1 Planning with Multiple Transmission and Storage Investment Options under Uncertainty: A Nested Decomposition Approach Paola Falugi, Member, IEEE, Ioannis Konstantelos, Member, IEEE, and Goran Strbac, Member, IEEE, Abstract—Achieving the ambitious climate change mitigation objectives set by governments worldwide is bound to lead to unprecedented amounts of network investment to accommodate low-carbon sources of energy. Beyond investing in conventional transmission lines, new technologies such as energy storage can improve operational flexibility and assist with the cost- effective integration of renewables. Given the long lifetime of these network assets and their substantial capital cost, it is imperative to decide on their deployment on a long-term cost- benefit basis. However, such an analysis can result in large-scale Mixed Integer Linear Programming (MILP) problems which contain many thousands of continuous and binary variables. Complexity is severely exacerbated by the need to accommodate multiple candidate assets and consider a wide range of exogenous system development scenarios that may occur. In this manuscript we propose a novel, efficient and highly-generalizable framework for solving large-scale planning problems under uncertainty by using a temporal decomposition scheme based on the principles of Nested Benders. The challenges that arise due to the presence of non-sequential investment state equations and sub-problem non-convexity are highlighted and tackled. The substantial com- putational gains of the proposed method are demonstrated via a case study on the IEEE 118 bus test system that involve planning of multiple transmission and storage assets under long- term uncertainty. The proposed method is shown to substantially outperform the current state-of-the-art. Index Terms—Energy storage, mixed integer-linear program- ming, nested Benders decomposition, stochastic programming, transmission planning I. NOMENCLATURE Note that throughout the paper, bold symbols are used to denote vectors and |C| indicates the cardinality of the set C . Ω B Set of demand blocks, indexed by b. Ω E Set of all stages, indexed by e. Ω G Set of generation units, indexed by g. Ω L Set of transmission lines, indexed by ℓ. Ω M Set of all nodes belonging to the scenario tree, in- dexed by m. Ω N Set of system buses, indexed by n. Ω m ς Set of scenarios to which node m belongs to. Ω E S Set of existing storage assets, indexed by s. Ω C S Set of storage candidates, indexed by s. Ω T S Set of storage technologies, indexed by s. Ω b T Set of periods in demand block b, indexed by t. Ω W ℓ Set of expansion options for line ℓ, indexed by w. P. Falugi, I. Konstantelos and G. Strbac are with the Department of Electrical and Electronic Engineering, Imperial College London, London, UK. This research was supported by EPSRC and by grant EP/N030028/1. m − Parent node of scenario tree node m. ε(m) Stage to which the node m belongs to. Φ k (m) Set of ancestor nodes of m from stage 1 to ε(m) − k. N + (m) Set of children nodes of m that belong to stage ε(m)+1. All the parameters involved in the optimization model can be represented by the vector ρ containing B n,g Bus-to-generation incidence matrix. I n,ℓ Bus-to-line incidence matrix. S n,s Bus-to-storage incidence matrix. W b Weight of demand block b. τ b t Time duration of demand period t, demand block b. σ b First period of demand block b. T b Last period of demand block b. D b m,t,n Demand at bus n for operating point (m, t). χ ℓ Line length in km. X ℓ Reactance of transmission line ℓ. u ℓ Sending bus for line ℓ. v ℓ Receiving bus for line ℓ. F 0 ℓ Initial capacity for line ℓ. F max ℓ,w Capacity provided by expansion option w for line ℓ. γ ℓ,w Build time for line ℓ and expansion option w. κ F ℓ,w Annual fixed investment cost for line ℓ, option w (£/(km yr)). κ V ℓ,w Annual variable investment cost for line ℓ, option w (£/(MW km yr)). ˜ h 0 s Initial condition of storage device s ∈ Ω E S . ¯ h s Maximum charge/discharge rate (MW) of storage s. ¯ η s Energy capacity (MWh) of storage s. ρ e s Storage efficiency of s. γ H s Build time of storage technology s. κ H s Annual capital cost of storage device s (£/yr). ¯ p m,g,t Maximum generation for g at operating point (m, t). κ G g Operation cost of generating unit g (£/MWh). Γ System balance penalty constant (£/MWh). r I e Cumulative discount factor for investment cost in epoch e. r O e Cumulative discount factor for operation cost in epoch e. All decision variables involved are represented by the vector x containing d m,t,n Curtailed demand at bus n and operating point (m, t). f m,t,ℓ Power flow on line ℓ at operating point (m, t).