2 \u03c4 0 u it \u03c4k t DT i k T 1 33 where UT i k and DT i k denote the minimum updown

2 τ 0 u it τk t dt i k t 1 33 where ut i k and dt i

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2 τ =0 u i t τ,k t = DT i k · · · T 1 (33) where UT i k and DT i k denote the minimum up/down times for i th category of thermal units in region k , respectively. As in- dicated in (32), the capacity starting up within the interval of [ t- UT i k , UT i k ] hours can not be shut down again at time t . Similar formulation of the minimal down time constraints is presented in (33). Employing the continuous variables and re-formulating sub- sequent flexibility constraints will significantly reduce modeling complexity. For each group of units, only 3 continuous variables are employed to describe the aggregated effect of the commit- ment status for each unit within the group at each time interval. In the procedure using ¯ p i t,k to approximate ˆ S O j ( t ) , a minor error will be introduced since the actual aggregated online capacity can only be valid for a discrete set of numbers. The error will be lower than the installed capacity of the smallest unit in the group, and it will be less significant as the size of the unit group increases. A detailed comparison of the hourly energy balance derived from the proposed model and the rigorous simulation model is presented later in Result section, part A. The error introduced in this process is minor compared with other uncer- tainties in the planning process. In sum, employing the above formulation for flexibility, the capacity expansion model adopts the objective function of (4)–(6), (11) and (24), constrained by (1)–(3) associated with investment and operational decisions, (7)–(10) for system bal- ancing and reliability requirements, (12)–(20) for energy stor- age, (21)–(23) and (25) for low carbon policies and (27)–(33) unit for flexibility. III. N ORTHWESTERN P OWER G RID OF C HINA The model is used to examine the the optimal generation mix with higher penetration of variable renewables, taking the north- western power grid of China as a representative application. We focus on the year of 2035, a target year far enough away to ex- amine a wider range of renewable targets. The generation mix and grid structure, renewable resource availability, and invest- ment costs and operational characteristics are summarized in this section. A. System Overview The following analysis is based on the northwestern power grid, a regional power grid covering 5 provinces in China. This regional power grid covers over 3 million square kilometers, similar in size to India. Wind and solar PV accounted for 10% of total power generation in 2015, with hydro power accounting for an additional 15% of annual power production. The peak power production in this region is expected to reach 190 GW in 2020, with 133 GW local demand and 57 GW of power export. The generation structure will still be dominant by the coal fired power units in 2020. The installed capac- ity for coal-fired power generators and CHP units will reach 126 GW and 38 GW respectively. The installed capacity of wind [23], solar [24] and hydro will reach 48 GW, 44 GW and 39 GW respectively in 2020. The generation mix and peak power de- mand for 2020 for different provinces of the northwestern China power grid are summarized in Table I. The five provinces are
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