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 [1–3]. Security-constrained optimal power flow (SCOPF) [3–5] 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) [6–8], 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:

where

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

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

• P̃_D uncertain active power system load level,

• P_L active power system losses,

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

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

• P_resmin minimum limit of active power reserve,

• P_G,imax maximum limit of active power generation,

• P_G,imin minimum limit of active power generation,

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

• P_Fc,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.

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, ΔP_D 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 ΔP_D. The change in the system load level has a fuzzy membership function value μ(ΔP_D) 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 ΔP_D 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:

where

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

• IC_i incremental cost of the i-th generator,

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

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

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

• ΔP_G,imax maximum limit of active power generation,

• ΔP_G,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:

where

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

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

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

• w_BRk penalty weight for z_BRk,

• w_CAm penalty weight for z_CAm,

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

• P_Fc,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, P_Go, 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, i ∈ SG.

• 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)}, i ∈ SG, 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)}, i ∈ SG, 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, P_Go,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.

Table 1. Simulation data for the dispatchable generators.

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

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