reserve is required to be provided by conventional units . Insuch cases, the limitation of downward reserve and correspond-ingly over-generation problem will be more pronounced withhigher renewable penetration .
CHENet al.: POWER SYSTEM CAPACITY EXPANSION UNDER HIGHER PENETRATION OF RENEWABLES CONSIDERING FLEXIBILITY6243In light of the system reliability requirement, the total in-stalled capacity should be no less than the required capacitylimit:Mki=1¯Iik+λwk·¯Iwk+λsk·¯Isk≥Dmaxk(10)whereλwkandλskare the capacity credits for wind and so-lar power in regionk, respectively;Dmaxkrepresents the totalcapacity requirement to maintain reliability standards. For sim-plicity in the modeling, we use fixed capacity credits for windand solar respectively, although the value of capacity credit forwind and solar is non-linear with respect to the total capacity.D. Modeling for the Operation and Investment for StorageA novel linear simulation model for energy storage is pre-sented in this section to represent the costs and constraints asso-ciated with their investment and operation, considering multiplegeographical areas and different storage types.The overall costs for storage systems include both amortizedinvestment and operational costs formulated as:Ces=Nak=1Ne sz=1ap,zk·¯Ip,zk+ae,zk·¯Ie,zk+ces,zk·Tt=1pdis,zt,k+pch,zt,k·Δt(11)whereap,zkandae,zkare the power-specific and energy-specificamortized investment costs for thezthcategory of energy stor-age in regionk. The power-specific cost is related for exampleto the rotating synchronous machines in a pumped hydro unit,or to the power electronic rectifier/inverters in a battery stor-age system, or to the costs for anodes and cathodes in the caseof flow battery . The energy-specific cost is related to thereservoir for a pumped hydro system  or to the cost for thesolution included in a flow battery.ces,zkis the operational costfor thezthcategory of energy storages in regionk. The opera-tional cost is proportional to both the charging and dischargingpower.¯Ip,zkand¯Ie,zkdefine the corresponding maximum powerand energy capacities for the newly installed storage systems.They satisfy:¯Ip,zk≥0,¯Ie,zk≥0(12)0≤pdis,zt,k≤¯Ip,zk(13)0≤pch,zt,k≤¯Ip,zk(14)The energy balances of electric storage systems are repre-sented by:ees,zt+1,k=ees,zt,k+γch,zes·pch,zt,k·Δt−1γdis,zes·pdis,zt,k·Δt−γself,zes·ees,zt,k(15)whereγch,zes,γdis,zesandγself,zesindicate respectively the charg-ing/discharging efficiency and energy loss (self-discharge) ratefor thezthcategory of energy storages;ees,zt,krepresents the en-ergy level at timetfor thezthcategory of energy storage inregionk, constrained by the installed capacity:ν−zes·¯Ie,zk≤ees,zt,k≤¯νzes·¯Ie,zk(16)whereν−zesand¯νzesrepresent the minimum and maximum levelsof residual energy in the storage system respectively. For certaintypes of storage systems, deep-discharge will significantly re-
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