An optimization model for regional micro-grid system management based on hybrid inexact stochastic-fuzzy chance-constrained programming

Introduction Electric power industry provides basic power for human activities and economic growth, and plays an essential role in social development. At the meantime, it is also a major source of carbon dioxide, sulfur dioxide, and nitric oxide (e.g. CO2, SO2, and NOx) emissions. Especially, the environmental pollution caused by traditional fossil generation becomes a serious problem, which gains more social concern accurately. With the ever-increasing energy demand, electric system develops fast and traditional fossil generation still constitutes a high proportion of electricity market. With increasing globe awareness of environmental protection, effective generation scheduling has been extensively discussed to reduce greenhouse gas and air pollutants emission. On the other hand, under the pressure of both resources and environment, renewable energy generation has become more and more popular. Many countries have set up the goals of renewable resources generation plan for future grid construction. Due to its advantages of convenience, flexibility and environmental friendly, micro-grid providing better platform for renewable energy generation has become popular and important in modern electricity system [1,2]. In order to deal with the interrelated and complex problems in the development of modern electric power system, many great efforts have been made on the optimal generation scheduling. Compared with historical researches, which mainly focus on thermoelectric power generators, most of recent studies are about 1026 L. Ji et al. / Electrical Power and Energy Systems 64 (2015) 1025–1039 micro-grid and distributed generation with renewable resources. For example, Yazawa and Shakouri (2013) analyzed the energy cost and optimization of thermoelectric power generators, which shows a lower initial cost compared with commercialized micro gas turbines but higher operating cost per energy due to moderate efficiency [3]. Guo et al. (2012) advanced the optimal generation dispatch with wind power generation and coal-fired generation embedded, and the optimal dispatch model was solved by a particle swarm optimization algorithm on an IEEE 30-bus system [4]. Moreover, the optimal goal has changed from simply minimizing cost or maximizing profit into multi objective optimization like minimizing carbon emissions and violation risks of uncertainties. Since those goals are usually conflict, the Pareto-optimal solutions are desired by decision makers. Considering variable penalty policy for CO2 emission, Tang and Che (2013) developed mixed integer nonlinear programming model to deal with the economic dispatch problem of thermal generation [5]. In the study by Buayai et al. (2012), the objective functions include real power loss, load voltage deviation and annualized investment cost, a pareto based non-dominated sorting genetic algorithm II was proposed to determine locations and sizes of the distributed generator units within micro-grid [6]. In fact, the electric system is extremely complex facing various uncertainties in generation side, demand side and market environment, which brings great challenge to the reliability of the electricity system [7–12]. As a result, relative research and on site projects are being carried out with a growing trend, a series of uncertainty methods have been proposed [13–20]. Among these methods, Two-stage stochastic programming (TSP) is a potential uncertainty technology and widely applied in electric generation planning and dispatch. Considering the power generation in a hydro-thermal generation system under uncertainty in demand and prices of fuel and delivery contracts, Nurnberg and Romisch (2002) developed a two-stage stochastic programming model for the shortor mid-term cost-optimal electric power production planning [21]. Nowak (2005) adopted a two-stage stochastic integer model for the simultaneous optimization of power production and day-ahead power trading [22]. Considering renewable energy generation, Hendrik van der Weijde (2012) developed cost-minimising TSP model and estimated the cost of ignoring uncertainty [23]. Combined with interval-parameter programming (IPP), TSP method was further developed into interval two-stage stochastic programming (ITSP) method, which can deal with uncertain optimization by interval and random numbers [24]. The traditional TSP or ITSP method cannot only provide an effective tool for energy policy scenarios analysis, but also handle the uncertain issue with certain probability [25,26]. However, in distributed energy generation system, the forecasting load of wind and solar power are usually obtained based on numerical weather prediction, which belongs to fuzzy information. If take them as deterministic parameters, it might easily mislead or bias the decision makers and lead to resource waste. In order to deal with the vague and obscure information, fuzzy set theory has provided a convenient formalism for classifying available renewable energy sources conditions. Fuzzy credibility constraints programming (FCCP) was proposed recently as a measure of confidence level in fuzzy environment to tackle uncertainties expressed as fuzzy sets. It was recognized as a competent measure of the confidence level regarding fuzzy constraints in optimization models [27,28]. Compared with other fuzzy programming approaches, the FCCP has a relatively low computational requirement and can obtain a series of solutions leading to high system benefits at allowable violation risk levels [29,30]. FCCP has been applied to many real-world cases due to its simplicity and efficiency in reflecting the fuzziness inherited with parameters associated with subjective consideration. Xue et al. (2012) developed an optimization model based on fuzzy credibility constraints programming for micro-grid operation with the uncertainties related to load and wind speed into consideration [31]. Based on mixed integer programming and FCCP, Zhang et al. (2012) developed integer fuzzy credibility constrained programming (IFCCP) to minimize the total cost of an independent regional power system [32]. Xu and Zhuan (2012) studied the optimization of wind power capacity for an electric power system with the system operation, economy and reliability emphasized, which is addressed by the FCCP approach [33]. Nevertheless, few previous studies were focused on development of inexact two-stage stochastic credibility constrained programming method through integrating IPP, TSP and FCCP into a general framework for electric schedule management within considering the pollutants and CO2 emission control. Therefore, the objective of this study is to develop an inexact stochastic-fuzzy chance-constrained optimization model for electric schedule management. Interval-parameter programming, two-stage stochastic programming and fuzzy credibility constrained programming methods are integrated into a general framework to manage pollutants and CO2 emissions under uncertainties presented as interval values, fuzzy possibilistic and stochastic probabilities. It shows the impact for accounting for both risk-averse and emission reduction goal in a two-stage stochastic optimization model. Within this framework, a new formulation is proposed to determine the operation of traditional and renewable resources generation over a 24-h optimization horizon with both economic and environmental considerations, where pollutant management should follow a real time and dynamic control strategy, and total amount control for CO2 emission. The remaining sections of this paper are organized as follows: Section ‘‘Methodology’’ introduces the main theory of interval two-stage stochastic programming and credibility constrained programming. The framework of electric system operation with wind and photovoltaic power is presented in Section ‘‘Case study’’. A case study and results analysis are illustrated in Section ‘‘Results analysis and discussion’’. Finally, some conclusions are provided in Section ‘‘Conclusion’’. Methodology A hybrid inexact stochastic-fuzzy chance-constrained programming (ITSFCCP) model was based on interval-parameter programming, two-stage stochastic programming, and fuzzy credibility constraints programming (as shown in Fig. 1). Each technique has its unique contribution in enhancing the ITSFCCP’s capacities for tackling the uncertainties and making the trade-offs between system economy and reliability. For example, in micro-grid system, the interval two-stage stochastic programming is used to reflect the uncertainty of energy market and technical parameters that expressed as intervals and the random characteristics of electric demand that expressed as stochastic numbers; and the system risk and the fuzzy availability of renewable energy sources were reflected through FCCP. Interval two-stage stochastic programming Two-stage stochastic programming (TSP) is effective for addressing problems where an analysis of policy scenarios is desired periodically over time and uncertain parameters are expressed as probability distribution functions (PDFs). A general TSP model can be formulated as follows [34]: Fig. 1. Schematic diagram of ITSFCCP model. L. Ji et al. / Electrical Power and Energy Systems 64 (2015) 1025–1039 1027 min f 1⁄4 cxþ XN s1⁄41 psQðy;xsÞ ð1aÞ subject to : ax 6 b ð1bÞ TðxsÞxþWðxsÞy 1⁄4 hðxsÞ ð1cÞ x P 0; yðxsÞ P 0 ð1dÞ where x is vector of first-stage decision variables; cx is first-stage benefits; x is random events after the first-stage decisions are made; s is the scenario of the happening of random events; ps is probability of event xs, P ps = 1; Q(y, xs) is system recourse at the second-stage under the occurrence of event xs; PN s1⁄41psQðy;xsÞ is expected value of the second-stage system penalties. The existing TSP methods are effective in handling probabilistic uncertainties in the model’s right-hand sides which are often related to resources availability; however they have difficulties in dealing with independent uncertainties of the model’s left-hand sides and cost coefficients. Interval-parameter programming (IPP) is an alternative for handling uncertainties in the model’s leftand/or right-hand sides as well as those that cannot be quantified as membership or distribution functions, since interval numbers are acceptable as its uncertain inputs. Let x be a set of intervals with crisp lower bound (e.g., x ) and upper bounds (i.e., x), but unknown distribution information. Let x be a set of closed and bounded interval numbers x [35]: x 1⁄4 1⁄2x ; xþ 1⁄4 tjx 6 t 6 xþ f g ð2Þ Through introducing interval parameters into Model 1, the ITSP model can be formulated as follows: min f 1⁄4 c x þ XN s1⁄41 psQ y ;x s ð3aÞ subject to : a x 6 b ð3bÞ Tðx s Þx þWðx s Þy 1⁄4 hðx s Þ ð3cÞ x P 0; y x s P 0 ð3dÞ Fuzzy credibility constrained programming Fuzzy credibility constrained programming (FCCP), which based on credibility conception, can be expressed as follows [36]: Min cjxj ð4aÞ Subject to : Cr Xn j1⁄41 aijxj 6 ~ bi; i 1⁄4 1;2; . . . ;m ( ) P ki ð4bÞ xj P 0; i 1⁄4 1; . . . ;n ð4cÞ where x = (x1, x2, . . ., xn) is a vector of non-fuzzy decision variables; cj are cost coefficients; aij are technical coefficients; ~ bi are right-hand side coefficients; Cr{ } denotes the credibility of the event { }; k is the confidence level. Let n be a fuzzy variable with membership function l, and let u and r be real numbers. Dubois and Prade proposed the following indices defined by possibility and necessity measures [36,37]: Pos n 6 r f g 1⁄4 sup u6r lðuÞ ð5aÞ Nec n 6 r f g 1⁄4 1 Posfn > rg 1⁄4 1 sup u>r l ð5bÞ The credibility measure Cr is the average of the possibility measure and the necessity measure: Crfn 6 rg 1⁄4 1 2 Posfn 6 rg þ Necfn 6 rg ð Þ ð6Þ Let the fuzzy variable n be fully determined by the triplet ðt; t; tÞ of crisp numbers with ðt < t < tÞ, whose membership function is given by lðrÞ 1⁄4 ðr tÞ=ðt tÞ if t 6 r 6 t; ð t rÞ=ð t tÞ if t 6 r 6 t; 0 otherwise: 8>< >: ð7Þ From the above definitions, the possibility, necessity, and credibility of r 6 n are provided as follows: Posfn 6 rg 1⁄4 0 if r 6 t r t t t if t 6 r 6 t 1 if r P t 8>< >: ð8aÞ Necfn 6 rg 1⁄4 0 if r 6 t r t t t if t 6 r 6 t 1 if r P t 8>< >: ð8bÞ Crðr 6 nÞ 1⁄4 0 if r 6 t r t 2ðt tÞ if t 6 r 6 t 2t t r 2ðt tÞ if t 6 r 6 t 1 if r P t >>< >>>: ð8cÞ Let Pn j1⁄41aijxj be replaced by si. Thus, the constraint (4b) can be represented as: Cr si 6 ~ bi; i 1⁄4 1; . . . ;m n o P ki; ð9Þ Normally, a significant credibility level should be greater than 0.5. Therefore, based on the definition of credibility, we have the following equation for each 1 P l~ti P ki P 0:5: 2bi bi si 2ðbi biÞ P ki ð10Þ where ~ bi are right-hand side coefficients fully determined by the triplet ðbi; bi; biÞ of crisp numbers with bi < bi < bi, whose membership function is l. Let Pn j1⁄41aijxj 1⁄4 si be the credibility constraints. The interval credibility levels, parameters and variables for such constraints can be formulated as: Crfsi 6 ~ bi; i 1⁄4 1; . . . ;mg P ki: ð11Þ 1028 L. Ji et al. / Electrical Power and Energy Systems 64 (2015) 1025–1039 Therefore, based on the definition of credibility, we have the following expression for each 1 P l~ti P ki P 0:5: 2bi bi si 2ðbi biÞ P ki ð12Þ Thus, the FCCP can be transformed to an equivalent model as follows: Min Xn j1⁄41 cjxj ð13aÞ Subject to : Xn j1⁄41 aijxj 6 bi þ ð1 2kiÞðbi biÞ ð13bÞ xj P 0; 8j ð13cÞ Inexact stochastic-fuzzy chance-constrained programming To tackle multi-type uncertainties, the ITSP and FCCP methods can be incorporated within a general optimization framework. Then an inexact stochastic-fuzzy chance-constrained programming model can be formulated as follows: Min f 1⁄4 Xn j1⁄41 c j x j þ Xn j1⁄41 Xm h1⁄41 pjhd j y jh ð14aÞ subject to : Xn j1⁄41 a ij x j 6 bi þ 1 2k i ðbi biÞ ð14bÞ Tðx s Þx j þWðx s Þy jh 1⁄4 hðx s Þ ð14cÞ x j P 0;x j 2 X ; j1⁄4 1;2; . . . ;n1 ð14dÞ y jh P 0;y jh 2 Y ; j1⁄4 1;2; . . . ;n2; h1⁄4 1;2; . . . ;m: ð14eÞ Model (14) can be transformed into two deterministic submodels that correspond to the lower and upper bounds of desired objective function value. This transformation process is based on an interactive algorithm, which is different from the best/worst case analysis [38]. The objective function value corresponding to f is desired first because the objective is to minimize net system costs. Based on the above solutions, the submodel for f can be formulated as follows: Min f 1⁄4 Xk1 j1⁄41 c j x j þ Xn j1⁄4k1þ1 c j x þ j þ Xk2 j1⁄41 Xm s1⁄41 psd j y js þ Xn j1⁄4k2þ1 Xm s1⁄41 psd j y þ js ð15aÞ subject to : Xk1 j1⁄41 a ij signða ij Þx j þ Xn j1⁄4k1þ1 a ij signða ij Þxj 6 bi þ ð1 2ki Þðbi biÞ ð15bÞ Xk1 j1⁄41 a rj sign a rj x j þ Xn1 j1⁄4k1þ1 a rj signða rjÞxj 6 b r ;8r ð15cÞ Xk1 j1⁄41 Tðx s Þx j þ Xn j1⁄4k1þ1 Tðx s Þxj þ Xk2 j1⁄41 Wðx s Þy js þ Xn j1⁄4k2þ1 Wðx s Þyjs 1⁄4 hðx s Þ8s ð15dÞ Xm s1⁄41 ps 1⁄4 1 ð15eÞ x j P 0; j 1⁄4 1;2; :::; k1 ð15fÞ xj P 0; j 1⁄4 k1 þ 1; k1 þ 2; :::;n ð15gÞ y js P 0; 8s; j 1⁄4 1;2; :::; k2 ð15hÞ yjs P 0; 8s; j 1⁄4 k2 þ 1; k2 þ 2; :::;n ð15iÞ where x j , j = 1, 2, . . ., k1, are interval variables with positive coefficients in the objective function; x j , j = k1 + 1, k1 + 2, . . ., n are interval variables with negative coefficients; y jh, j = 1, 2, . . ., k2 and h = 1, 2, . . ., v, are random variables with positive coefficients in the objective function; y jh, j = k2 + 1, k2 + 2, . . ., n and h = 1, 2, . . ., v, are random variables with negative coefficients [28,34,35]. Solutions of x j opt (j = 1, 2, . . ., k1), x þ j opt (j = k1 + 1, k1 + 2, . . ., n), y js opt (j = 1, 2, . . ., k2), and yjs opt (j = k2 + 1, k2 + 2, . . ., n) can be obtained through submodel (15). Based on the above solutions, the second submodel for f can be formulated as follows: Min f 1⁄4 Xk1 j1⁄41 cj x þ j þ Xn j1⁄4k1þ1 cj x j þ Xk2 j1⁄41 Xm s1⁄41 psd þ j y þ js þ Xn j1⁄4k2þ1 Xm s1⁄41 psd þ j y js ð16aÞ subject to : Xk1 j1⁄41 a ij signða ij Þxj þ Xn j1⁄4k1þ1 a ij signða ij Þx j 6 bi þ ð1 2k i Þðbi biÞ ð16bÞ Xk1 j1⁄41 a rj signða rjÞxj þ X n1 j1⁄4k1þ1 a rj signða rjÞx j 6 br ;8r ð16cÞ Xk1 j1⁄41 Tðxs Þxj þ Xn j1⁄4k1þ1 Tðxs Þx j þ Xk2 j1⁄41 Wðxs Þyjs þ Xn j1⁄4k2þ1 Wðxs Þy js 1⁄4 hðxs Þ8s ð16dÞ Xm s1⁄41 ps 1⁄4 1 ð16eÞ xj P x jopt P 0; j 1⁄4 1;2; . . . ; k1 ð16fÞ xj opt P x j P 0; j 1⁄4 k1 þ 1; k1 þ 2; . . . ;n ð16gÞ yjs P y js opt P 0;8s; j 1⁄4 1;2; . . . ; k2 ð16hÞ yjs opt P y js P 0;8s; j 1⁄4 k2 þ 1; k2 þ 2; . . . ;n ð16iÞ Solutions of x j opt (j = 1, 2, . . ., k1), x þ j opt (j = k1 + 1, k1 + 2, . . ., n), y þ js opt (j = 1, 2, . . ., k2), and y js opt (j = k2 + 1, k2 + 2, . . ., n) can be obtained through submodel (16). Through integrating solutions of submodels (15) and (16), interval solution for model (14) can be obtained as f opt 1⁄4 1⁄2f opt; f þ opt , x j opt 1⁄4 1⁄2x j opt ; xj opt , and y js opt 1⁄4 1⁄2y js opt; yjs opt . Case study Overview of the study system The electric power system in this study is a micro-grid consisted of coal-fired generation, gas-fired generation, wind power and photovoltaic power generation. These conventional and renewable recourses are served for the regional electric demand. Besides, coal-fired power has a residual capacity of 2.5 GW, natural gas-fired power has a residual capacity of 1.5 GW, wind power and photovoltaic power generations have installed capacity of 0.95 and 1.15 GW. It supposes that the shortage of electricity would be satisfied by electricity purchased from the main grid, while that the electric power generated in micro-grid is self-consumed and not allowed to be sold to main grid. In order to encourage efficient forecasting and dispatch, there are penalties for deviations between the real time delivery and pre-designed schedules. From the aspect of environment protection, regional pollution emission control policy is considered in operation management. Extra pollutant treatment cost would be necessary to meet environmental demand. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 lo ad d em an d (G W h) low medium high (a) lower bound