A new nonparametric density estimation for probabilistic security-constrained economic dispatch

Abstract

Security-constrained economic dispatch (SCED) which is used to minimize the operation cost of the committed units with the constraints of power balance, ramp rate, and unit capacity is one of the routine challenges in power system operation. In this paper, a nonparametric estimation method based on kernel density and linear diffusion is proposed to obtain continuous probability density functions for probabilistic SCED outcomes. It is assumed that the probabilistic SCED problem is the second stage of a two-stage problem, while stochastic security-constrained unit commitment is the first stage. To evaluate the efficacy of the proposed method, a 6-bus test system and IEEE 118-bus system are used as case studies. Implementing the proposed method on these case studies demonstrates the accuracy of the proposed method for large scale power systems.

Keywords

Benders’ decomposition kernel density estimation linear diffusion method probabilistic security-constrained economic dispatch security-constrained unit commitment

1 Introduction

Economic dispatch is one of the basic routine challenges in power system operation. Conventional economic dispatch is defined and formulated to minimize operation costs of the committed units with the constraints of power balance, ramp rate, and unit capacity. Economic efficiency and reliability of a power system are important challenges of the restructured energy market; however, these two aims inherently oppose each other. Efficiency forces the maximum use of available tie line capacity, whereas reliability necessitates an absolute degree of precaution in the application of transmission capacity to be prepared for uncertainties such as wind power forecasting errors and load forecasting errors and contingencies. To make a proper balance between goals, system operators basically solve a security-constrained economic dispatch (SCED) model [1]. Security constraints require the economic dispatch solution to coincidentally address a feasible power flow under a list of scenarios, named uncertainty. A practical load flow is one, which does not result in overloads in lines.

The deterministic approach for security assessment has been widely used [2]. The system security evaluation is accomplished in the case of thermal loading of system elements [3], as well as voltage and frequency variations for both transient and steady states [4]. A new modeling of economic dispatch problem for gas turbine combined-cycle, in which fuel consumption is minimized for a period of operation in order to accomplish optimal load dispatch among units, was presented in reference [5].In [6], a new heuristic optimization method, called bee algorithm, was proposed for emission and security-constrained economic dispatch (ESCED). In [7], thermostatically controlled loads (TCLs) are candidates for a short term demand response, and a SCED model is used for optimizing output of generators and TCL energy consumption. An SCED problem subject to frequency response and transmission line constraints was proposed in [8].

Wind energy is considered a clean and renewable source of energy and is widely applied in power system generation. However, probabilistic characteristic of wind energy causes important challenges to the power system operation and control. Besides, load variations institute another origin of uncertainty in power system scheduling and operation. Therefore, using probabilistic evaluations methods for today’s power system secure operation is inevitable. A considerable number of remarkable studies have been recently done on probabilistic SCED. In [9], economic dispatch problem was considered a two-stage convex program for operational decision under uncertainty and the L-shaped method was applied to decompose the large stochastic program into smaller sub-problems. According to Benders’ decomposition (BD) method, series algorithmic enhancements were implemented in [9] in order to mitigate the computational difficulty. A risk-based security constrained economic dispatch (RB-SCED) for real-time electricity market was proposed in [10]. In [11], a new formulation of a hybrid stochastic-robust optimization was presented and used to calculate a look ahead security-constrained optimal power flow. It was designed to decrease the carbon dioxide (CO₂) emissions by efficiency accommodating of renewable energy sources. In [12], unavailability and uncertainty of wind speed, solar radiation, and power demand were chosen as random variables for probabilistic economic dispatch model. Authors of [13] modelled an economic environmental power dispatch as a probabilistic multiobjective problem. In the latter reference, the problem is solved based on expected value of random variables. Then, required reserve for compensating uncertainties is developed using cumulative distribution function (CDF) of each random variables.

In conventional studies, a stochastic SCUC problem was solved and scheduling of committed units was determined according to the results. Previously, the effect of uncertainties on the real time output power of generation units and line flows was neglected; due to the fact that it was not necessary according to structure of day-ahead scheduling in the electricity market. Although, in recent studies, e.g. [10, 13], a probabilistic SCED was implemented as the second stage of stochastic SCUC in order to gain information about effect of uncertainties on the real time operation. These recent studies presented the results of probabilistic SCED in a discrete way, i.e. data sets, histograms, etc. Having a continuous probability density function (PDF) for output real power of generation units and transmission line flow, in which the effect of uncertainties are considered, is so beneficial in some applications such as required reserve analysis, risk assessment, and other power system probabilistic assessments. In this paper, in order to meet maximum accuracy for final estimated PDFs, a new nonparametric density estimation was proposed and the efficiency and accuracy of the proposed method were investigated.

The result of the probabilistic analysis of a SCED problem as expected is a PDF. In this study, a nonparametric density estimation method is applied for estimating the PDF of probabilistic SCED results. The advantage of the nonparametric density estimation method is that it offers greater flexibility for modeling a given data set [14]. There are numerous approaches for estimating nonparametric density, the most popular of which is kernel density estimation (KDE). It has been illustrated that this method is useful in Monte Carlo simulation (MCS) methods such as smoothed bootstrap method [15], particle filter method [16], probabilistic load flow [17], and analyzing transient recovery voltage induced across circuit breaker poles during clearing faults [18].

In KDE method, low bandwidth leads to discrete PDF and high bandwidth leads to over-smooth PDF. Therefore, a technique should be used to find the optimum bandwidth. Since finding the optimal bandwidth is very essential in KDE, in this paper a linear diffusion method is used for this issue [19].

The rest of this paper is organized as follows. Section 2 presents the concept of KDE and its utilization in this study. Probabilistic SCED problem formulation and solution methodology are given in Section 3 and Section 4, respectively. Two distinct case studies are reported in Section 5. Finally, Section 6 presents a brief conclusion about this study.

2 Kernel density estimation

Nonparametric density estimation techniques enjoy particular advantages and are very popular tools in the statistical analysis of data. Complete discussion about the advantages of nonparametric techniques was presented in [14]. The term “nonparametric” includes techniques which allow a functional form of fit on the data to be obtained without relying on any prior information about the data [20, 21]. KDE is one of the most popular nonparametric techniques for density estimation.

2.1 Gaussian kernel density estimation

Assume that A_N ={ X₁, X₂, …, X_N } is a set of independent variables that are distributed based on an unknown continuous PDF. Then, Gaussian kernel density estimator is defined as (1). $\hat{g} (x; t) = \frac{1}{N} \sum_{i = 1}^{N} φ (x, X_{i}; t) x \in R$ (1) $φ (x, X_{i}; t) = \frac{1}{\sqrt{2 π t}} e^{\frac{- (x - X_{i})^{2}}{2 t}}$ (2)

Where (2) is a Gaussian PDF with location X_i and scale t^1/2. The scale parameter is also known as bandwidth.

Performance of $\hat{g} (x; t)$ as the estimator depends on the optimal value of t. Finding the optimal bandwidth is so important; therefore, many studies have been concentrated on this regard. A complete discussion about KDE concept, different types of KDE, and some methods for finding optimal bandwidth was presented in [14].

2.2 Diffusion estimator

Since finding the optimal bandwidth is very essential in KDE, in this paper, a linear diffusion method is used for this issue [19].

Mean integrated squared error (MISE), which is shown (3), is a useful criterion to obtain optimal t. $MISE {\hat{g}} (t) = E_{f} \int {[\hat{g} (x; t) - f (x)]}^{2} dx$ (3)

For the sake of simplicity and without loss of accuracy, instead of MISE, its first-order asymptotic approximation, which is known as AMISE, can be used. AMISE can be calculated using of (4) and (5). $AMISE {\hat{g}} (t) = \frac{1}{4} t^{2} | | f^{″} | |^{2} + \frac{1}{2 N \sqrt{π t}}$ (4) $| | f^{″} | |^{2} = \int (f^{″} (x))^{2} dx$ (5)

The asymptotically optimal value of bandwidth, (6), can be obtained by minimizing AMISE. It should be noted that (6) depends on the unknown PDF. $t_{opt} = (2 N \sqrt{π} | | f^{″} | |^{2})^{- 2 / 5}$ (6)

A single stage plug-in bandwidth selector t based on the smoothing properties of the linear diffusion partial differential equation (PDE) was presented in [19]. The PDE is given in (7) where the linear differential operator, L, is defined as (8). In this equation, a (x) and p (x) are arbitrary positive functions on X. $\frac{\partial}{\partial t} g (x; t) = L (g (x; t)) x \in X, t > 0$ (7) $L (.) = (\frac{1}{2}) (\frac{d}{dx}) a (x) (\frac{d}{dx}) (\frac{.}{p (x)})$ (8)

3 SCED problem formulation

The formulation of SCED problem is almost similar to that of security-constrained unit commitment (SCUC) formulation [22]. Therefore, in this section, first, SCUC formulation is presented. Then, SCED formulation will be introduced.

3.1 SCUC formulation

Formulation of the deterministic SCUC is given in (9–18). The objective function (9) consists of generation costs, startup costs, and shutdown costs of thermal units. The cost function, F_ci (P_it), is modeled by a quadratic equation. Therefore, the problem is considered non-linear programming (NLP). Constraints of SCUC problem are given in (10–18). Equation (10) represents the system power balance. System spinning and operating reserves are shown in (11) and (12), respectively. Ramping up/down limits are presented in (13) and (14). Minimum ON/OFF time limits are shown in (15) and (16). Equation (17) represents unit generation limits. Finally, DC network security constraints, which models the transmission system limits, is given in (18). $min \sum_{i = 1}^{NG} \sum_{t = 1}^{NT} [F_{ci} (P_{it}) . I_{it} + {SU}_{it} + {SD}_{it}]$ (9) $\sum_{i = 1}^{NG} P_{it} . I_{it} + \sum_{i = 1}^{NW} P_{w, it}^{f} = P_{D, t}^{f}; (t = 1, \dots, NT)$ (10) $\sum_{i = 1}^{NG} R_{S, it} . I_{it} \geq R_{S, t}; (t = 1, \dots, NT)$ (11) $\sum_{i = 1}^{NG} R_{O, it} . I_{it} \geq R_{O, t}; (t = 1, \dots, NT)$ (12)

$\begin{matrix} P_{it} - P_{i (t - 1)} \leq [1 - I_{it} (1 - I_{i (t - 1)})] \cdot {UR}_{i} \\ I_{it} + (1 - I_{i (t - 1)}) . P_{i}^{\min}; (t = 1, \dots, NT; \\ i = 1, \dots, NG) \end{matrix}$ (13)

$\begin{matrix} P_{i (t - 1)} - P_{it} \leq [1 - I_{i (t - 1)} (1 - I_{it})] \cdot {DR}_{i} \\ + I_{i (t - 1)} (1 - I_{it}) . P_{i}^{\min}; \\ (t = 1, \dots, NT; i = 1, \dots, NG) \end{matrix}$ (14) $\begin{matrix} [X_{i (t - 1)}^{on} - T_{i}^{on}] . [I_{i (t - 1)} - I_{it}] \geq 0 \\ ; (t = 1, \dots, NT; i = 1, \dots, NG) \end{matrix}$ (15) $\begin{matrix} [X_{i (t - 1)}^{off} - T_{i}^{off}] . [I_{it} - I_{i (t - 1)}] \geq 0; \\ (t = 1, \dots, NT; i = 1, \dots, NG) \end{matrix}$ (16)

$\begin{matrix} P_{i}^{\min} . I_{it} \leq P_{it} \leq P_{i}^{\max} . I_{it}; \\ (t = 1, \dots, NT; i = 1, \dots, NG) \end{matrix}$ (17) ${- PL}^{max} \leq SF {\cdot (K}_{P} {. P}_{t} {- K}_{D} {. P}_{D} t) \leq {PL}^{max}$ (18)

3.2 SCED formulation

Formulation of SCED problem is almost similar to that of SCUC formulation. Only two differences should be considered. First, the binary statue indices of thermal units, denoted by I_it, are constants in SCED. In fact, SCED problem is only concerned with the value of generation power of thermal units, P_it, and ON/OFF statue of thermal units is previously determined by SCUC [23]. Second, constraints related to the statue of thermal units are not considered in SCED. As a result, minimum ON/OFF time limits, Equations (15) and (16), and terms SU_it and SD_it in the objective function should be omitted.

Consequently, the SCED formulation can be modeled by (10–18) subject to the above considerations.

4 Solution methodology

SCUC problem modeled in Equations (9–18) is non-convex, large scale, and mixed integer non-linear programming (MINLP). However, SCED is an NLP problem due to the fact that the binary statue indices of thermal units, I_it, are fixed to a predetermined value.

In this paper, we assume that probabilistic SCED problem is the second stage of a two-stage problem, while the stochastic SCUC is the first stage. The flowchart of such a problem is shown in Fig. 1.

As demonstrated in Fig. 1, at the first stage, the stochastic SCUC problem is solved via scenario-based MCS and BD. This stage has been widely reported in previous studies [22, 24]. Outcomes of the first stage are binary statue indices of thermal units (I_it), total operation cost (Z), and output power of thermal units (P_it). Due to power system uncertainties, I_it is the only operative outcome. In other words, there is no guarantee in day-ahead scheduling that the forecasting values of uncertain input variable such as price-based loads, wind generation, etc. will be met in real time operation. Therefore, a probabilistic SCED should run to obtain the output PDFs for total operation cost, generation power of thermal units, and transmission line flows. These PDFs can be used for risk-based assessments and more valid prediction of total operation costs.

Probabilistic SCED is presented at the second stage of the flowchart (Fig. 1). According to Fig. 1, the probabilistic SCED problem is solved as a deterministic problem for each Monte Carlo scenario. Outcomes of this problem are the vectors which include probabilistic results for total operation cost, generation power of thermal units, and transmission line flows. Finally, PDF estimation methods should be applied to the output vectors in order to have a continuous PDF. In this paper, as mentioned in Section 2, a KDE via diffusion was used for PDF estimation [19]. It will be illustrated in the case study section that a KDE in which the optimal bandwidth is obtained via diffusion is more accurate than the normal PDF estimation for probabilistic SCED results.

5 Case studies

Two distinct cases, a 6-bus system and the IEEE 118-bus system, were studied to illustrate the efficacy of the proposed method. The accuracy of KDE for estimating output PDFs in probabilistic SCED problem was demonstrated by simulation results of the case studies. In both cases, the estimated PDFs obtained by KDE via diffusion were compared with the histogram and normal PDFs. Moreover, the effect of the number of final scenarios in scenario reduction step and value of bandwidth in KDE wereinvestigated.

5.1 6-bus test system

As the first case, the proposed method was implemented on the 6-bus system shown in Fig. 2. The system had three thermal generators, seven transmission line, three load points, and a wind farm located on the third bus. At each hour, the loads L1, L2, and L3 consumed 20%, 40%, and 40% of the total hourly load of the system, respectively.

Information data of thermal units and transmission lines are given in Tables 1 and 2, respectively. Table 3 presents the hourly forecasted load and wind power generation of the system.

The wind power is one of the most important uncertainties in power systems [25]. Therefore, in this study, the penetration level of wind power (ratio of the wind power generation to the total generation) was increased by 40% at some hours. As a result, a large amount of generation was probabilistic at those hours, which demonstrated the validity of the proposed method in the systems with high volatility of power generation. Figure 3 presents the hourly penetration level of the wind power generation [26].

Two cases are studied on the 6-bus test system:

Case 1) Deterministic SCUC and SCED (base case).

Case 2) Stochastic SCUC and SCED (with uncertainties).

Case 1) In the first case, all uncertain inputs were fixed to their forecasted values. Therefore, no probabilistic analysis was considered in this case. Total operations cost was 95421.21 $. Unit commitment, generation scheduling, and line flow scheduling are presented in Tables 4–6, respectively. As given in Table 6, the power flows through line 6 at hour 16 to 21 was fixed to its upper limit due to network security constraint, which happened due to the fact that, at these hours, the wind generation decreased drastically; therefore, the load connected to bus 3 had to be supplied by other units through line 6 (refer to Figs. 2 and 3).

Case 2) In the second case, the probabilistic nature of power system uncertainties was considered. Three load points and one wind farm were considered the hourly uncertainties and modeled by truncated normal PDFs with means of equal to the forecasted values and standard deviations of 10% of the corresponding mean values. The problem was solved based on the flowchart proposed in Fig. 1. According to PDFs of input uncertain parameters, 10000 initial scenarios were generated by MCS. Then, the initial scenarios were reduced to 500 final scenarios using GAMS scenario reduction library (SCENRED) [27]. As presented in the flowchart, after the second stage, the outcome vectors were used for PDF estimation by KDE via diffusion.

The PDFs of power generations at hours 7, 8, and 11 for units 1, 2, and 3 are given in Figs. 4–6, respectively. According to Figs. 4c, 5c, and 6b, the generations of all the three units were almost similar to those of the normal PDF at hour 11 (peak load hour) and KDE presented the appropriate estimation. Moreover, in the situation where histogram showed a multimodal PDF such as Figs. 5a, 5b, and 6a, KDE could estimate a more exact PDF than the normal estimation.

For estimating multimodal PDFs, such as those shown in Figs. 5a, 5b, and 6a, KDE via diffusion is more accurate. To illustrate it, a very popular method for finding optimum bandwidth of KDE is tested. Finding optimum for estimation normaldensities method [28], is very well-known, since it has been chosen as the default method in MATLAB (Distribution Fitting toolbox) for calculating bandwidth of KDE. In this study, diffusion is compared with this method in multimodal situation. As shown in Figs. 5a, 5b, and 6a, the bandwidths obtained from linear diffusion are 0.995, 0.828, and 0.294, while MATLAB default method bandwidths are 2.019, 1.760, and 0.723, respectively. It can be observed that MATLAB default method estimation is rather over-smooth.

According to Fig. 4a and 4b, the generation of the first unit was almost fixed at 100 (MW). KDE estimation could track the extreme mode of the histogram better than the normal estimation. However, in this situation, estimating a continuous PDF may not be a good option and the output could be considered a constant value.

A comparison between different numbers of final scenarios and different bandwidths is presented in Figs. 8 and 9, respectively. As shown in Fig. 8, the number of final scenarios should be enough to have a valid PDF. It should be noticed that the probabilistic SCED is an NLP problem, while the stochastic SCUC is an MINLP problem with computational burden and high simulation time. In contrast to stochastic SCUC, solving time was not high in probabilistic SCED. Therefore, implementing the second stage of the flowchart with a high number of final scenarios and having an accurate PDF for the outcomes would not be too time consuming.

As illustrated in Fig. 9, the optimal bandwidth for KDE was so essential. Bandwidth of less than the optimal one led to non-smooth PDF and bandwidth of more than the optimal one made over-smooth PDF. In this study, as mentioned in Section 2, a linear diffusion method was used to obtain the optimal bandwidth [19].

5.2 IEEE 118-bus test system

In order to evaluate the proposed method in a large scale problem, the 118-bus system of [26] with a large number of uncertain variables was used. The system included 54 thermal units, 186 transmission lines, and 91 load points. It was modified by adding three wind farms on buses 36, 77, and 69. System characteristics and wind power generations could be found in [26].

All the 91 load points and three wind farms were considered the uncertain input variables and modeled by truncated normal distributed functions, in which means were the forecasted values and standard deviations were 10% of the means. Initially, 10000 Monte Carlo scenarios were generated and reduced to 500 final scenarios by SCENRED [27]. Total operation cost for the base case and for the stochastic SCUC with 500 final scenarios was 736011$ and 714355$, respectively. The unit commitments for both base case and stochastic SCUC could be found in the authors’ previous work [26].

As presented in [26], unit 43 was ON during 24 hours; however, unit 28 was ON only at 9–23. Density estimation based on KDE for the output power of unit 28 at hour 12 and unit 43 at hour 14 is provided in Figs. 10 and 11, respectively. Capability of KDE via diffusion for density estimation in such large scale system is illustrated in these figures.

6 Conclusion

In this study, a nonparametric PDF estimation method based on kernel density and linear diffusion was proposed for probabilistic SCED. In this paper, the probabilistic SCED problem was assumed as the second stage of a two-stage problem, while the stochastic SCUC was the first stage. Then, the problem was solved via the proposed two-stage flowchart and the vectors, including the probabilistic values of outcomes were used for PDF estimation. To evaluate the capability and efficacy of the proposed method, a 6-bus test system and IEEE 118-bus system were used as the case studies. Implementing the proposed method on 118-bus system with 95 uncertain inputs, as a large scale high uncertain problem, demonstrated the accuracy of the proposed method. Multiple figures of the estimated PDF by the proposed method were presented to validate the accuracy and efficacy of the method. As illustrated in the case study section, via diffusion, KDE is capable of estimating appropriate continuous PDFs for probabilistic SCEDproblem.

Footnotes

Appendix: Nomenclature

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