The widely recognized procedure to formulate the flexibility constraints for

The widely recognized procedure to formulate the

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The widely recognized procedure to formulate the flexibility constraints for grouped units, pioneered by [11] and developed importantly by [22], is to assume an average P m ax within the group and to then combine binary variables into an integer vari- able x(t) . For some unit group in [11], the integer requirements of x(t) are relaxed. This procedure is enriched in [22] by elabo- rating reserve constraints. For the modeling formulation targeting larger geographical areas as in this paper, a different representation is adopted. We introduce a continuous variable ¯ p i t,k to approximate the integer variable of ˆ S O i ( t ) , indicating the aggregated nameplate capacities that is online at time t for the i th unit group. In addition, the two other continuous variables s i t,k and u i t,k are introduced as the start-up, and shut-down capacity for the i th category of thermal units in region k at hour t . By definition, they satisfy: ¯ p i t,k ¯ p i t 1 ,k = s i t,k u i t,k (27)
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CHEN et al. : POWER SYSTEM CAPACITY EXPANSION UNDER HIGHER PENETRATION OF RENEWABLES CONSIDERING FLEXIBILITY 6245 where s i t,k and u i t,k must be no larger than the total aggregated nameplate capacities for unit group k: 0 s i t,k , u i t,k ¯ I i k (28) After introducing the above continuous variables, flexibility constraints for the unit group could be formulated equivalently as follows. The aggregated power output p i t,k for the i th category of generation units at k th region satisfies: μ i k · ¯ p i t,k p i t,k ¯ μ i k · ¯ p i t.k (29) where μ i k and ¯ μ i k are the minimum and maximum output ratios for the i th category of thermal units in region k at hour t , respec- tively. For instance, the values of μ i k and ¯ μ i k for the coal fired units in China are 0.5 and 1 respectively. The power output for p i t,k will range between 50% and 100% of the online capacity for ¯ p i t,k at time t . Incorporating the continuous variables p i t,k , s i t,k and u i t,k , the ramping constraints for the unit group are formulated as in (30) and (31): p i t,k ¯ μ i k · p i t,k s i t,k u i t +1 ,k ) + V S i k · s i t,k + V D i k · u i t +1 ,k (30) p i t,k p i t 1 ,k RU i k · p i t,k s i t,k ) + V S i k · s i t,k μ i k · u i t,k p i t 1 ,k p i t,k RD i k · p i t,k s i t,k ) μ i k · s i t,k + V D i k · u i t,k (31) where RU i k and RD i k denote the ratios of upward/downward ramping for the i th category of thermal units in region k , re- spectively. V S i k and V D i k are the start-up and shut-down ramp limits for the i th category of thermal units in region k . For sim- plicity of the analysis, we assume hereafter they both are equal to the minimum power output level μ i k . In this case, the power output for the second time period after startup will be equal to the minimum output level. Employing the above continuous variables, minimum on/off time constraints are formulated as follows: 0 u i t +1 ,k ¯ p i t,k t 1 τ =0 s i t τ,k t = 1 · · · UT i k 1 0 u i t +1 ,k ¯ p i t,k U T i k 2 τ =0 s i t τ,k t = UT i k · · · T 1 (32) 0 s i t +1 ,k ¯ I i k ¯ p i t,k t 1 τ =0 u i t τ,k t =1 · · · DT i k 1 0 s i t +1 ,k ¯ I i k ¯ p i t,k DT i k
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