Probabilistic Locational Marginal Price by Monte Carlo Simulation with Latin Hypercube Sampling and Scenario Reduction Techniques

Nội dung text: Probabilistic Locational Marginal Price by Monte Carlo Simulation with Latin Hypercube Sampling and Scenario Reduction Techniques

1. JST: Smart Systems and Devices Volume 31, Issue 1, May 2021, 041-049 Probabilistic Locational Marginal Price by Monte Carlo Simulation with Latin Hypercube Sampling and Scenario Reduction Techniques Pham Nang Van Department of Electric Power Systems, Hanoi University of Science and Technology, Hanoi, Vietnam Email: van.phamnang@hust.edu.vn Abstract Leveraging Monte Carlo Simulation (MCS) combined with simple random sampling (SRS) to evaluate probabilistic locational marginal price (P-LMP) requires long computation time and large computer storage. The paper proposes the joint usage of Latin hypercube sampling (LHS) with sample reduction techniques called the fast forward selection (FFS) algorithm into Monte Carlo simulation for calculation of the P-LMP. This fast forward selection algorithm is needed to cut down the number of samples while keeping most of the stochastic information embedded in such samples. The LHS-FFS-based P-LMP is investigated using IEEE 6-bus and 24-bus systems. This method is compared with SRS and LHS only. The LHS-FFS approach is found to be efficient and flexible; therefore, it has the potential to be applied in many power system probabilistic problems. Keywords: Latin hypercube sampling (LHS), Monte Carlo simulation (MCS), probabilistic locational marginal price (P-LMP), fast forward selection (FFS) algorithm, uncertainty. 1. Introduction being integrated with higher and higher amounts into electric energy systems throughout the world [1]. Currently, many*countries around the world, However, wind power is also an intermittent source. including Vietnam, have been operating wholesale The integration of a significant amount of wind electricity markets. In the wholesale electricity power into a power system results in critical market, the market participants are generation operational challenges, which, in turn, originate companies (GENCOS) and distribution companies alternations in locational marginal prices (LMPs) [2]. (DISCOS). The market operator collects generating offers by producers, load bids by consumers and LMP is the additional cost when the load clears the market by maximizing social welfare. To increases at a specific node. The LMP-based make payments in the electricity market, locational approach has been dominant to determine electricity marginal price (LMP) is calculated. The difference in prices and manage transmission congestion in power LMPs between two nodes of a branch is due to markets. Locational marginal prices may encompass congestion and losses on that branch. The locational three components: marginal energy price, marginal marginal pricing methodology is widely used in congestion price, and marginal loss price. The electricity markets to determine the electricity prices optimal power flow (OPF) has been applied in the and evaluate the transmission congestion cost. power industry to calculate LMP [3]. Furthermore, the issue of climate change has The uncertainty and variability of wind required the pressing need for limiting industrial generation could lead to LMP variations [4]. emissions of greenhouse gases. In addition to the Evaluating the impacts of the uncertain parameters on tragic consequences of climate change, there is an LMP is of most importance in power system planning energy crisis in many countries in the world due to and operation. Different techniques, such as the the depletion of fossil fuels. Therefore, renewable probabilistic approach [5-6], possibilistic method [7], energy has been prominent in most industrialized hybrid possibilistic-probabilistic strategies [8], countries with the aim of decarbonizing in the information gap decision theory [9], and robust electricity sector as well as meeting the rising demand optimization [4], have been developed to deal with for energy and safeguarding the security of the energy uncertainties. Among these methods, probabilistic supply. Solar energy, wind, geothermal, biomass, techniques are more appropriate for the impact waves and hydrogen energy are major renewables. assessment of renewable energy sources [10]. Wind power is an economically attractive renewable Different approaches have been introduced to for producing electricity. Therefore, wind power is determine the probabilistic locational marginal price (P-LMP) from solving the probabilistic optimal ISSN: 2734-9373 power flow (P-OPF). In Monte Carlo (MC) based techniques, samples from random input variables are Received: 28 September 2018; accepted: 21 February 2021 41
2. JST: Smart Systems and Devices Volume 31, Issue 1, May 2021, 041-049 generated, and then the deterministic problem is each interval and is forced to recreate the input solved for each sample [6]. Simple random sampling probability distribution. With Latin hypercube, a (SRS) is one of the most popular MC techniques in sample is drawn from each interval; therefore, the which samples are randomly generated from input samples more accurately reflect the distribution of variables' distribution functions. While the SRS can values in the input probability distribution. provide highly accurate results, it suffers from the It is assumed that there are the K input random drawback of heavy computation time and the high variables in a probabilistic formulation, including X1, storage required for many repeated calculations. The X2, , XK. The cumulative distribution function of Xk P-OPF problem can be solved by employing the Latin is expressed as follows: hypercube sampling (LHS) approach [6]. Although the number of Latin hypercube samples is reduced Yk= FX kk( ) ∀= k1,2, , K compared with the simple random samples, the number of these samples is generally large to If the sample size is N, then the range of Yk is represent the uncertainty involved accurately. This separated into N intervals that are not overlapping. may render the P-OPF problem intractable for Each interval has a length of 1/N. One sampling value electricity market operation [11]. As a result, a of Yk is selected from the midpoint or random point of scenario reduction is applied to decrease an initial each interval. After that, the inverse function of the LHS sample size. There are several methods available cumulative distribution function is employed to to reduce scenarios to be used in the P-OPF problem determine the sampling values of Xk. [12]. These methods seek to obtain a reduced number As illustrated in Fig. 1, if the midpoint of each of scenarios that best retain the essential features of a interval is adopted, the nth sample of Xk can be given original scenario set according to a probability calculated as follows: distance. In [13], a scenario reduction algorithm based on submodular function optimization is −1 n − 0.5 xFkn= k ∀=k1,2, , K employed to optimize the number of scenarios and N rank these scenarios. Then, a row of the sampling matrix In this paper, P-LMP is calculated using MC with LHS combined with scenario reduction [xxk1 kn x kN ] is formed from the sample values of techniques, namely Fast Forward Selection (FFS) Xk. When all the K input random variables are Algorithm. This paper's main contribution is that P- sampled, a sampling matrix X (K ì N) can be LMP determined using three different approaches, constructed. particularly simple random sampling, Latin In this paper, Latin hypercube samples are hypercube sampling, and Latin hypercube sampling generated using MATLAB software [14]. combined scenario reduction techniques, are compared and analyzed. The paper's remainder is presented as follows: Section 2 presents Latin hypercube sampling; Section 3 presents the FFS sample reduction technique. The LMP calculation method based on the ACOPF market-clearing model is presented in Section 4. Section 5 describes the MC simulation procedure for calculating the probabilistic locational marginal price. The calculation examples and the comparisons of different P-LMP approaches are presented in Section 6. The conclusion is given in Section 7. 2. Latin Hypercube Sampling Latin hypercube sampling is a recent development in sampling technology designed to Fig. 1. Latin hypercube sampling technology accurately reflect the input distribution through 3. Sample reduction technique sampling in fewer iterations when compared with the simple random sampling method. The key to Latin 3.1 Algorithm hypercube sampling is the stratification of the input probability distributions [5]. Stratification divides the A scenario reduction methodology seeks to cumulative curve into equal intervals on the downsize a scenario set while still keeping as intact cumulative probability scale (0 to 1.0). A sample is as possible the stochastic information embedded in it. then randomly taken from each interval of the input Next, we briefly explain a scenario reduction distribution. Sampling is forced to represent values in procedure, which relies on the concept of probability 42
3. JST: Smart Systems and Devices Volume 31, Issue 1, May 2021, 041-049 distance, namely the Fast Forward Selection (FFS) Choose ω1 ∈ arg min dω Algorithm. ω∈Ω Update the set ΩJ ←Ω\ ω1 Step n Compute ν[n] ( ωω,') where ν[n] ( ωω,') = [nn−−11] [ ] [n−1] =minν( ωω ,',) ν( ωω ,nJ−1 ) ∀ ωω ,' ∈Ω [nn] [ ] [ n−1] = n−1 πν ωω∀ ω ∈Ω dωω∑ωω∈Ω[ ] ' ( ', ) J J \ n Choose ωn ∈ arg min dω [nn] [ −1] Update the set ΩJ ←Ω Jn\ ω N +1 Step S N *ΩJ Ω=ΩJJ;\ Ω=ΩΩ S J * ππωω= + ∑ π ω' ωω'∈J ( ) Fig. 2. Flow diagram of the Fast Forward Selection where Algorithm to reduce scenario * Jj(ωω) =' ∈ΩJ ω = ( ω') such that The FFS algorithm is an iterative greedy process j ω'= arg min νω ", ω ' ( ) * ( ) starting with an empty set. In each iteration, from the ω "∈ΩS set of non-selected scenarios, the scenario which * * minimizes the Kantorovich distance between the where ΩJ is the final set of deleted scenarios and ΩS reduced and original sets is selected. Then, this is the set of selected scenarios after the scenario- scenario is included in the reduced set. The algorithm reduction process. stops if either a specified number of scenarios or a certain Kantorovich distance is attained. 3.2 Illustrative Example The Kantorovich distance can be equivalently It is assumed that the four power scenarios determined as: Pω ,ω = 1, ,4 with associated probabilities πω can statistically represent the generating output of a 100 DK ( QQ,') = ∑ πω min ν( ωω , ') (1) ω '∈ΩS MW wind farm in a given period. ω∈Ω/ ΩS Table 1. Four power scenarios of illustrative example where ν( ωω,') is a cost function, which is the vector Scenario 1 2 3 4 distance between scenario ω and ω'; Q and Q' are the probability distributions in the initial scenario set Ω Pω (MW) 5 40 60 100 and selected scenario set ΩS, respectively; πω is the π probability of scenario ω. ω 0.25 0.2 0.2 0.35 FFS algorithm is depicted in Fig. 2. A step-by- Suppose that the number of wind power step explanation of the algorithm [15]-[16] is also scenarios being reduced is equal to 2. The fast provided in the following: forward selection algorithm works as follows: Step 0 Step 0: Calculate the cost function [1] Compute cost function ν( ωω,') for each pair WW ν( ωω, ') = ||PPωω −' ||, ∀ωω , ' ∈Ω scenarios ω and ω '. where Ω={1,2,3,4} Step 1 NΩ The values of function ν can be cleverly [1] Compute dωω=∑πν( ωω,') ∀ ω ∈Ω organized into a symmetric matrix whose diagonal ω =1 elements equal zero: 43
4. JST: Smart Systems and Devices Volume 31, Issue 1, May 2021, 041-049 0 35 55 95 Given that:  35 0 20 60 Ω* ν = MW Scenario 3 in S is the closest one to scenario 1  55 20 0 40 Ω* ν = ν =  in J ( (1, 3) 55 while (1, 4) 95 ) and 95 60 40 0 Scenario 3 in Ω* is the closest one to scenario 2 Step 1: Scenario ω that minimizes the resulting S Ω* ν = ν = Kantorovich distance between the reduced and in J ( (2,3) 20 while (2, 4) 60 ). original sets is chosen: It follows: d12=πν(1, 2) ++ πν 3 (1, 3) πν 4 (1, 4) = 51, 25 MW * π3=++= πππ 3210,65 d21=++=πν(2,1) πν 3 (2,3) πν 4 (2, 4) 33,75 MW * ππ44= = 0,35 =++=πν πν πν d31(3,1) 2 (3,2) 4 (3,4) 31,75 MW * To sum up, a reduced scenario set Ω=S {3, 4} d41=++=πν(4,1) πν 2 (4, 2) πν 3 (4,3) 43,75 MW * * with associated probabilitiesπ 3 = 0,65 and π 4 = 0,35 Therefore, is provided. Ω=[1] {3,} 4. Market-Clearing Model S Ω=[1] {1,2,4} The clearing of an electricity market involves J two primary tasks, namely, determining the Step 2: The cost matrix is updated as follows: production (consumption) level of every producer (consumer) and settling the locational marginal price [2] ν(1, 2)= min{ νν (1, 2), (1, 3)} = 35 MW (LMP) at which every producer (consumer) is paid ν[2] (1, 4)= min{ νν (1, 4), (1, 3)} = 55 MW (charged) for its energy production (consumption). In this paper, we assume that an independent system [2] ν(2,1)= min{ νν (2,1), (2,3)} = 20 MW operator (ISO) is in charge of conducting these tasks ν[2] (2, 4)= min{ νν (2, 4), (2,3)} = 20 MW by solving the following problem: ν[2] (4,1)= min{ νν (4,1), (4,3)} = 40 MW 4.1 Objective Function ν[2] (4, 2)= min{ νν (4, 2), (4,3)} = 40 MW The objective of the market-clearing problem is to maximize the total social welfare (SW), as shown Consequently, in Equation (2) below: N 0 35 55 55 ND Dj NNG Gi  MaximizeSW= ∑∑λλDjk .PP Djk − ∑∑ Gib. Gib (2) 20 0 20 20 jk=11 = ib=11 = ν [2] = MW 55 20 0 40  where λGib is the price of the energy block b offered 40 40 40 0 by generating unit i, PGib is the power of the energy block b offered by generating unit i, λ is the price Considering the new cost matrix ν [2] , the Djk [1] of the energy block k bid by demand j, PDjk is the scenario ω selected from ΩJ is the one that power block k bid by demand j. minimizes the Kantorovich distance between the [2] subsequent reduced set ΩS and the original set Ω : 4.2 Constraints [2] [2] [2] 4.2.1 Power balance d12=+=πν(2,1) πν 4 (4,1) 18 MW [2] [2] [2] The active power and reactive power injected d =+=πν(1,2) πν (4,2) 22,75 MW 21 4 into bus i is subjected to the following power flow [2] [2] [2] d41=+=πν(1,4) πν 2 (2,4) 17,75 MW equations: Hence, PPi=+− Gi P Wi P Di n Ω[2] =Ω= *  SS{3, 4} = Ui UG j( ijcosδδ ij+ B ij sin ij ) ∑ [2] * j=1 ΩJJ =Ω={1, 2} (3) QQi=+− Gi Q Wi Q Di Step 3: The scenario reduction algorithm stops n =  δδ− with the optimal transfer of probabilities from the set Ui∑ UG j( ijsin ij B ij cos ij ) = * * j 1 of non-selected scenarios ΩJ to selected ones ΩS . 44
5. JST: Smart Systems and Devices Volume 31, Issue 1, May 2021, 041-049 i where PQGi, Gi are real and reactive power generated ∆P GSF = k (9) by the conventional generators at bus i, respectively; ki− ∆Pi PQWi, Wi are real and reactive power generated by the The GSFk-i above depends solely on the wind turbine at bus i, respectively; PDi, QDi are the active and reactive power of demand at node i, structure of electrical networks; therefore, these   factors can be calculated offline using sparse matrix respectively; Ui and U j are the magnitudes of techniques. However, the GSFk-i depends on the voltage at node i and j, respectively; Gij and Bij are location of the voltage reference node. the real and imaginary part of element ij of the Note that wind production is treated as a admittance matrix, respectively; δij is the voltage negative demand that can be spilled. This is the case angle difference between nodes i and j; n is the in which wind power is treated in most energy number of buses in power systems. systems worldwide. Moreover, this is equivalent to assuming that wind producers offer their energy 4.2.2 The active power limit of each generation production at zero price and are not penalized in the block of conventional producer real-time market for their energy imbalances. max 0,≤≤PGib P Gib ( ∀ ib) (4) 4.3 LMP Calculation and Components where Pmax is the MW size of block b offered by The locational marginal price (LMP) of Gib electricity at a location is defined as the least cost to generating unit i. supply the next incremental of demand at that 4.2.3 Power limit of the generating units location consistent with all power system operating constraints. The active power LMP at each bus i is For a generating unit, its active power is simply the Lagrange multiplier related to that bus's subjected to the constraint (5), as follows: real power balance constraint. min max PGi≤≤ PP Gi Gi ( ∀ i) The locational marginal price consists of the (5) max following components [4]: 0 ≤≤PPWq Wq ( ∀ q) LMPi=−+ LMP E LF i LMP E ∑ GSFki− à k (10) 4.2.4 Limits on the price-sensitive loads k In a wholesale power market, the loads are where LMPE is the marginal energy price, LFi is the considered to consist of two components: fixed load loss factor for node i, μk is the shadow price of and price-sensitive load. The demand curve of price- transmission constraint on the kth line. sensitive loads can consist of several blocks, each with a lower and an upper limit, as shown in (6)-(7). The loss factor can be computed as follows: E min E E max M 2 PDj≤≤ PP Dj Dj ( ∀ j) (6) Ploss = ∑ PRkk (11) k =1 E E max 0≤≤PPDjk Djk ( ∀ j,k) (7) M ∂Ploss ∂ 2 E E max = PR (12) where P is elastic power of demand j and P is ∑ kk Dj Djk ∂∂PPiik =1 the MW size of block k bid by demand j. nn 4.2.5 Branch flow limits Pk =∑∑ GSFk−− i( P Gi +− P Wi P Di ) = GSFk i ì P i (13) ii=11= The branch flow can be expressed by a function of injected active power via the power distribution where Rk is the resistance of line k, Ploss is the total factors [3]. loss of the power system, and M is the number of lines. n max max −Pk ≤= P k ∑ GSFk− i( P Gi + P Wi − P Di) ≤ P k (8) Equation (12) can be expanded further as i=1 follows: where GSFk-i is the sensitivity of branch power flow k M ∂Ploss ∂ 2 max = PR with respect to injected power i and P is the power ∑ ( kk) k ∂∂PPiik =1 flow limit on branch k. M ∂P k =∑ RPkk ìì2 Sensitivity of branch power flow k with respect k =1 ∂Pi to injected power i is defined as a power flow Mn increase on the kth line when the power injected in ith =∑∑2 ììRk GSF k−− i ì GSFk i( P Gi+ P Wi − P Di ) node increases by 1 MW and is computed as follows: ki=11= (14) 45
6. JST: Smart Systems and Devices Volume 31, Issue 1, May 2021, 041-049 The market-clearing model (2)-(8) is a nonlinear 00v < v or v v used as inputs for the market-clearing model  out in  vv− described in Section 4. P= Pin v≤≤ vv (15) W Wr in r - Solve the market-clearing model for each Latin  vvr− in P v≤≤ vv hypercube sample by leveraging  Wr r out POWERWORLD software [22] and obtain the Locational Marginal Prices. where PWr is the rated power of wind turbine, v is wind speed, vin is cut-in wind speed, vout is cut-out - Perform statistical analysis to obtain the wind speed, v is the rated wind speed. statistical property of the P-LMP, such as PDF r and mean value, after solving N problems for The probability density function (PDF) of the market-clearing. wind speed can be described accurately by the The Monte Carlo simulation will stop when a Weibull distribution [21]: predefined convergence threshold ε has been reached. The stopping criterion is mathematically shown as follows: 46
7. JST: Smart Systems and Devices Volume 31, Issue 1, May 2021, 041-049 Table 1. Line data for 6-bus system σ E( LMP) ≤ ε (17) E( LMP) From To R (pu) X (pu) B (pu) Rate (MVA) where E() LMP is the mean value of LMP, σ is the 1 2 0.1 0.2 0.04 100 variance of a random variable. The mean value is 1 4 0.05 0.2 0.04 100 determined as the following expressions: 1 5 0.08 0.3 0.06 100 N 2 3 0.05 0.25 0.06 60 ∑ LMPk k =1 E( LMP) = 2 4 0.05 0.1 0.02 60 N where N is the number of samples. 2 5 0.1 0.3 0.04 60 6. Case Studies 2 6 0.07 0.2 0.05 60 A series of P-OPF studies with SRS, LHS, and 3 5 0.12 0.26 0.05 60 LHS-FFS are carried out on 6-bus and IEEE 24-bus test systems, respectively. 3 6 0.02 0.1 0.02 60 6.1 IEEE 6-Bus System 4 5 0.2 0.4 0.08 60 A six-bus system is considered, which is shown 5 6 0.1 0.3 0.06 60 in Fig.3. The branch data and bus data are presented in Table 1 and Table 2, respectively. Generator offers Table 2. Bus data for 6-bus system are provided in Table 3. max min PD QD PG PG To parameters of the system as per unit, the Number Type power base of the six-bus system is set at 100 MVA. (MW) (MVAr) (MW) (MW) The voltage base of this system is set at 230 kV. 1 3 200 50 A wind farm is assumed to have been constructed at bus 6. This wind farm consists of 2.5 2 2 150 37.5 MW wind generators, model Nordex N80/2500 with 3 2 180 45 a hub height of 105 m. The power curve of this turbine model is supplied by the Danish Wind 4 1 100 15 Industrial Association [21]. The Weibull distribution with shape parameter (k), and scale parameter (c) 5 1 100 15 equal to 1.6 and 9.7, respectively, is deployed to 6 1 100 15 model wind speed. Type 3 = swing bus, type 2 = generator bus, type 1 = load bus Fig. 3. Six-bus system topology Fig. 4. 24-bus power system 47
8. JST: Smart Systems and Devices Volume 31, Issue 1, May 2021, 041-049 Table 3. Generator offers (VND/kWh) for 6-bus system Generator 1 Generator 2 Generator 3 Block 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 Power 50 50 50 30 20 37.5 30 30 20 32.5 45 40 30 25 40 (MW) Price 500 800 1200 1500 2000 200 800 1200 1600 2500 600 1000 1300 1800 2300 (đ/kWh) Table 4. The mean LMP (VND/kWh) of six-bus system Bus 1 2 3 4 5 6 SRS (5000 samples) 1196.898 1214.634 1217.834 1269.807 1278.744 1249.164 LHS (1000 samples) 1196.762 1214.38 1217.388 1269.585 1278.411 1248.612 LHS-FFS (100 samples) 1196.775 1214.366 1216.895 1269.547 1278.014 1247.022 40 LMP ($/MWh) 35 30 25 20 15 10 5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Bus SRS (5000 samples) LHS (500 samples) LHS-FFS (100 samples) Fig. 5. The mean LMP of IEEE 24-bus system Using the proposed P-OPF model, the 6.2 IEEE 24-Bus System probabilistic locational marginal prices are obtained This subsection presents the calculated results of through Monte Carlo simulation with three different mean LMP using the IEEE 24-bus system [21]. The sampling methods, namely simple random sampling diagram of this 24-bus system is revealed in Fig. 4. (SRS), Latin hypercube sampling (LHS), and Latin The resulting LMPs from this system are depicted in hypercube sampling combined with Fast Forward Fig. 5. Selection Algorithm (LHS-FFS). Fig. 5 shows that the P-OPF problem with three The obtained results that are depicted in Table 4 sampling methods all results in identical solutions. show that the solution of P-OPF with three sampling On the other hand, there is a significant decrease in techniques is very similar. However, the the computational time for solving the P-OPF with computational time and computer storage for solving LHS-FFS (around 5 seconds with 100 samples), the P-OPF problem with LHS-FFS are considerably compared to approximately 4 minutes when applying reduced. SRS (5000 samples). 48
9. JST: Smart Systems and Devices Volume 31, Issue 1, May 2021, 041-049 7. Conclusion [9] A. Soroudi and M. Ehsan, IGDT based robust decision-making tool for DNOs in load procurement This paper presents methods for calculating under severe uncertainty, IEEE Trans. Smart Grid, 4 probabilistic locational marginal price (LMP) using (2013) 886–895. Monte Carlo Simulation integrated with different sampling techniques. Three sampling approaches are [10] N. B. Hargreaves, S. M. Pantea, and G. A. Taylor, compared, including simple random sampling (SRS), Large Scale Renewable Power Generation (Advances Latin hypercube sampling (LHS), and LHS combined in Technologies for Generation, Transmission, and with the fast forward selection (FFS) algorithm to Storage), 2014. reduce the number of samples. The results show that [11] J. M. Morales, S. Pineda, A. J. Conejo, and M. the mean LMP obtained with these sampling Carrion, Scenario reduction for futures market trading procedures is very similar; however, the in electricity markets, IEEE Trans. Power Syst., 24 computational performance such as time and (2009) 878–888. computer storage is markedly improved when applying LHS-FFS. This contributes to the efficient [12] K. C. Sharma, P. Jain, and R. Bhakar, Wind Power operation of electricity markets. Scenario Generation and Reduction in Stochastic Programming Framework, Electr. Power Components References Syst., 41 (2013) 271–285. [1] Z. Zhao and L. Wu, Impacts of high penetration wind generation and demand response on LMPs in day- [13] Y. Wang, Y. Liu, and D. S. Kirschen, Scenario ahead market, IEEE Trans. Smart Grid, 5 (2014) 220– Reduction with Submodular Optimization, IEEE 229. Trans. Power Syst., 32(2016) 2479-2480. [2] S. Liu, L. Trinh, J. Zhu, and M. Moore, [14] Comprehensive wind power interconnection [15] N. Grửwe-Kuska, H. Heitsch, and W. Rửmisch, evaluation method based on LMP market simulation, Scenario reduction and scenario tree construction for 1st IEEE-PES/IAS Conf. Sustain. Altern. Energy, power management problems, 2003 IEEE Bol. SAE 2009 - Proc., 2009. PowerTech - Conf. Proc., 3 (2003) 152–158. [16] H. Heitsch and W. Rửmisch, Scenario reduction [3] F. Li, S. Member, R. Bo, and S. Member, DCOPF- algorithms in stochastic programming, Comput. Based LMP Simulation : Algorithm, comparison with Optim. Appl., 24 (2003) 187–206. ACOPF and sensitivity, IEEE Trans. Power Syst., 22 (2007) 1475–1485. [17] H. W. Dommel and W. F. Tinney, Optimal Power Flow Solutions, IEEE Trans. Power Appar. Syst., PAS-87 (1968) 1866–1876. [4] X. Fang, Y. Wei, and F. Li, Evaluation of LMP Intervals Considering Wind Uncertainty, IEEE Trans. Power Syst., 31 (2016) 2495–2496. [18] D. I. Sun, B. Ashley, B. Brewer, A. Hughes, and W. F. Tinney, Optimal power flow by Newton approach, IEEE Trans. Power Appar. Syst., PAS-103 (1984) [5] H. Yu, C. Y. Chung, K. P. Wong, H. W. Lee, and J. 2864–2880. H. Zhang, Probabilistic load flow evaluation with hybrid Latin hypercube sampling and Cholesky decomposition, IEEE Trans. Power Syst., 24 (2009) [19] O. Alsaỗ, J. Bright, M. Prais, and B. Stott, Further 661–667. developments in lp-based optimal power flow, IEEE Trans. Power Syst., 5 (1990) 697–711. [6] M. Hajian, W. D. Rosehart, and H. Zareipour, Probabilistic power flow by Monte Carlo simulation [20] L. S. Vargas, V. H. Quintana, and A. Vannelli, A with Latin supercube sampling, IEEE Trans. Power tutorial description of an interior point method and its Syst., 28 (2013)1550–1559. applications to security-constrained economic dispatch, IEEE Trans. Power Syst., 8 (1993) 1315– 1324. [7] A. Soroudi, M. Ehsan, R. Caire, and N. Hadjsaid, Possibilistic evaluation of distributed generations impacts on distribution networks, IEEE Trans. Power [21] J. M. Morales and J. Pộrez-ruiz, Simulating the Syst., 26 (2011) 2293–2301. Impact of Wind Production on Locational Marginal Prices, IEEE Trans. Power Syst., 26 (2011) 820-828. [8] A. Soroudi, Possibilistic-scenario model for DG impact assessment on distribution networks in an [22] uncertain environment, IEEE Trans. Power Syst., 27 (2012) 1283–1293. 49