Electric Power Systems Research xxx (2017) xxx-xxx Contents lists available at ScienceDirect Electric Power Systems Research journal homepage: www.elsevier.com Multi-period stochastic security-constrained OPF considering the uncertainty sources of wind power, load demand and equipment unavailability Hossein Sharifzadeh a, Nima Amjady a, Hamidreza Zareipour b, a University of Semnan, Iran b University of Calgary, Canada ARTICLE INFO Article history: Received 2 September 2016 Received in revised form 2 January 2017 Accepted 5 January 2017 Available online xxx Keywords: Optimal power flow Security constraints Uncertainty sources Wind generation Load demand Nomenclature Sets and indices S ABSTRACT The uncertainty sources of the intermittent generation and load demand as well as transmission line unavailability threaten the security of power systems. In this paper, to treat these uncertainties, a new stochastic optimal power flow considering the security constraints is proposed. A scenario generation method is also presented to model the uncertainties of wind generations and load demands considering their correlations. In the proposed model, the uncertainties are coped with through combination of optimal here-and-now and wait-and-see decisions. The effectiveness of the proposed model is shown on the well-known IEEE 24-bus test system. Higher effectiveness of the proposed model compared with four deterministic methods and one other stochastic method to determine procured reserve and after-the-fact conditions is numerically illustrated. Additionally, the impact of the number of scenarios on the performance of the proposed model is evaluated by means of a sensitivity analysis. It has also been shown that the scenarios generated considering correlations have more smooth variations and can more effectively capture the uncertain behavior of load. 2016 Published by Elsevier Ltd. Set of network buses Bus-generator incidence matrix Bus-branch incidence matrix Set of non-islanding branch contingencies Branch contingency index Set of conventional generating units Index of conventional generating units Network bus indices The set of scenarios including the same realizations of the uncertain variables from hour 1 to hour Set of wind generation and load demand (W&L) scenarios S Set of trial scenarios s, s 1, s 2 W&L scenario indexes s Trial scenario index T Parameters Number of time steps in the scheduling horizon Index of time steps in the scheduling horizon Index of wind generating units Corresponding author. Fax: +1 4032826855. Email address: hzareipo@ucalgary.ca (H. Zareipour) Offer cost for up and down spinning reserve capacity by conventional unit g for hour t, respectively Maximum flow limit of the branch connecting bus i and bus j Minimum and maximum generation limits of conventional unit g Specified load demand of bus i in hour t and W&L scenario s, obtained through the scenario generation method Specified generation for wind unit w in hour t and W&L scenario s, obtained through the scenario generation method Maximum available down/up spinning reserve capacity of conventional unit g in hour t Up and down ramp rate limit, respectively Load shedding cost for load i in hour t Branch reactance between buses i and j Offer cost for energy and deployed up and down spinning reserve by conventional unit g for hour t, respectively Offer cost of conventional unit g for up and down generation shift in hour t, respectively Probability of contingency c, W&L scenario s, and trial scenario s', respectively Variables http://dx.doi.org/10.1016/j.epsr.2017.01.011 0378-7796/ 2016 Published by Elsevier Ltd.
2 Electric Power Systems Research xxx (2017) xxx-xxx 1. Introduction Power flow of the branch connecting bus i and bus j in hour t and W&L scenario s, in normal state and post-contingent state c, respectively Scheduled generation of conventional unit g for hour t in normal state Generation of conventional unit g in hour t, W&L scenario s, and post-contingent state c Generation of conventional unit g in hour t, W&L scenario s, and normal state Load shed of bus i in hour t, W&L scenario s, and post-contingent state c Scheduled generation of wind unit w in hour t, W&L scenario s and normal state Scheduled generation of wind unit w in hour t, W&L scenario s and post-contingent state c Scheduled up and down reserve capacity of conventional unit g for hour t, respectively Up and down reserve of conventional unit g in hour t, W&L scenario s, and post-contingent state c, deployed in addition to, respectively, to cope with the contingency Deployed up and down reserve of conventional unit g in hour t, W&L scenario s, and normal state, respectively, to cope with the W&L uncertainty Total deployed up and down reserve of conventional unit g in hour t of trial scenario s Up and down generation shift of conventional unit g in hour t of trial scenario s, respectively Load shed of bus i in hour t of trial scenario s Generation of conventional unit g in hour t of trial scenario s Wind spillage of wind unit w in hour t, W&L scenario s and normal state to cope with the W&L uncertainty Wind spillage of wind unit w in hour t, W&L scenario s and post-contingent state c, deployed in addition to to cope with the contingency Voltage angle of bus i in hour t and W&L scenario s, in normal state and post-contingent state c, respectively Continuously increasing fossil fuel prices and atmosphere pollution as well as depletion of fossil fuel energy sources causes that renewable green energy sources have a growing share in supplying human s energy demands. Wind power is the most widely used renewable energy source in today s electricity generation. However, the uncertain and variable nature of wind generation together with traditional load forecasting error dramatically add the uncertainty sources and seriously challenge the optimal and secure operation of power systems [1]. Moreover, load growth, change of generation pattern caused by renewable resources, and increased energy transactions in the environment of competitive electricity markets increase the probability of overload in the electric network, which in turn jeopardize the system security [2]. Therefore, power system operators should make secure their systems against different unexpected generation/ load patterns and network configurations. However, the uncertainty sources seriously challenge the applicability of traditional optimal power flow (OPF), which usually determines the operating state of power system assuming only its most likely operating conditions [3]. A few approaches have been presented in the literature to handle wind/load uncertainties in OPF models. The most important methods include probabilistic techniques [4], chance constrained programming [5], robust optimization [6], and stochastic programming [7]. Probabilistic methods tend to present the probability density function characteristics of desired outputs instead of one aggregated optimal solution. In chance constrained programming, constraints are satisfied with some predetermined probabilities. Chance constrained programming leads to nonlinear problems, which are not easy to handle. While robust optimization (RO) is a professional uncertainty handling method for non-deterministic optimization problems involving low-frequency and non-random uncertain variables, setting the appropriate uncertainty sets for RO may be a challenging task. Moreover, when the uncertainty sources of non-deterministic optimization problem are in the form of high-frequency uncertain variables, RO may not necessarily be able to effectively employ the whole statistical information of these variables. Apart from the foregoing problems, none of the mentioned research works considers the required preventive/ corrective actions together with their associated costs, e.g., when the so-called N 1 criterion is employed. Inclusion of security constraints in the OPF problem leads to a more effective operation tool, usually known as the security constrained OPF (SCOPF). However, SCOPF is a more computationally complex optimization problem than OPF as it includes significantly more constraints. Different numerical optimization approaches, such as iterative-based solution methods [8], Benders decomposition [9], and parallelism techniques [10], have been proposed to handle the high dimensional SCOPF problem. Although various evolutionary algorithms have also been presented for solving SCOPF [11], these methods usually suffer from high computation burden to solve SCOPF for practical power systems. Moreover, the effectiveness of evolutionary algorithms depends on the initial population and selected values for their settings, while usually there is no analytical method to fine-tune these settings for solving SCOPF. The main contributions of this paper can be summarized as: 1) A new multi-period security-constrained stochastic optimal power flow (MPSC-SOPF) model taking into account load and wind generation uncertainties as well as uncertainties associated with equipment unavailability is presented. The proposed model, based on two-stage stochastic programming, determines the optimal operating point and required corrective actions considering probability of different load/wind generation scenarios and contingency states. By simultaneously handling the corrective actions required for coping with different uncertainty sources (including the uncertainties of load and wind as well as the uncertainties of equipment unavailability), a more effective reserve procurement strategy is achieved. This is the key issue which distinguishes the proposed MPSC-SOPF model from the previous stochastic OPF models which determine the corrective actions required for handling different uncertainty sources separately. 2) A new scenario generation method composed of Latin hypercube sampling (LHS) and rank correlation is proposed. The proposed method can model the correlations between wind generations and load demands, correlations between generations of different wind farms, and inter-temporal dependencies. 3) A new out-of-sample analysis to evaluate the performance of stochastic models for after-the-fact conditions in a sufficiently long run is presented. In this way, the performance of a stochastic model can be better evaluated for unseen scenarios and realizations.
Electric Power Systems Research xxx (2017) xxx-xxx 3 2. The proposed MPSC-SOPF model The objective function of the proposed MPSC-SOPF model including the costs of here-and-now, i.e. first stage decisions, and wait-and-see, i.e. second stage decisions, in the both normal and contingency states is as follows: The objective function includes the cost of energy and up/down reserve capacity procurement, i.e. the first summation term, plus the expected cost of corrective actions, i.e. the second summation term. The first summation term includes the cost of first stage decisions, i.e. the decision variables that are independent of the scenarios. The second part of Eq. (1) denotes the costs of second stage variables, which depend on the realized scenario. The first summation in the second part of Eq. (1) represents deployed reserves cost in normal state to cope with the uncertainties stemmed from load/wind forecast errors. The second summation of this part is the cost of corrective actions, including deployed reserves and load shedding, to eliminate overloads in post-contingent states of branch outages. In the second part of Eq. (1), the scenarios modeling the wind generation and load demand (W&L) uncertainties are weighted by their probabilities. Similarly, the scenarios pertaining to branch contingencies are weighted by their probabilities. The considered linear offer cost in the objective function is consistent with the today s electricity market practices in which participants submit their offering curves usually in single or multiple linear energy price blocks. The presented objective function considers single energy price blocks but can be extended to multiple energy price blocks. The constraints of the proposed model associated with the normal and post-contingent states are as follows. A.) Normal state constraints - Variables bounds (1) (2) (3) (4) (5) - Ramp rate limits - Power flow equality constraints - Branch flow limits B.) Post-contingent states constraints - Variables bounds (6) (7) (8) (9) (10) (11) (12) (14) (15) (16) (17) (18) (19) (20) (21)
4 Electric Power Systems Research xxx (2017) xxx-xxx - Power flow equality constraints - Line flow limits C.) Non-anticipativity constraints (22) (24) (25) (26) (27) (28) (29) (30) (31) (32) Constraints (2) (10) and (16) (22) bound the variables to their allowable ranges in normal and post contingency states, respectively. Wind spillage is considered in Eq. (19) as a down reserve to enhance feasibility of serious extra-generation scenarios and decrease total operating cost by reducing downward reserve procurement [12]. Constraints (11) and (12) model the ramp rate limits of conventional generators. DC power flow constraints are shown in Eqs. (13) and (14) and (23) (24) in normal and post-contingent states, respectively. Constraints (15) and (25) express the branch flow limits in normal and post-contingent states, respectively. The non-anticipativity constraints of (26) (32) avoid from unreasonable decisions made in the stochastic program. These conditions state that two scenarios with the same history of the realized uncertain variables from hour 1 to hour t must have the same decisions made in hour t [13]. The proposed MPSC-SOPF includes the first stage decision variables of, and, and second stage decision variables of,,,,,, and. The corrective actions of the proposed model, appeared as the second stage decision variables, are divided to two categories. The first category includes the corrective actions required for dealing with the uncertainty sources of wind generations and load demands, which consist of,, and. The second one contains,,, and, adopted to cope with the uncertainty source of contingencies (load shed is only allowed for treating contingencies). An important advantage of the proposed MPSC-SOPF model is that both categories of the second stage decision variables are simultaneously optimized along the first stage decision variables within it leading to lower required corrective actions and their associated costs. For instance, both categories of the deployed reserves, including (, ) and (, ) are simultaneously optimized along the capacity of reserves ( and ) within the proposed MPSC-SOPF model. This leads to lower required total reserve deployment and capacity compared to the other works that separately determine the reserves required for coping with the uncertainties of W&L and contingencies. An OPF tool usually determines the final generation levels of committed generation units which their off/on statuses have already been determined in the unit commitment (UC) problem [14]. What is critical in the multi-period OPF compared to the single-period OPF is ramp rate limits of generating units [6], which have been considered in the proposed model as the constraints (11) and (12). However, if the on/off states of conventional units are considered, the model changes into the security constrained UC (SCUC) model which is not the focus of this paper. Additionally, the proposed MPSC-SOPF model is based on DC-OPF and thus control targets only involve generator and load aspects instead of switched capacitors/reactors and transformer taps. As reviewed in Ref. [15], none of the ISOs in USA use the AC optimal power flow. Thus, DC optimal power flow is more consistent with the current industry practice. The proposed MPSC-SOPF model can be used as the sub-problem of the UC problem. Moreover, it can be used as the rescheduling tool to fine tune the UC decisions when more exact forecasts are available in upcoming hours. Finally, it can be directly used as the clearing mechanism in the OPF-based markets. 3. Proposed scenario generation and reduction The proposed scenario generation approach produces scenarios based on two important criteria: 1) The produced scenarios should be consistent with the stochastic behavior of the uncertain variables. 2) Correlation should be considered in the scenario generation approach to avoid producing infeasible scenarios. Despite the recent developments in the load and wind power forecast methods, prediction error is an indispensable aspect of these forecast processes. These errors have important effects on the operational decisions of power systems. As the prediction error values are not known in advance, load demand and wind power generation appear as uncertain variables. In the stochastic programming (SP) framework, the uncertain variables are modeled using their possible realizations called scenarios. Owning to the key role of scenarios on the quality of SP model, various researchers have focused on the scenario generation techniques. Among these techniques, sampling methods have some prominent advantages, such as simplicity and the ability to resemble the probability distribution of stochastic variables. However, generally, they need to a large number of samples to effec
Electric Power Systems Research xxx (2017) xxx-xxx 5 tively represent the original distribution. Among the sampling-based methods, stratified sampling techniques, such as Latin hypercube sampling (LHS), have higher efficiency and require lower number of scenarios with respect to random sampling [16,17]. Accordingly, this method is adopted in this paper to generate the scenarios for the MPSC-SOPF model. The main feature of LHS technique is to divide the probability space of stochastic variables to some intervals, so-called strata, and sample from these intervals to efficiently cover the probability space of stochastic variables. However, despite the previous LHS-based sampling methods [16,17], the proposed scenario generation approach does not rely on the predetermined probability distributions for the uncertain variables. Instead, the proposed approach uses the relative frequency of the stochastic variables. In this way, the proposed approach can avoid from the errors of approximating the uncertain nature of stochastic variables by predetermined probability distributions. Additionally, the performance of LHS method depends on the selected strata, which should be consistent with the relative frequency of the uncertain variables. The proposed scenario generation approach can directly produce such intervals. On the other hand, the previous LHS-based sampling methods, which fit the data to a predetermined distribution and then partition it, may produce some intervals that are inconsistent with the relative frequency of the uncertain variables. The proposed LHS-based scenario generation method can be summarized as the following algorithm, its flowchart is shown in the Appendix A: 1) Suppose that the number of the uncertain variables is NV. A counter nv is initialized as: nv = 1. Also, let be the number of required scenarios. 2) The probability domain of the uncertain variable nv, i.e. [0,1], is partitioned to n equal intervals. 3) The uncertain variable nv is clustered using k-means clustering technique based on its historical data [18]. The relative frequency of each produced cluster is computed. Using these clusters and their relative frequencies, a discrete cumulative relative frequency distribution is constructed for the uncertain variable nv. 4) Produce random numbers with uniform distribution in the range of [0,1]. Using each produced, probability of the sample associated with kth probability interval, denoted by, is determined as follows: (33) Note that is a random number with uniform distribution in the kth probability interval, i.e.. 5) The intersection point of each with the discrete cumulative relative frequency distribution (constructed in step 3) is determined. Suppose that the intersection point belongs to the hth interval of the distribution. The kth sample of the uncertain variable nv, denoted by, is generated as below: (34) where is a random number with uniform distribution in the range [0,1], separate from. In this way, n samples for the uncertain variable nv are generated such that each sample is randomly generated with uniform distribution within one interval of the probability domain. Accordingly, the proposed scenario generation method can effectively cover the uncertainty spectrum of every uncertain variable. 6) Increment nv: nv = nv + 1. If nv > NV go to the next step. Otherwise go back to step 2. 7) The n samples generated for every uncertain variable are randomly arranged in one column of a matrix, denoted by R, with the dimensions of n NV. Each row of the matrix R presents one scenario produced by the proposed approach for the stochastic program. Totally, n rows or n scenarios are given by R. Due to the correlation between wind farm generations as well as the correlation between wind farm generations and power system loads, mostly because of their dependency on the weather conditions, their associated random variables have statistical correlation. Additionally, loads and wind generations of successive hours have correlation. Ignoring these correlations in the scenario generation process may lead to infeasible generated scenarios [12]. One of the well-known approaches to consider correlation in scenario generation is based on the Cholesky decomposition [19]. However, as discussed in Ref. [19], this method may distort strata made by LHS technique. Moreover, this method cannot effectively reproduce the original distribution of stochastic variables in non-normal distributions. To cope with this problem, an effective technique called rank correlation (having advantages, such as simplicity, independency of distribution, and applicability to any sampling approach) is adopted in the proposed scenario generation method. The proposed LHS-based scenario generation approach, incorporating rank correlation (RC) technique to consider correlation between stochastic variables, denoted by LHS-RC, can be described as the following steps: 1) Let the correlations between the columns of the matrix, generated by the LHS algorithm, is denoted by the matrix. Also, suppose that the matrix illustrates the actual correlation matrix between the uncertain variables. Note that the samples of the uncertain variables, i.e. the columns of the matrix R, are independently generated by the LHS algorithm and so their possible correlations, represented by the matrix A, do not necessarily correspond to the actual correlations included in the matrix B. 2) Let and be the lower triangular matrix of and obtained by Cholesky decomposition, i.e. and. 3) Compute. The columns of the matrix will have the same correlations represented by the matrix (i.e., the actual correlations) [19]. 4) Sort the elements of each column in based on the rank (order) of the elements of the corresponding column in the matrix. It can be mathematically shown that the sorted columns of R will have the same correlations of the matrix B [20]. The scenarios generated by the proposed LHS-RC can effectively represent both the individual stochastic behavior of the uncertain variables (based on the uniform distribution of the samples within the uncertainty spectrum of every uncertain variable) as well as the correlations between them (based on the RC technique). Note that the RC technique does not apply any transformation to the matrix R and only changes the sorting of its columns. The proposed LHS-RC generates scenarios that model the individual and joint stochastic behaviors of W&L uncertainties, i.e. the scenarios. To model the uncertainty source of branch unavailability, the scenarios pertaining to branch contingencies or are generated based on criterion and reduced using the contingency filtering approach of Ref. [9]. This contingency filtering approach finds a subset of important contingencies among the set of all considered contingencies (here, the set ) and filters out the remaining un
6 Electric Power Systems Research xxx (2017) xxx-xxx necessary contingencies. To do this, the performance of the contingency filtering approach can be summarized as the following steps: Step 1) The contingency filtering approach first sets the subset of important contingencies to null set. Step 2) Solve the MPSC-SOPF problem given in Eqs. (1) (32) only considering the subset of important contingencies. Step 3) Check the solution obtained from Step 2 for all considered contingencies. If the obtained solution from Step 2 does not lead to any branch overload for all considered contingencies, the contingency filtering approach is terminated and the obtained solution with the subset of important contingencies is adopted; otherwise go to the next step. Step 4) Among the contingencies that lead to branch overloads, select the contingencies that lead to the highest branch overloads for at least one branch and filter out the other ones. Add the selected contingencies to the subset of important contingencies and go back to Step 2. 4. Simulation results Performance of the proposed MPSC-SOPF model and LHS-RC scenario generation method is evaluated on the IEEE 24-bus test system whose data is presented in Ref. [12]. Two wind farms are also connected to the buses 1 and 2 of the test system. The required data for load demand and wind generation levels as well as their forecasts are obtained from Ref. [21] and scaled down based on the peak load ratios to fit to the IEEE 24-bus test system. One system and two wind power generations have been considered due to available forecast data [21]. Time steps of the scheduling horizon, the number of scenarios and number of contingencies are assumed to be,, and, respectively. The IEEE 24-bus test system has 38 branches. Since the outage of one of these 38 branches leads to islanded network, we have omitted this branch from the contingency set. Thus, the total number of sample scenarios included in the MPSC-SOPF model is. Considering one system load, two wind generations and, there are (1 + 2) 6 = 18 correlated uncertain variables. Thus, the correlation matrices A and B of the proposed LHS-RC for this test case become 18 18. It is noted that the proposed MPSC-SOPF model can be used for longer scheduling horizons, such as. The up and down spinning reserve capacities of conventional units are limited to 10% of their output power ranges, i.e.. The load shedding cost is considered to be [12]. Cost of up/down deployed reserve of each unit is adopted as its offer cost for energy, i.e.. Up and down reserve capacity cost is assumed to be 10% of the offer cost for energy, namely. The LP problem of the proposed MPSC-SOPF model is solved using CPLEX solver within GAMS software package [22]. The run time of LHS-RC and MPSC-SOPF, measured on the simple hardware set of Core i7 2.20 GHz laptop computer with 8 GB RAM memory, is about 3 s and 11 s, respectively, for this test case. These run times are acceptable within a six-hour-ahead decision making framework for the IEEE 24-bus test system. Three solutions have been considered in the proposed framework to decrease its computation burden: 1) The proficient method of LHS has been used to generate scenarios [16,17]. 2) The generated scenarios are decreased to the desired number using a Backward/Forward process available as the SCENRED tool in GAMS [22] to reduce the problem size. 3) An efficient solution approach based on a non-dominated contingency filtering concept [9] is implemented to decrease the problem size and solve the proposed model in an acceptable time. To show the effectiveness of the proposed model, the following experimental test cases are considered: Case 0 All uncertainty sources are ignored (i.e. deterministic counterpart). Case 1 Only branch availability uncertainty is included. Case 2 Only load demand uncertainty is considered. Case 3 Only wind generation uncertainty is included. Case 4 Wind generation and load demand uncertainty sources are taken into account. Case 5 All three uncertainty sources, i.e. wind generation, load demand, and branch availability, are considered. The results obtained for the objective function, i.e. total cost, and sum of all procured reserve volume for h, i.e., in these experimental test cases are shown in Table 1. It can be seen that considering the uncertainty sources increases the operation costs due to changing the operating point and additional reserves required. Comparison of the objective function values demonstrates an additional incurred cost of 10.55% in case 5, indicated by bold font in Table 1, with respect to case 0. One of the key advantages of the proposed model can be inferred from the procured reserve capacity reported in case 5 in which all three mentioned uncertainty sources are taken into account. If the reserve requirements are determined by three separate optimization process of cases 1 3, the total reserve requirement will be the sum of 3.285 + 5.431 + 7.651 = 16.376 p.u. Note that each of case 1, case 2, and case 3 deal with a different uncertainty source including branch availability uncertainty in case 1, load demand uncertainty in case 2, and wind generation uncertainty in case 3. Each of these uncertainty sources requires its own reserve. In other words, we may encounter some load forecast error and some wind generation forecast error, and at the same time some branches of the system become unavailable due to contingencies. Thus, we may need the sum of reserves determined for case 1, case 2, and case 3, simultaneously. This is what we typically encounter in real-world power systems. This reserve requirement is considered as a constraint in the optimization problem. In other words, the total reserve requirement of 16.376 p.u. should be provided. This is significantly higher than the procured reserve by the Table 1 Results obtained in different experimental test cases. Test case Objective function ($) Procured reserve (p.u.) Case 0 83545.49 Case 1 85991.97 3.285 Case 2 85496.37 5.431 Case 3 85647.03 7.651 Case 4 89634.37 11.822 Case 5 92355.56 12.810
Electric Power Systems Research xxx (2017) xxx-xxx 7 proposed MPSC-SOPF, i.e. case 5 (12.810 p.u.), which simultaneously optimizes all reserves required for coping with the uncertainty sources. As an alternative approach, if the reserves required for treating the W&L uncertainty sources are determined simultaneously by an optimization process, as given in case 4, and then added to the branch failure reserve requirement, obtained in case 1, the total procured reserve will decrease to 3.285 + 11.822 = 15.107 p.u., which is still considerably more than the required reserve in case 5, i.e. 12.810 p.u. Apart from the procured reserve capacity, the proposed MPSC-SOPF approach determines a more optimal operating point compared to the other approaches that rely on predetermined reserve requirements in their formulation or separately handle different uncertainty sources. This issue is discussed in the following. Fig. 1 shows the scheduled generation as well as preventive/corrective actions determined by the proposed MPSC-SOPF for the first hour of the scheduling period in case 5. Preventive actions include applied generation shifts compared to case 0 (without uncertainty) and corrective actions consist of procured reserve capacities. All of these control variables are simultaneously optimized within the proposed approach considering the uncertainty sources. In order to show the effectiveness of the proposed MPSC-SOPF model for determining optimal operating point and reserve requirements considering the uncertainty sources of power system, it is compared with four other deterministic models and one other stochastic model. Unlike the proposed stochastic method, deterministic methods cannot model uncertainty sources and determine the required reserves based on them. Thus, these methods should have a criterion to determine the required reserve. Handling W&L uncertainty sources in deterministic approaches is often based on the forecast error standard deviation [23]. Higher forecast error standard deviation means higher dispersion of the forecast error probability density function and so higher forecast errors may be occurred in practice. Thus, more reserve is required for higher forecast error standard deviation. In this paper, to implement these four deterministic reserve procurement methods, the standard deviation of the forecast error pertaining to net load, i.e. total load demand minus sum of the wind generations, is computed and denoted by. Then, the four deterministic approaches, denoted by DA1, DA2, DA3, and DA4, are constructed adopting,,, and as W&L spinning reserve capacity, respectively. Among these four methods, DA1 illustrates the least conservative deterministic method and DA4 illustrates the most conservative deterministic method. In this way, we can compare the proposed stochastic method with different deterministic methods with different levels of conservativeness. The comparative stochastic approach (SA) is implemented based on Ref. [24]. This SA can only model W&L uncertainty. Thus, the contingency reserve capacity requirement for this SA as well as DA1, DA2, DA3, and DA4 is determined using corrective SCOPF [9] such that the system remains secure for all single branch contingencies. Total reserve capacity of SA as well as DA1, DA2, DA3, and DA4 is obtained as the sum of their W&L and contingency reserve requirements. The objective function value and total reserve capacities obtained by DA1, DA2, DA3, DA4, SA, and proposed MPSC-SOPF are shown in Table 2. From this table it is seen that while the procured reserve of the proposed approach, indicated by bold font in Table 2, is lower than DA2, DA3, and DA4 as well as SA, its objective function value is higher than all methods. The reason can be explained as follows. The deterministic methods only consider the cost of first stage decision variables, i.e.,, and. In other words, the objective function of these methods only includes the first summation term of Eq. (1). The objective function of SA also does not consider the cost of second stage decision variables associated with the post-contingent states. On the other hand, the proposed non-deterministic MPSC-SOPF considers the cost of both the first and second stage decision variables, i.e. the first and second summation terms of Eq. (1). However, the objective function values reported in Table 2 are related to before-the-fact' conditions, i.e., when the uncertain variables have not been realized. As the real system conditions are uncertain, to have a more realistic evaluation for the effectiveness of different approaches, their performance for after-the-fact conditions (i.e. when the uncertainty sources are realized) should be evaluated in a sufficiently long run. This evaluation is implemented here through an out-of-sample analysis. In the before-the-fact analysis reported in Table 2 both the first and second stage decision variables are optimized, while only the second stage decision variables are optimized in the after-the-fact or out-of-sample analysis (the first stage decisions have been fixed). A large number of trial scenarios (here, 10,000) are generated by LHS-RC. These trial scenarios are different from the 296 sample scenarios produced by LHS-RC in the previous section and used to implement the MPSC-SOPF. In other words, the trial scenarios of the after-the-fact or out-of-sample analysis are unseen for MPSC-SOPF. Thus, the performance of the proposed model to optimize the first stage decisions is evaluated by unseen scenarios, which gives a better assessment of the model s performance. Additionally, by means of a large number of trial scenarios generated by LHS-RC, the out-of-sample analysis can simulate the individual and joint stochastic behaviors of the uncertain variables in a long run. Fig. 1. Scheduled generation and preventive/corrective actions in t = 1, obtained by the proposed MPSC-SOPF (case 5).
8 Electric Power Systems Research xxx (2017) xxx-xxx Table 2 Results of four deterministic models, one stochastic model, and proposed MPSC-SOPF model. Approach Objective function ($) Procured reserve (p.u.) DA1 84655.25 9.455 DA2 85308.18 13.180 DA3 86205.17 16.905 DA4 88152.92 20.630 SA 89929.88 15.406 MPSC-SOPF 92355.56 12.810 For each trial scenario, the following optimization problem should be solved in the out-of-sample analysis: Subject to (36) (37) (38) (39) (40) (41) (42) (43) (45) The first summation term in Eq. (35) is a constant term as the first stage decision variables have already been fixed. It is only included in Eq. (35) to compute the total cost of each trial scenario s. However, the second and third summation terms include the trial scenario dependent decision variables of,,,, and that should be optimized to minimize the cost of the trial scenario s, i.e. given in Eq. (35). Here, /, /, and represent the deployed up/down reserves, up/down generation shifts, and load sheds, respectively, employed to cope with the realization of the uncertain variables (i.e. deviation of W&L from their forecasts and occurred contingencies) in trial scenario s. The up/down generation shifts of / are procured for security purposes [25]. After solving (35) (46) for all trial scenarios (i.e. aggregated cost, denoted by AC, is obtained as follows: (46) ), their The AC results of DA1, DA2, DA3, DA4, SA and proposed MPSC-SOPF (bold font) for different costs of the up/down generation shifts (i.e., when and equal 2 4 times of ) are shown in Table 3. It is seen that SA by considering W&L uncertainty leads to lower AC results than DA1-DA4 since it more effectively determines W&L reserves. Table 3 shows that the proposed MPSC-SOPF obtains lower AC results than all other deterministic and stochastic methods for all values of the generation shifts costs. Moreover, the highest AC result of MPSC-SOPF (obtained for generation shifts' costs equal ) is lower than the lowest AC result of DA1 DA4 and SA (obtained for as the generation shifts costs). These comparisons illustrate higher effectiveness of MPSC-SOPF compared to the deterministic and stochastic approaches to cope with various realizations of the uncertainty sources in a long run. In other words, in after-the-fact conditions, when the uncertain variables are realized, the proposed MPSC-SOPF performs better than DA1 to DA4 as well as SA, since MPSC-SOPF simultaneously optimizes all W&L and contingency reserves. To evaluate the convergence of the out-of-sample analysis, the convergence coefficient is used [26]: where and represent the standard deviation of the AC results and number of trial scenarios, respectively. The results obtained Table 3 The results of out-of-sample analysis (AC results). Approach / DA1 96486.62 99119.54 100329.9 DA2 96155.28 97546.02 97956.58 DA3 95958.71 96461.34 96561.25 DA4 96569.97 96674.68 96693.35 SA 95432.84 96242.111 96477.763 MPSC-SOPF 92702.89 92920.35 93101.51 (47) (48)
Electric Power Systems Research xxx (2017) xxx-xxx 9 for in all cases of Table 3 are well below 0.01 indicating acceptable convergence of the out-of-sample analysis [26]. The set of selected scenarios is effective on the performance of a stochastic model. Usually, by increasing the number of scenarios, we can more accurately model the stochastic problem, since we will have more realizations of the uncertain variables, but at the expense of higher computation burden. To study this aspect, a sensitivity analysis is performed on the test system, which evaluates the impact of the number of scenarios on the performance of the MPSC-SOPF model. The results of this sensitivity analysis are shown in Table 4. In this sensitivity analysis, is changed from 3 to 12. For each W&L scenario, all non-islanding single-contingencies by the number of are considered. Thus, the number of sample scenarios, produced by the proposed method of Section 3 and included in the MPSC-SOPF model, is changed from to in this sensitivity analysis. Similarly, all of these scenarios are different from the 10,000 trial scenarios of the out-of-sample analysis. For each number of sample scenarios, the computation time, before-the-fact results (i.e. the objective function value), and after-the-fact results (i.e. the AC results for 2 4 times of as the cost of generation shift) obtained by the MPSC-SOPF model are shown. Table 4 shows that by increasing the number of scenarios, the computation time of the MPSC-SOPF model rises, since the number of second stage variables and constraints increases. Before-the-fact/after-the-fact costs increases/decreases by increasing the number of scenarios, since a more accurate evaluation of the total cost is obtained by the MPSC-SOPF model when more realizations of the uncertain variables are considered. In other words, the before-the-fact and after-the-fact costs goes toward each other, when the stochastic model becomes more accurate. For instance, the difference between the after-the-fact cost with and the before-the-fact cost reaches from 93111.91 90112.2 = 2999/71 $ for to 92689.05 92362.85 = 326.2 $ for, i.e. about ten times decreases. Additionally, it is seen that after the before-the-fact and after-the-fact costs do not change significantly, while the computation time rises rapidly. For this reason, indicated by bold font in Table 4, which corresponds to 296 sample scenarios, is considered in the other numerical experiments of this paper. In Tables 5 and 6, the effects of considering correlation among the uncertain variables (here, 18 correlated uncertain variables) are shown. The scenarios without/with considering correlation are generated by LHS and LHS-RC, respectively. Table 5 shows that without considering correlation, both the required reserve and objective function value decreases in the before-the-fact conditions. However, in the after-the-fact conditions, considering correlation leads to lower AC results for all values of up/down generation shifts costs as shown in Table 6. In other words, considering correlation leads to a more realistic evaluation of the after-the-fact conditions. This in turn leads to a lower incurred cost encountering various realizations of the uncertain variables. In Figs. 2 and 3, scenarios generated by LHS and LHS-RC (i.e. without/with considering correlation among the uncertain vari Table 4 Sensitivity analysis with respect to the number of scenarios. ables) for system load are shown (in the figure legends Sn stands for scenario). While LHS-RC generates scenarios considering correlations among loads of different hours, LHS ignores these inter-temporal correlations. In Figs. 2 and 3, is taken into account, the same as the previous numerical experiments. In these figures, the forecast and actual loads, denoted by Forecast and Actual, are shown by thick red and thick blue lines, respectively, in addition to the generated scenarios. The actual load is not available when the applied scenarios are generated to simulate the likely realizations of the upcoming load level. The actual load has been shown in Figs. 2 and 3 to compare it with the generated scenarios using LHS and display the quality of the generated scenarios. The eight scenarios of Fig. 3 have a more smooth behavior compared to the eight scenarios of Fig. 2, due to considering inter-temporal correlation by LHS-RC for scenario generation. This avoids from sudden changes. Thus, the generated scenarios by considering correlation better simulate the load behavior (i.e. Sn 1 8 in Fig. 3 better follow Actual load pattern compared to Sn 1 8 in Fig. 2), since the actual system load has inertia and cannot change suddenly. Additionally, Sn 1 8 in Fig. 3 better cover the Actual load compared to Sn 1 8 in Fig. 2. Accordingly, the scenarios generated with considering correlation can more effectively capture the uncertain behavior of load compared to the scenarios generated without considering correlation. Similar results have been obtained for the two other uncertain variables of this test case, i.e. generations of the wind farms 1 and 2. 5. Conclusion To cope with the uncertainty sources of wind farm generations, power system load and transmission equipment unavailability in an optimal power flow framework, a new MPSC-SOPF model as well as LHS-RC scenario generation approach has been proposed. The proposed MPSC-SOPF model simultaneously optimizes preventive and corrective actions considering the uncertainty sources. The LHS-RC scenario generation method produces scenarios taking into account correlation among wind farm generations, and wind generations and system load as well as inter-temporal correlations. The comparative results of the proposed MPSC-SOPF model with other deterministic models show that the proposed approach may obtain higher before-the-fact costs than other deterministic models. Its reason is that deterministic methods only consider the cost of the first stage decision variables, while the proposed MPSC-SOPF considers the cost of both the first and second stage decision variables. However, MPSC-SOPF leads to lower after-the-fact operation costs considering various realizations of the uncertain variables in a long run. In other words, in after-the-fact conditions, when the uncertain variables are realized, the proposed approach performs better than deterministic models as well as another stochastic model, since the proposed MPSC-SOPF simultaneously optimizes all W&L and contingency reserves. Additionally, considering correlation among the uncertain variables may result in higher before-the-fact costs and procured reserves. However, it can improve the after-the-fact performance by more realistically modeling the uncertain behaviors of loads and wind generations. The pro 3 4 5 6 7 8 9 10 11 12 Time (s) 5.1 5.8 6.8 8.1 9.6 11.2 13.6 16.5 20.2 25 Before-the-fact 90112.2 91832.71 92188.27 92324.33 92341.69 92355.56 92358.91 92360.08 92361.55 92362.85 After-the-fact 93111.91 93004.22 92819.46 92739.19 92710.03 92702.89 92698.48 92694.11 92691.62 92689.05 93633.04 93243.97 93095.08 92966.11 92938.27 92920.35 92912.3 92906.09 92903.61 92900.75 94120.16 93581.01 93238.72 93197.92 93143.05 93101.51 93084.68 93071.73 93064.25 93058.84
10 Electric Power Systems Research xxx (2017) xxx-xxx Table 5 Results of before-the-fact conditions without/with considering correlation (results of with correlation are indicated by bold font). Scenario generation Objective function ($) Procured reserve(p.u.) Without correlation 91237.94 10.06 With correlation 92355.56 12.81 Table 6 AC results without/with considering correlation for after-the-fact conditions (results of with correlation are indicated by bold font). Scenario generation / Without correlation 92739.31 93422.54 93942.28 With correlation 92702.89 92920.35 93101.51 Fig. 2. Eight scenarios generated by LHS (without considering correlation) with T = 6. (For interpretation of the references to color in the text, the reader is referred to the web version of this article.) Fig. 3. Eight scenarios generated by LHS-RC (with considering correlation) with T = 6. (For interpretation of the references to color in the text, the reader is referred to the web version of this article.) posed MPSC-SOPF model focuses on the optimization of the expected value of the operation cost. However, if the operators of a power system tend to hedge operation costs against the worst case scenarios, other methods such as robust optimization can be proposed. Additionally, the current study considers the contingencies and their impacts. Because of increasing pressure on transmission network in the some current power systems leading to the risk of instability problems, considering stability constraints such as voltage stability constraints can be also proposed. Both the mentioned challenges can be considered as the subjects of future works. Appendix A.
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