The widely recognized procedure to formulate the flexibilityconstraints for grouped units, pioneered by  and developedimportantly by , is to assume an averagePmaxwithin thegroup and to then combine binary variables into an integer vari-ablex(t). For some unit group in , the integer requirementsofx(t)are relaxed. This procedure is enriched in  by elabo-rating reserve constraints.For the modeling formulation targeting larger geographicalareas as in this paper, a different representation is adopted.We introduce a continuous variable¯pit,kto approximate theinteger variable ofˆSOi(t), indicating the aggregated nameplatecapacities that is online at timetfor theithunit group.In addition, the two other continuous variablessit,kanduit,kare introduced as the start-up, and shut-down capacity for theithcategory of thermal units in regionkat hourt. By definition,they satisfy:¯pit,k−¯pit−1,k=sit,k−uit,k(27)
CHENet al.: POWER SYSTEM CAPACITY EXPANSION UNDER HIGHER PENETRATION OF RENEWABLES CONSIDERING FLEXIBILITY6245wheresit,kanduit,kmust be no larger than the total aggregatednameplate capacities for unit group k:0≤sit,k, uit,k≤¯Iik(28)After introducing the above continuous variables, flexibilityconstraints for the unit group could be formulated equivalentlyas follows.The aggregated power outputpit,kfor theithcategory ofgeneration units atkthregion satisfies:μ−ik·¯pit,k≤pit,k≤¯μik·¯pit.k(29)whereμ−ikand¯μikare the minimum and maximum output ratiosfor theithcategory of thermal units in regionkat hourt, respec-tively. For instance, the values ofμ−ikand¯μikfor the coal firedunits in China are 0.5 and 1 respectively. The power output forpit,kwill range between 50% and 100% of the online capacityfor¯pit,kat timet.Incorporating the continuous variablespit,k,sit,kanduit,k, theramping constraints for the unit group are formulated as in (30)and (31):pit,k≤¯μik·(¯pit,k−sit,k−uit+1,k) +V Sik·sit,k+V Dik·uit+1,k(30)⎧⎨⎩pit,k−pit−1,k≤RUik·(¯pit,k−sit,k) +V Sik·sit,k−μ−ik·uit,kpit−1,k−pit,k≤RDik·(¯pit,k−sit,k)−μ−ik·sit,k+V Dik·uit,k(31)whereRUikandRDikdenote the ratios of upward/downwardramping for theithcategory of thermal units in regionk, re-spectively.V SikandV Dikare the start-up and shut-down ramplimits for theithcategory of thermal units in regionk. For sim-plicity of the analysis, we assume hereafter they both are equalto the minimum power output levelμ−ik. In this case, the poweroutput for the second time period after startup will be equal tothe minimum output level.Employing the above continuous variables, minimum on/offtime constraints are formulated as follows:0≤uit+1,k≤¯pit,k−∑t−1τ=0sit−τ,kt= 1· · ·UTik−10≤uit+1,k≤¯pit,k−∑U Tik−2τ=0sit−τ,kt=UTik· · ·T−1(32)0≤sit+1,k≤¯Iik−¯pit,k−∑t−1τ=0uit−τ,kt=1· · ·DTik−10≤sit+1,k≤¯Iik−¯pit,k−∑DTik
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