−2τ=0uit−τ,kt=DTik· · ·T−1(33)whereUTikandDTikdenote the minimum up/down times forithcategory of thermal units in regionk, respectively. As in-dicated in (32), the capacity starting up within the interval of[t-UTik,UTik]hours can not be shut down again at timet. Similarformulation of the minimal down time constraints is presentedin (33).Employing the continuous variables and re-formulating sub-sequent flexibility constraints will significantly reduce modelingcomplexity. For each group of units, only 3 continuous variablesare 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¯pit,kto approximateˆSOj(t), a minor errorwill be introduced since the actual aggregated online capacitycan only be valid for a discrete set of numbers. The error willbe lower than the installed capacity of the smallest unit in thegroup, and it will be less significant as the size of the unit groupincreases. A detailed comparison of the hourly energy balancederived from the proposed model and the rigorous simulationmodel is presented later in Result section, part A. The errorintroduced 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 withinvestment 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. NORTHWESTERNPOWERGRID OFCHINAThe model is used to examine the the optimal generation mixwith higher penetration of variable renewables, taking the north-western power grid of China as a representative application. Wefocus on the year of 2035, a target year far enough away to ex-amine a wider range of renewable targets. The generation mixand grid structure, renewable resource availability, and invest-ment costs and operational characteristics are summarized inthis section.A. System OverviewThe following analysis is based on the northwestern powergrid, a regional power grid covering 5 provinces in China. Thisregional 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 accountingfor an additional 15% of annual power production.The peak power production in this region is expected to reach190 GW in 2020, with 133 GW local demand and 57 GW ofpower export. The generation structure will still be dominantby the coal fired power units in 2020. The installed capac-ity for coal-fired power generators and CHP units will reach126 GW and 38 GW respectively. The installed capacity of wind, solar  and hydro will reach 48 GW, 44 GW and 39 GWrespectively in 2020. The generation mix and peak power de-mand for 2020 for different provinces of the northwestern Chinapower grid are summarized in Table I. The five provinces are
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