optimization formulations for the two stages. The formulation of the UED solution is as follows:

$min∑i∈SGICiΔPG,iU(fd),$$s.t.∑i∈SGΔPG,iU(fd)=ΔPD(fd),$$ΔPres,min≤-∑i∈SGΔPG,iU(fd),$$ΔPG,imin≤ΔPG,iU(fd)≤ΔPG,imax, i∈SG, fd∈{1,2,3},$

where

• fd break point number of the fuzzy model for system load level,

• ICi incremental cost of the i-th generator,

• $ΔPG,iU(fd)$ UED generation dispatch of the i-th generator at the fd-th break point,

• $ΔPD(fd)$ system load level at the fd-th break point,

• ΔPres,min minimum limit of active power reserve change,

• ΔPG,imax maximum limit of active power generation,

• ΔPG,imin minimum limit of active power generation.

In the objective function of the UED, the incremental cost of each generator is determined from the current operating point. In Eq. (8), $ΔPD(1),ΔPD(2)$ and $ΔPD(3)$ correspond to $ΔPDpre-α,ΔPDpre$ and $ΔPDpre+β$, respectively. The formulation of the CED solutions is as follows:

$min∑i∈SGICiΔPG,iC(fd)+∑k∈SCBRwBRkzBRk+∑m∈SCCAwCAmzCAm,$$s.t.∑i∈SGΔPG,iC(fd)=ΔPD(fd),$$ΔPres,min≤-∑i∈SGΔPG,iC(fd),$$ΔPG,imin≤ΔPG,iC(fd)≤ΔPG,imax, i∈SG,$$∑i∈SGSnikΔPG,iC(fd)-zBRK≤PF,kmax-PFn,ko k∈SCBR, fd∈{1,2,3},$$∑i∈SGScimΔPG,iC(fd)-zCAm≤PF,mmax-PFc,mo m∈SCCA, fd∈{1,2,3},$

where

• $ΔPG,iC(fd)$CED generation dispatch of the i-th generator at the fd-th break point,

• zBRk slack variable for the k-th branch flow constraint,

• zCAm slack variable for the m-th contingency branch flow constraint,

• wBRk penalty weight for zBRk,

• wCAm penalty weight for zCAm,

• PFn,ko initial active power flow for the k-th branch flow constraint,

• PFc,mo initial active power flow for the m-th contingency branch flow constraint,

• SCBR set of critical branches in the normal state,

• SCCA set of critical branches in the contingency state.

From these two formulations, one notices that the decision-making vector and the generation dispatch patterns for the UED and CED are affected only by the corresponding load level discrete point. It should be noted that the difference between these two formulations is in whether security constraints exist. The security constraints can be derived from branch flow constraints or contingency branch flow constraints, and they are in the form of linearized constraints in the CED formulation. Further, they mainly use branch flow sensitivities, which are obtained based on the state estimation data. Figure 3 illustrates a conceptual structure of the solution technique.

As shown in Figure 3, the UED begins with the initial generation pattern, PGo, which is exactly the same as that in the measurement used in the state estimation. This determines the generation dispatch vectors $ΔPGU(1), ΔPGU(2)$ and $ΔPGU(3)$, only following the load level discrete points. In real SCED applications, it is desirable to reduce the number of security constraints in the CED owing to the convergence issue. For this purpose, the generation dispatch vectors from the UED were examined in terms of possible violations of the security constraints, and critical (contingency) branch flow constraints were chosen. Then, CED problems were formulated, including the selected security constraints, and the constrained generation dispatch vectors $ΔPGC(1), ΔPGC(2)$ and $ΔPGC(3)$ were obtained.

Finally, the secure generation limits for all AGC participating generators were determined from the constrained generation dispatch vectors. Within the secure generation limits determined by SCED, a real-time economic dispatch module provides the AGC base point, and the generator resource limits were set using the ramp-up/down rates and a 1-minute time period. The system can then be operated within secure regions using the SCED results. Therefore, the decision on the secure generation limits with the outputs from the UED and CED is important, and it should be made considering the operational philosophy of the power system. In this study, it was assumed that system security is the top priority. Based on this assumption, the decision was made by applying the following procedure:

• Step 1 $PG,imaxAGC←PG,imax$ and $PG,iminAGC←PG,imin$, iSG.

• Step 2 If $ΔPG,iU(fd)<0$ and $ΔPG,iC(fd)>ΔPG,iU(fd)$, then $PG,iminAGC←max{PG,iminAGC, PGo,i+PG,iC(fd)}$, iSG, fd ∈ {1, 2, 3}.

• Step 3 If $ΔPG,iU(fd)>0$ and $ΔPG,iC(fd)<ΔPG,iU(fd)$, then $PG,iminAGC←max{PG,imaxAGC, PGo,i+ΔPG,iC(fd)}$, iSG, fd ∈ {1, 2, 3}.

• Step 4 If $PG,imaxAGC≤PG,iminAGC$, then $PG,imaxAGC←PGo,i$ and $PG,iminAGC←PGo,i$.

In the above procedure, PGo,i denotes the active power output of the generator at the previous time point; $PG,imaxAGC$ and $PG,iminAGC$ stand for the secure upper and lower limits of the generator; Δt is the time step of the SCED run. The main idea is that there are no violations of the security limits in the normal and contingent states within the secure generation limits, considering the range of the uncertain load level parameter.

The proposed method was applied to a test system with 43 buses, which was modified from the 39-bus test system in [11], and Figure 4 illustrates a one-line diagram of the system. The total load of the base case was 6, 020.73 MW and the total generation was 6, 069.99 MW. The locations connected to the three generators were used to represent the combined cycle (CC) generators. It should be noted that the ramp-up and ramp-down rates for CC generators are usually high. In the base case, the marginal power plant is that with CC generators at buses 38, 42, and 43.

In this simulation, a strict scenario was examined, in which the forecasted level of the system load was assumed to have changed by −100 MW from the initial load level. The target time point for the load forecasting was 10 minutes at the time of the initial load level. During system operation, there are some cases in which the load level changes exhibit nonlinearity.

If the actual load level changes follow the example shown in Figure 5, network security might not be assured because of load level uncertainty. To adequately deal with this uncertainty, inferior and superior dispersion values were adopted in this study, and they were set to −10 and 150 MW, respectively, as shown in Figure 6.

For the initial system condition, network analysis was performed, including the normal and contingent states, and generation shift sensitivities were evaluated for the critical branches in the normal and contingent states. Four credible contingencies were analyzed in the simulation. Using the network analysis results, the procedure enters the SCED, considering the load-level uncertainty. Table 1 lists the simulation data for the dispatchable generators. It should be noted that one critical branch exists for the contingency of line 26–29, and that the critical branch is line 28–29. As shown in Figure 4, one can notice that the generators at buses 38, 42, and 43 are only sensitive to relieve the loading on critical line 26–29. Tables 2 and 3 show the simulation results obtained by applying the proposed SCED procedure.

In Tables 2 and 3, UED(k) and CED(k) represent the generation dispatch cases obtained by the UED and CED modules for the k-th break point of the load level change, respectively. The most sensitive generators for the critical line reduced their generation amounts in the UED stage for the breakpoints with −110 and −100MWin load change. The dispatch directions alleviated the loading on the critical line. Thus, no UED dispatch pattern caused any security constraint violations, as shown in Table 3. However, for a break point of +50 MW, those generators increased their outputs in the UED stage to follow the load change; hence, the critical line in the contingent state might exceed the security constraint limit of 712.5 MW, as shown in Table 3. Subsequently, the CED stage was performed to obtain the dispatch pattern for securing the limit.

As shown in Table 1, the generator at bus 30 with the highest IC increased its output by more than 50 MW and the most sensitive generators reduced their outputs. From Table 2, one can see that the dispatch pattern satisfies the security constraint for the 3rd break point. Following the procedure for determining the secure generation limits, the upper limits of the generators at buses 38, 42, and 43 must be set to 257.0155, 257.0155, and 253.0295 MW, respectively. Other dispatchable generators do not necessarily use secure limits. If the secure generation limits proposed by the SCED are considered, it is possible to maintain the security of the system with load level uncertainty in an economic dispatch for AGC.

A SCED method considering an uncertainty model for a system load level change is described in this paper. The conventional SCED adopts a load-forecasting module, and if the forecasting error is rather high or the load changes in a nonlinear manner, the SCED might expose the system to an insecure situation. Herein, a fuzzy model with three breakpoints was employed for the load level change in the SCED. To resolve the difficulty in determining the reference generation pattern from solutions when using this model, an extended procedure is applied for determining the secure upper and lower output limits for dispatchable generators. Using the secure limits for generators sensitive to critical lines can be an effective option in real-time operations, allowing the system to operate within the secure region.

This work was supported by the Seoul National University of Science and Technology.

Fig. 1.

Fuzzy model for change in the system load level.

Fig. 2.

Two-stage SCED procedure and related modules.

Fig. 3.

Conceptual structure of the solution technique.

Fig. 4.

One-line diagram of the 43-bus test system.

Fig. 5.

One-line diagram of the 43-bus test system.

Fig. 6.

One-line diagram of the 43-bus test system.

Table. 1.

Table 1. Simulation data for the dispatchable generators.

BusPGo (MW)PGmax (MW)PGmin (MW)Rdn/up (MW/m)IC (₩/kW)
3010020010040141.96
31527527280550.30
34527527280648.20
36527527280448.90
3721221214626.874.41
4021221214626.874.41
411321321093074.41
382582721313476.37
422582721313476.37
4325426813140.676.37

Table. 2.

Table 2. Dispatch results for the dispatchable generators (unit: MW).

BusUED(1)UED(2)UED(3)CED(3)
300.00.00.052.9395
310.00.00.00.0
340.00.00.00.0
360.00.00.00.0
370.00.00.00.0
400.00.00.00.0
410.00.00.00.0
38*−36.8235−33.346117.0804−0.9845
42*−36.8235−33.346117.0804−0.9845
43*−36.3529−33.307915.8392−0.9704

Table. 3.

Table 3. Simulation results applying the proposed SCED procedure (unit: MVA).

CaseP28–29Q28–29S28–29P29–28Q29–28S29–28Smax
UED(1)−601.321.6601.7606.03.2606.1712.5
UED(2)−611.021.3611.4615.95.3616.0712.5
UED(3)−756.010.4756.0763.847.8765.3712.5
CED(3)−705.015.1705.1711.730.9712.3712.5

1. Stott, B, Alsac, O, and Monticelli, AJ (1987). Security analysis and optimization. Proceedings of the IEEE. 75, 1623-1644. https://doi.org/10.1109/PROC.1987.13931
2. Zhu, J (2015). Optimization of Power System Operation. Hoboken, NJ: John Wiley & Sons
3. Wood, AJ, Wollenberg, BF, and Sheble, GB (2014). Power Generation, Operation, and Control. Hoboken, NJ: John Wiley & Sons
4. Stott, B, and Alsac, O . Optimal power flow: basic requirements for real-life problems and their solutions., Proceedings of SEPOPE XII Symposium, 2012, Rio de Janeiro, Brazil.
5. Capitanescu, F (2016). Critical review of recent advances and further developments needed in AC optimal power flow. Electric Power Systems Research. 136, 57-68. https://doi.org/10.1016/j.epsr.2016.02.008
6. Elacqua, AJ, and Corey, SL (1982). Security constrained dispatch at the New York power pool. IEEE Transactions on Power Apparatus and Systems. 101, 2876-2884. https://doi.org/10.1109/TPAS.1982.317613
7. Bacher, R, and Van Meeteren, HP (1988). Real-time optimal power flow in automatic generation control. IEEE Transactions on Power Systems. 3, 1518-1529. https://doi.org/10.1109/59.192961
8. Papalexopoulos, AD (1996). Challenges to on-line OPF implementation. IEEE Tutorial Course Optimal Power Flow: Solution Techniques, Requirements, and Challenges. Piscataway, NJ: IEEE
9. Zhang, H, and Li, P (2011). Chance constrained programming for optimal power flow under uncertainty. IEEE Transactions on Power Systems. 26, 2417-2424. https://doi.org/10.1109/TPWRS.2011.2154367
10. Vrakopoulou, M, Katsampani, M, Margellos, K, Lygeros, J, and Andersson, G . Probabilistic security-constrained AC optimal power flow., 2013 IEEE Grenoble Conference, 2013, Grenoble, France, Array, pp.1-6. https://doi.org/10.1109/PTC.2013.6652374
11. Padiyar, KR (2008). Power System Dynamics: Stability and Control. Chichester, UK: Wiley

BenJeMar-Hope Flores received his degree in electrical engineering from the University of the Philippines, Diliman in 2011. He finished his M.S. degree in Seoul National University of Science and Technology in 2014 and is currently a Ph.D. candidate in the same university. He is a registered electrical engineer and has also served as a senior science research specialist for the Department of Science and Technology in the Philippines. His research interests include power system planning and operation, grid resilience, and reliability.

E-mail: floresbenjie@seoultech.ac.kr

Hwachang Song received his B.S., M.S., and Ph.D. degrees in electrical engineering from Korea University in 1997, 1999, and 2003, respectively. He was a post-doctoral scholar at Iowa State University from 2003 to 2004. He worked as a faculty member in the School of Electronic and Information Engineering, Kunsan National University, from 2005 to 2008. Currently, he is a professor in the Department of Electrical and Information Engineering and the Department of Smart Energy System Engineering, Seoul National University of Science and Technology. His recent research interests include optimization with uncertain parameters, system modeling, system operation and control, and renewable energy.

E-mail: hcsong@seoultech.ac.kr

### Article

#### Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2022; 22(2): 128-134

Published online June 25, 2022 https://doi.org/10.5391/IJFIS.2022.22.2.128

## Security-constrained Economic Dispatch (SCED) Considering Load Level Uncertainty

BenJeMar-Hope Flores1 and Hwachang Song2

1Department of Electrical and Information Engineering, Seoul National University of Science and Technology, Seoul, Korea
2Department of Smart Energy System Engineering, Seoul National University of Science and Technology, Seoul, Korea

Correspondence to:Hwachang Song (hcsong@seoultech.ac.kr)

Received: September 24, 2021; Revised: September 24, 2021; Accepted: January 10, 2022

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In power system operation, security-constrained economic dispatch (SCED) is used to provide a reference point for the next time period for each dispatchable generator, and load forecasting needs to be involved in the SCED procedure. In real-time operation, forecasting errors and the nonlinearity of the load change might cause undesirable dispatch patterns in terms of system security. This study presents an SCED that considers the uncertainty of the system load level. For this purpose, a fuzzy model was employed to express the uncertainty in the load level change, and a practical two-stage SCED solution method was adopted. In addition, to resolve the difficulty in determining the reference generation pattern from solutions using this model, a procedure is required to determine the secure upper and lower limits for the dispatchable generators as the final SCED outcome. The numerical results were obtained using a 43-bus test system to test the feasibility of the proposed method.

Keywords: Economic dispatch, Load level, Generation pattern, Security constraints, Uncertainty model

### 1. Introduction

The optimal generation dispatch must be determined to operate power systems securely and economically, so that the system can operate within the secure operation region while supplying the corresponding load, and electrical power is supplied based on a predefined pattern [13]. Security-constrained optimal power flow (SCOPF) [35] is a tool for the optimal adjustment of power generation to prevent the system from securing region violations under normal and contingency conditions. However, in real-time system operation, a security-constrained economic dispatch (SCED) [68], which is a simplified form of SCOPF, is employed instead. SCED considers several linearized security constraints, including branch flow limits and reserve power constraints, and needs to use real-time state estimation data and network analysis results. The optimal generation correction is given to generators that have adjustable ramp-up and ramp-down rates. The energy management system adopts an external module of automatic generation control (AGC) for transmitting the correction signal.

The system load level critically affects the optimal generation dispatch obtained by the SCED. The target time point of the SCED application is usually a few minutes after the current time, so another external module for short-term load forecasting is needed, whose prediction results are passed to the SCED. In addition, conventional SCED relies entirely on the forecasting results, and the solutions are determined to satisfy the change in the predicted load level within the target time snapshot. Thus, it is desirable to increase the accuracy of the load prediction. If the actual load level deviates from the forecasted load level, the determined generation pattern may cause violations of the system security limits in real-time operation. To avoid this, increasing the accuracy of short-term load forecasting can be considered as an applicable countermeasure. However, it is obvious that there is a limit to improving the accuracy. Therefore, for secure system operation, it is important to properly deal with the uncertainty in load prediction during the SCED solution process.

In this study, an SCED formulation that considers the uncertainty of the system load level and its solution method are proposed. To express the nonlinearity and forecasting error in the load change of a real system operation, a fuzzy model with a set of three break points is adopted herein. However, when the uncertainty model for load change is applied, it might be difficult to determine the final solution for one reference generation point because it is expressed in the same form as the SCED solution. To manage this difficulty, a procedure for determining the secure upper and lower limits for dispatchable generators as the final outcome of SCED is proposed in this study. Furthermore, a two-stage solution technique corresponding to a practical SCED solution is employed. To test the feasibility of the proposed method, this paper includes numerical results, which were obtained using a modified 43-bus test system.

### 2. Problem Description

An SCED formulation with uncertain system load level parameters can be expressed as follows:

$min∑i∈SGfi(PG,i),$$s.t.∑i∈SGPG,i=P˜D+PL,$$Presmin≤∑i∈SG(PG,imax-PG,i),$$PG,imin≤PG,i≤PG,imax, i∈SG,$$∣PF, k∣≤PF,kmax, k∈SBR,$$|PF,kc| ≤PF,kmaxc, k∈SBR, c∈SCA,$

where

• fi (•) cost function for the i-th generator,

• PG,i active power generation of the i-th generator,

• D uncertain active power system load level,

• PL active power system losses,

• PF,k active power flow at the k-th branch,

• $PF,kc$ active power flow at the k-th branch during contingency c,

• Presmin minimum limit of active power reserve,

• PG,imax maximum limit of active power generation,

• PG,imin minimum limit of active power generation,

• PFn,kmax maximum limit of active power flow at the k-th branch,

• PFc,kmax maximum limit of active power flow at the k-th branch during contingency c,

• SBR set of branches,

• SCA set of contingencies,

• SG set of generators.

From the power balance in Eq. (2), it can be seen that an uncertainty parameter is included for the system load level. The uncertainty in the deviation of the actual and forecasted load-level parameters affects the optimal generation dispatch and system security. The above formulation includes the branch flow constraints for the normal and contingent states that require power flow calculation; therefore, an external power flow calculation module is needed to obtain branch flows for the states.

### 3. Solution Technique for SCED with Load Level Uncertainty

Various models have been proposed to express uncertain parameters in power systems, as in [2,5,9,10], but in this study a fuzzy model is adopted for the change in the system load level from the previous value. The model is illustrated in Figure 1. In Figure 1, ΔPD and $ΔPDest$ represent the actual change in the system load level from the one time point to the next one and its predicted value, respectively; α and β represent the inferior and superior dispersion values for the uncertainty in ΔPD. The change in the system load level has a fuzzy membership function value μPD) in the range of [0, 1].

The conventional SCED problem considers only one forecasted load level. Therefore, a two-stage solution technique is employed for practical solutions to this problem. The simple procedure for the conventional technique is shown in Figure 2. The first stage in the solution procedure, which is called unconstrained economic dispatch (UED), is to obtain the generation change of each AGC participating generator and follow ΔPD by considering only the corresponding ramp-up and ramp-down rates. The second stage, called constrained economic dispatch (CED), corrects the generation dispatch from the first stage to remove possible violations of the security constraints. Because the dispatch pattern after the first stage is different from that at the current time point, there should be another process for the re-evaluation of the branch and interface line loading between the UED and CED.

When obtaining a solution that considers the uncertainty model of the system load level change, as described above, it is necessary to determine the modified optimization formulations for the two stages. The formulation of the UED solution is as follows:

$min∑i∈SGICiΔPG,iU(fd),$$s.t.∑i∈SGΔPG,iU(fd)=ΔPD(fd),$$ΔPres,min≤-∑i∈SGΔPG,iU(fd),$$ΔPG,imin≤ΔPG,iU(fd)≤ΔPG,imax, i∈SG, fd∈{1,2,3},$

where

• fd break point number of the fuzzy model for system load level,

• ICi incremental cost of the i-th generator,

• $ΔPG,iU(fd)$ UED generation dispatch of the i-th generator at the fd-th break point,

• $ΔPD(fd)$ system load level at the fd-th break point,

• ΔPres,min minimum limit of active power reserve change,

• ΔPG,imax maximum limit of active power generation,

• ΔPG,imin minimum limit of active power generation.

In the objective function of the UED, the incremental cost of each generator is determined from the current operating point. In Eq. (8), $ΔPD(1),ΔPD(2)$ and $ΔPD(3)$ correspond to $ΔPDpre-α,ΔPDpre$ and $ΔPDpre+β$, respectively. The formulation of the CED solutions is as follows:

$min∑i∈SGICiΔPG,iC(fd)+∑k∈SCBRwBRkzBRk+∑m∈SCCAwCAmzCAm,$$s.t.∑i∈SGΔPG,iC(fd)=ΔPD(fd),$$ΔPres,min≤-∑i∈SGΔPG,iC(fd),$$ΔPG,imin≤ΔPG,iC(fd)≤ΔPG,imax, i∈SG,$$∑i∈SGSnikΔPG,iC(fd)-zBRK≤PF,kmax-PFn,ko k∈SCBR, fd∈{1,2,3},$$∑i∈SGScimΔPG,iC(fd)-zCAm≤PF,mmax-PFc,mo m∈SCCA, fd∈{1,2,3},$

where

• $ΔPG,iC(fd)$CED generation dispatch of the i-th generator at the fd-th break point,

• zBRk slack variable for the k-th branch flow constraint,

• zCAm slack variable for the m-th contingency branch flow constraint,

• wBRk penalty weight for zBRk,

• wCAm penalty weight for zCAm,

• PFn,ko initial active power flow for the k-th branch flow constraint,

• PFc,mo initial active power flow for the m-th contingency branch flow constraint,

• SCBR set of critical branches in the normal state,

• SCCA set of critical branches in the contingency state.

From these two formulations, one notices that the decision-making vector and the generation dispatch patterns for the UED and CED are affected only by the corresponding load level discrete point. It should be noted that the difference between these two formulations is in whether security constraints exist. The security constraints can be derived from branch flow constraints or contingency branch flow constraints, and they are in the form of linearized constraints in the CED formulation. Further, they mainly use branch flow sensitivities, which are obtained based on the state estimation data. Figure 3 illustrates a conceptual structure of the solution technique.

As shown in Figure 3, the UED begins with the initial generation pattern, PGo, which is exactly the same as that in the measurement used in the state estimation. This determines the generation dispatch vectors $ΔPGU(1), ΔPGU(2)$ and $ΔPGU(3)$, only following the load level discrete points. In real SCED applications, it is desirable to reduce the number of security constraints in the CED owing to the convergence issue. For this purpose, the generation dispatch vectors from the UED were examined in terms of possible violations of the security constraints, and critical (contingency) branch flow constraints were chosen. Then, CED problems were formulated, including the selected security constraints, and the constrained generation dispatch vectors $ΔPGC(1), ΔPGC(2)$ and $ΔPGC(3)$ were obtained.

Finally, the secure generation limits for all AGC participating generators were determined from the constrained generation dispatch vectors. Within the secure generation limits determined by SCED, a real-time economic dispatch module provides the AGC base point, and the generator resource limits were set using the ramp-up/down rates and a 1-minute time period. The system can then be operated within secure regions using the SCED results. Therefore, the decision on the secure generation limits with the outputs from the UED and CED is important, and it should be made considering the operational philosophy of the power system. In this study, it was assumed that system security is the top priority. Based on this assumption, the decision was made by applying the following procedure:

• Step 1 $PG,imaxAGC←PG,imax$ and $PG,iminAGC←PG,imin$, iSG.

• Step 2 If $ΔPG,iU(fd)<0$ and $ΔPG,iC(fd)>ΔPG,iU(fd)$, then $PG,iminAGC←max{PG,iminAGC, PGo,i+PG,iC(fd)}$, iSG, fd ∈ {1, 2, 3}.

• Step 3 If $ΔPG,iU(fd)>0$ and $ΔPG,iC(fd)<ΔPG,iU(fd)$, then $PG,iminAGC←max{PG,imaxAGC, PGo,i+ΔPG,iC(fd)}$, iSG, fd ∈ {1, 2, 3}.

• Step 4 If $PG,imaxAGC≤PG,iminAGC$, then $PG,imaxAGC←PGo,i$ and $PG,iminAGC←PGo,i$.

In the above procedure, PGo,i denotes the active power output of the generator at the previous time point; $PG,imaxAGC$ and $PG,iminAGC$ stand for the secure upper and lower limits of the generator; Δt is the time step of the SCED run. The main idea is that there are no violations of the security limits in the normal and contingent states within the secure generation limits, considering the range of the uncertain load level parameter.

### 4. Numerical Results

The proposed method was applied to a test system with 43 buses, which was modified from the 39-bus test system in [11], and Figure 4 illustrates a one-line diagram of the system. The total load of the base case was 6, 020.73 MW and the total generation was 6, 069.99 MW. The locations connected to the three generators were used to represent the combined cycle (CC) generators. It should be noted that the ramp-up and ramp-down rates for CC generators are usually high. In the base case, the marginal power plant is that with CC generators at buses 38, 42, and 43.

In this simulation, a strict scenario was examined, in which the forecasted level of the system load was assumed to have changed by −100 MW from the initial load level. The target time point for the load forecasting was 10 minutes at the time of the initial load level. During system operation, there are some cases in which the load level changes exhibit nonlinearity.

If the actual load level changes follow the example shown in Figure 5, network security might not be assured because of load level uncertainty. To adequately deal with this uncertainty, inferior and superior dispersion values were adopted in this study, and they were set to −10 and 150 MW, respectively, as shown in Figure 6.

For the initial system condition, network analysis was performed, including the normal and contingent states, and generation shift sensitivities were evaluated for the critical branches in the normal and contingent states. Four credible contingencies were analyzed in the simulation. Using the network analysis results, the procedure enters the SCED, considering the load-level uncertainty. Table 1 lists the simulation data for the dispatchable generators. It should be noted that one critical branch exists for the contingency of line 26–29, and that the critical branch is line 28–29. As shown in Figure 4, one can notice that the generators at buses 38, 42, and 43 are only sensitive to relieve the loading on critical line 26–29. Tables 2 and 3 show the simulation results obtained by applying the proposed SCED procedure.

In Tables 2 and 3, UED(k) and CED(k) represent the generation dispatch cases obtained by the UED and CED modules for the k-th break point of the load level change, respectively. The most sensitive generators for the critical line reduced their generation amounts in the UED stage for the breakpoints with −110 and −100MWin load change. The dispatch directions alleviated the loading on the critical line. Thus, no UED dispatch pattern caused any security constraint violations, as shown in Table 3. However, for a break point of +50 MW, those generators increased their outputs in the UED stage to follow the load change; hence, the critical line in the contingent state might exceed the security constraint limit of 712.5 MW, as shown in Table 3. Subsequently, the CED stage was performed to obtain the dispatch pattern for securing the limit.

As shown in Table 1, the generator at bus 30 with the highest IC increased its output by more than 50 MW and the most sensitive generators reduced their outputs. From Table 2, one can see that the dispatch pattern satisfies the security constraint for the 3rd break point. Following the procedure for determining the secure generation limits, the upper limits of the generators at buses 38, 42, and 43 must be set to 257.0155, 257.0155, and 253.0295 MW, respectively. Other dispatchable generators do not necessarily use secure limits. If the secure generation limits proposed by the SCED are considered, it is possible to maintain the security of the system with load level uncertainty in an economic dispatch for AGC.

### 5. Conclusion

A SCED method considering an uncertainty model for a system load level change is described in this paper. The conventional SCED adopts a load-forecasting module, and if the forecasting error is rather high or the load changes in a nonlinear manner, the SCED might expose the system to an insecure situation. Herein, a fuzzy model with three breakpoints was employed for the load level change in the SCED. To resolve the difficulty in determining the reference generation pattern from solutions when using this model, an extended procedure is applied for determining the secure upper and lower output limits for dispatchable generators. Using the secure limits for generators sensitive to critical lines can be an effective option in real-time operations, allowing the system to operate within the secure region.

### Fig 1.

Figure 1.

Fuzzy model for change in the system load level.

The International Journal of Fuzzy Logic and Intelligent Systems 2022; 22: 128-134https://doi.org/10.5391/IJFIS.2022.22.2.128

### Fig 2.

Figure 2.

Two-stage SCED procedure and related modules.

The International Journal of Fuzzy Logic and Intelligent Systems 2022; 22: 128-134https://doi.org/10.5391/IJFIS.2022.22.2.128

### Fig 3.

Figure 3.

Conceptual structure of the solution technique.

The International Journal of Fuzzy Logic and Intelligent Systems 2022; 22: 128-134https://doi.org/10.5391/IJFIS.2022.22.2.128

### Fig 4.

Figure 4.

One-line diagram of the 43-bus test system.

The International Journal of Fuzzy Logic and Intelligent Systems 2022; 22: 128-134https://doi.org/10.5391/IJFIS.2022.22.2.128

### Fig 5.

Figure 5.

One-line diagram of the 43-bus test system.

The International Journal of Fuzzy Logic and Intelligent Systems 2022; 22: 128-134https://doi.org/10.5391/IJFIS.2022.22.2.128

### Fig 6.

Figure 6.

One-line diagram of the 43-bus test system.

The International Journal of Fuzzy Logic and Intelligent Systems 2022; 22: 128-134https://doi.org/10.5391/IJFIS.2022.22.2.128

Simulation data for the dispatchable generators.

BusPGo (MW)PGmax (MW)PGmin (MW)Rdn/up (MW/m)IC (₩/kW)
3010020010040141.96
31527527280550.30
34527527280648.20
36527527280448.90
3721221214626.874.41
4021221214626.874.41
411321321093074.41
382582721313476.37
422582721313476.37
4325426813140.676.37

Dispatch results for the dispatchable generators (unit: MW).

BusUED(1)UED(2)UED(3)CED(3)
300.00.00.052.9395
310.00.00.00.0
340.00.00.00.0
360.00.00.00.0
370.00.00.00.0
400.00.00.00.0
410.00.00.00.0
38*−36.8235−33.346117.0804−0.9845
42*−36.8235−33.346117.0804−0.9845
43*−36.3529−33.307915.8392−0.9704

Simulation results applying the proposed SCED procedure (unit: MVA).

CaseP28–29Q28–29S28–29P29–28Q29–28S29–28Smax
UED(1)−601.321.6601.7606.03.2606.1712.5
UED(2)−611.021.3611.4615.95.3616.0712.5
UED(3)−756.010.4756.0763.847.8765.3712.5
CED(3)−705.015.1705.1711.730.9712.3712.5

### References

1. Stott, B, Alsac, O, and Monticelli, AJ (1987). Security analysis and optimization. Proceedings of the IEEE. 75, 1623-1644. https://doi.org/10.1109/PROC.1987.13931
2. Zhu, J (2015). Optimization of Power System Operation. Hoboken, NJ: John Wiley & Sons
3. Wood, AJ, Wollenberg, BF, and Sheble, GB (2014). Power Generation, Operation, and Control. Hoboken, NJ: John Wiley & Sons
4. Stott, B, and Alsac, O . Optimal power flow: basic requirements for real-life problems and their solutions., Proceedings of SEPOPE XII Symposium, 2012, Rio de Janeiro, Brazil.
5. Capitanescu, F (2016). Critical review of recent advances and further developments needed in AC optimal power flow. Electric Power Systems Research. 136, 57-68. https://doi.org/10.1016/j.epsr.2016.02.008
6. Elacqua, AJ, and Corey, SL (1982). Security constrained dispatch at the New York power pool. IEEE Transactions on Power Apparatus and Systems. 101, 2876-2884. https://doi.org/10.1109/TPAS.1982.317613
7. Bacher, R, and Van Meeteren, HP (1988). Real-time optimal power flow in automatic generation control. IEEE Transactions on Power Systems. 3, 1518-1529. https://doi.org/10.1109/59.192961
8. Papalexopoulos, AD (1996). Challenges to on-line OPF implementation. IEEE Tutorial Course Optimal Power Flow: Solution Techniques, Requirements, and Challenges. Piscataway, NJ: IEEE
9. Zhang, H, and Li, P (2011). Chance constrained programming for optimal power flow under uncertainty. IEEE Transactions on Power Systems. 26, 2417-2424. https://doi.org/10.1109/TPWRS.2011.2154367
10. Vrakopoulou, M, Katsampani, M, Margellos, K, Lygeros, J, and Andersson, G . Probabilistic security-constrained AC optimal power flow., 2013 IEEE Grenoble Conference, 2013, Grenoble, France, Array, pp.1-6. https://doi.org/10.1109/PTC.2013.6652374
11. Padiyar, KR (2008). Power System Dynamics: Stability and Control. Chichester, UK: Wiley