A. Mathematical Formulation

A.

Mathematical Formulation

In this section the mathematical formulations for the perfect information and min-max regret planning problem are given. The mathematical nomenclature is shown in Section A.1. We then introduce the perfect information problem formulation, where a single scenario is considered and proceed with the min-max regret formulation expressed in terms of the perfect information problem solutions. Note that all problems presented in the paper were programmed using the commercial software (FICO Xpress, 2009) and solved using the in-built branch-and-bound algorithm with a stopping criterion of finding an integer solution within 0.1% of the best available bound.

A.1 Nomenclature Sets and indices Note that cardinality (number of elements) of a set Ω is denoted |Ω|. e(m) Epoch e associated with scenario tree node m (an epoch is a set of years) Set of countries in EU, indexed c. |Ω𝐶 | = 8. Ω𝐶 Set of epochs, indexed 𝑒. |Ω𝐶 | = 5. Ω𝐸 Set of onshore generators, indexed 𝑔. |Ω𝑔 | = 256. Ω𝐺 Set of all transmission corridors, indexed 𝑙. |Ω𝐿 | = 5. Ω𝐿 Ω𝐿𝐴𝐶

𝐴𝐶

Set of onshore AC transmission corridors, indexed 𝑙. |Ω𝐿 | = 28. 𝐷𝐶

Ω𝐷𝐶 Set of offshore DC transmission corridors, indexed 𝑙. |Ω𝐿 | = 106. 𝐿 Set of scenario tree nodes, indexed 𝑚. For a single scenario, |Ω𝑀 | = 5. Ω𝑀 Set of system buses, indexed 𝑛. |Ω𝑁 | = 16. Ω𝑁 Set of scenarios, indexed 𝑠. Ω𝑆 Set of demand periods, indexed 𝑡. |Ω 𝑇 | = 200. Ω𝑇 Set of offshore wind clusters, indexed 𝑤. |Ω𝑊 | = 32. Ω𝑤 Set that contains all parent nodes of node 𝑚, including 𝑚 Φ𝑚 Input parameters Susceptance of AC transmission corridor 𝑙 (p.u.) 𝑏𝑙 𝐷𝑡,𝑛 Demand at bus 𝑛 in period 𝑡 (MW) 𝐼𝑛,𝑙 Bus-to-corridor incidence matrix 0 Initial capacity of transmission corridor 𝑙 (MW) 𝐹𝑙 Maximum capacity per DC cable that can be added to corridor 𝑙 (MW) 𝐹̅𝑙 𝐵 Annuitized fixed cost of reinforcing corridor 𝑙 (€/year) 𝜅𝑙 ̅ Annuitized fixed cost of a DC undersea cable used in corridor 𝑙 (€/year) 𝜅𝑙𝐵 𝐹 Annuitized variable cost of reinforcing corridor 𝑙 (€/MW-year) 𝜅𝑙 𝜅𝑔𝐺 Operating cost of generating unit 𝑔 (€/MWh) Cost of offshore wind curtailment (€/MWh) 𝜅𝑤 𝑝̅𝑒,𝑔 Generation capacity for onshore plant g at epoch 𝑒 (MW) 𝐼 Cumulative discount factor for investment cost in epoch e (p.u.) 𝑟𝑒 Cumulative discount factor for operation cost in epoch e (p.u.) 𝑟𝑒𝑂 Start node for corridor l 𝑢𝑙 Destination node for corridor l 𝑣𝑙 Wind availability factor at period 𝑡 (p.u.) 𝑊𝑡 ̅ 𝜉𝑚,𝑤 Generation capacity for offshore wind plant g under scenario node 𝑚 (MW) Time duration of period 𝑡 (hours) 𝜏𝑡 Decision variables Binary variable signifying the choice of reinforcement scenario node 𝑚 for AC 𝐵𝑚,𝑙 corridor 𝑙 𝐵̅𝑚,𝑙 Integer variable signifying the number of DC undersea cables used in reinforcement 1

scenario node 𝑚 for DC corridor 𝑙 Investment cost under scenario node 𝑚 (€) Operational cost under scenario node 𝑚 (€) Capacity added to corridor l in scenario node 𝑚 (MW) State variable of aggregate extra capacity available to corridor l in scenario node 𝑚 (MW) Power flow on corridor 𝑙 at operating point (𝑚, 𝑡) (MW) Output of onshore generation unit 𝑔 at operating point (𝑚, 𝑡) (MW) Bus angle of 𝑛 at operating point (𝑚, 𝑡) (rad) Output of offshore cluster 𝑤 at operating point (𝑚, 𝑡) (MW) Curtailed output of offshore cluster 𝑤 at operating point (𝑚, 𝑡) (MW)

𝐼 𝐶𝑚 𝑂 𝐶𝑚 𝐹𝑚,𝑙

𝐹̃𝑚,𝑙 𝑓𝑚,𝑡,𝑙 𝑝𝑚,𝑡,𝑔 𝜃𝑚,𝑡,𝑛 𝜉𝑚,𝑡,𝑤 𝐶 𝜉𝑚,𝑡,𝑤

A.2 Mathematical Formulation – Perfect Information The perfect information problem for scenario 𝑠, comprised of all scenario tree nodes 𝑚 ∈ 𝑠, is a Mixed Integer-Linear Program (MILP) defined as follows. 𝑂 𝐼 𝐼 𝑂 ωs = 𝐦𝐢𝐧 { ∑ (𝑟𝑒(𝑚) 𝑐𝑚 + 𝑟𝑒(𝑚) 𝑐𝑚 )}

(1)

∀𝑚∈𝑠

where 𝐼 𝑐𝑚 = ∑ 𝐵𝑚,𝑙 𝜅𝑙𝐵 + ∑ 𝐵̅𝑚,𝑙 𝜅𝑙𝐵̅ + ∑ 𝐹𝑚,𝑙 𝜅𝑙𝐹 𝑙∈Ω𝐴𝐶 𝐿

𝑙∈Ω𝐷𝐶 𝐿

𝐶 𝑂 𝑐𝑚 = ∑ 𝜏𝑡 (∑ 𝑝𝑚,𝑡,𝑔 𝜅𝑔𝐺 + ∑ 𝜉𝑚,𝑡,𝑤 𝜅𝑤 ) ∀𝑡

∀g

,∀𝑚 ∈ 𝑠

(2)

,∀𝑚 ∈ 𝑠

(3)

𝑙∈Ω𝐿

∀𝑤

Investment constraints ∈ Ω𝐿𝐴𝐶 ∈ Ω𝐷𝐶 𝐿 ∈ Ω𝐿𝐴𝐶 ∈ Ω𝐷𝐶 𝐿

𝐵𝑚,𝑙 ∈ {0,1} 𝐵̅𝑚,𝑙 ∈ ℤ 𝐹𝑚,𝑙 ≤ 𝐵𝑚,𝑙 𝑀 𝐹𝑚,𝑙 ≤ 𝐵̅𝑚,𝑙 𝐹𝑙

,∀𝑚 ∈ 𝑠, 𝑙 ,∀𝑚 ∈ 𝑠, 𝑙 ,∀𝑚 ∈ 𝑠, 𝑙 ,∀𝑚 ∈ 𝑠, 𝑙

𝐹̃𝑚,𝑙 = ∑ 𝐹𝑗,𝑙

∀𝑚 ∈ 𝑠, 𝑙 ∈ Ω𝐿

(8)

0 ≤ 𝑝𝑚,𝑡,𝑔 ≤ 𝑝̅𝜀(𝑚),𝑔 ̅ 𝑊𝑡 0 ≤ 𝜉𝑚,𝑡,𝑤 ≤ 𝜉𝑚,𝑤 𝐶 ̅ 𝑊𝑡 𝜉𝑚,𝑡,𝑔 + 𝜉𝑚,𝑡,𝑤 = 𝜉𝑚,𝑤 𝑓𝑚,𝑡,𝑙 = 𝑏𝑙 (𝜃𝑚,𝑡,𝑢𝑙 − 𝜃𝑚,𝑡,𝑣𝑙 ) −(𝐹𝑙0 + 𝐹̃𝑚,𝑙 ) ≤ 𝑓𝑚,𝑡,𝑙 ≤ 𝐹𝑙0 + 𝐹̃𝑚,𝑙

,∀𝑚 ∈ 𝑠, 𝑡 ,∀𝑚 ∈ 𝑠, 𝑡 ,∀𝑚 ∈ 𝑠, 𝑡 ,∀𝑚 ∈ 𝑠, 𝑡 ,∀𝑚 ∈ 𝑠, 𝑡

(9) (10) (11) (12) (13)

∑ 𝑝𝑚,𝑡,𝑔 + ∑ 𝜉𝑚,𝑡,𝑤 + ∑ 𝐼𝑛,𝑙 𝑓𝑚,𝑡,𝑙 = 𝐷𝑡,𝑛

,∀𝑚 ∈ 𝑠, 𝑡 ∈ Ω𝑇 , 𝑛 ∈ Ω𝑁

(14)

,∀𝑚 ∈ 𝑠, ∀𝑐 ∈ Ω𝐶

(15)

𝑗∈Φ𝑚

(4) (5) (6) (7)

Operation constraints

∀𝑔∈𝑛

∀𝑤∈𝑛

∀𝑙

∑ 𝜏𝑡 ( ∑ 𝑝𝑚,𝑡,𝑔 + ∑ 𝜉𝑚,𝑡,𝑤 − ∑ 𝐷𝑡,𝑛 ) = 0 ∀𝑡

∀𝑔∈𝑐

∀𝑤∈𝑐

∈ Ω𝑇 , 𝑔 ∈ Ω𝐺 ∈ Ω𝑇 , 𝑤 ∈ Ω𝑊 ∈ Ω𝑇 , 𝑤 ∈ Ω𝑊 ∈ Ω𝑇 , 𝑙 ∈ ΩAC 𝐿 ∈ Ω𝑇 , 𝑙 ∈ Ω𝐿

∀𝑛∈𝑐

The objective function (1) minimizes the discounted system cost over the study horizon, defined as the sum of investment and operation costs accrued over all scenario tree nodes. In all presented studies, a discount rate of 5% has been assumed, which corresponds to the Transmission Owner companies’ weighted average cost of capital (WACC) and this is aligned with (National Grid, 2017). 2

Equation (2) defines the investment cost corresponding to scenario tree node 𝑚 as the summation of capital and fixed costs across all line reinforcements. Equation (3) defines the operation cost for scenario tree node 𝑚 as the summation of generation costs and curtailed wind production. Constraint (4) defines decision variables 𝐵𝑚,𝑙 , signifying the reinforcement of an onshore-to-onshore corridor, as binary. Constraint (5) defines decision variables 𝐵̅𝑚,𝑙 , signifying the number of times fixed cost has been paid for the reinforcement of a DC offshore corridor, as integer. Constraint (6) applies to AC corridors, where M is a large positive number enabling planners to reinforce by a large amount while subject to only one fixed cost payment. Constraint (7) applies to all subsea corridors and states that the amount of transmission capacity that can be built at node 𝑚 is bounded by the times fixed cost has been paid. Particularly in the case of offshore-offshore corridors this reflects the fact that individual offshore plants proceed with their connection-to-shore plans while coordinating with neighboring projects of the same cluster in order to reduce fixed cost elements such as permitting and use of cablelaying vessels (𝐹̅𝑙 being very large). However, under the ‘Radial’ policy choice, 𝐹̅𝑙 for offshoreonshore corridors is equal to 500 MW (rather than a very large number, which is used in the other policies so as to pay the fixed cost only once per epoch, if there is investment in that epoch) meaning that corridors are reinforced in lumps of 500 MW, essentially capturing the effects of offshore noncoordination in terms of increased fixed costs. Constraint (8) defines the investment state variables 𝐹̃𝑚,𝑙 as the summation of the corresponding control variables 𝐹𝑗,𝑙 . Note that commissioning delays are not considered in the present model and reinforcements are assumed to materialize instantaneously. Operation constant (9) provides limits on dispatch of conventional generation. In a similar vein, operation constraint (10) limits output of offshore wind clusters according to the period-variable wind factor 𝑊𝑡 . Equation (11) computes the amount of curtailed offshore wind as the difference between available and dispatched power. Equation (12) describes how power is distributed over all onshore links according to the dc, linear power flow formulation. Note that power flows over cross-border, onshore-offshore and offshore-offshore links are assumed to be fully controllable HVDC lines and not depend on bus angles and line reactance. Due to the complexity of the presented model, the susceptance value 𝑏𝑙 in (12) does not change with network reinforcements and this is suggested to be further developed. For this purpose, the disjunctive approach explained in (Binato et al., 2001) arises as a particularly interesting method to be explored and potentially applied within the presented model. Constraint (13) bounds line flows according to the existing and commissioned transmission capacity of the corridor. The system balance equation (14) states that at each bus, the local generation from conventional technologies and offshore wind and net incoming/outgoing power flows satisfy local demand. Equation (15) is activated solely for the energy neutral policy choices. It enforces energy neutrality at the member-state level, ensuring that year-round energy produced by domestic onshore and offshore generators is equal the country’s energy demand. Table 1 shows the changes in the above MILP model in each policy choice.

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Policy choice

Model features

Radial

(1)-(15) Offshore-to-offshore investment prevented 𝐹𝑙 = 500 MW (1)-(15) Offshore-to-offshore investment prevented 𝐹𝑙 →large number (1)-(15) Offshore-to-offshore investment allowed 𝐹𝑙 →large number (1)-(14) [(15) relaxed] Offshore-to-offshore investment allowed 𝐹𝑙 →large number (1)-(14) [(15) relaxed] Offshore generation investment co-optimized with transmission investment Extra constraints to ensure that net installed capacity of offshore wind per epoch corresponds to those in S1-S4 Offshore-to-offshore investment allowed 𝐹𝑙 →large number

Hub

Energy-neutral integrated

Fully integrated

Proactive

Table 1: Features to model each policy choice

We indicatively mention that the MILP planning problem when applied to a single scenario spanning five epochs has 670 integer variables, over 470,000 continuous variables and over 600,000 constraints.

A.3 Mathematical Formulation – Minimax Regret Although perfect information problem formulations constitute a useful tool for exploring optimal system development under different scenarios, they disregard some aspects of real-world decisionmaking that become critical when the decisions to be taken are capital-intensive and irreversible. The first aspect relates to the lack of perfect information and the need to endogenously consider the possibility for alternative future developments. In view of this, a flexible decision framework enables a planner to move beyond the concept of a static investment plan and instead identify the optimal investment strategy which encapsulates a range of contingent courses of actions to be taken according to all possible paths of uncertainty evolution. Furthermore, a flexible decision framework must also be accompanied by a specific decision criterion that describes the planner’s attitude towards risk and ultimately determines which commitments optimally balance cost efficiency and risk exposure. In general, there are three main classes of decision criteria when facing uncertainty; stochastic (also known as probabilistic), risk-constrained and robust. Stochastic planning is the case where each scenario node is attributed a probability of occurrence; the planner’s objective is the minimization of expected system cost over all realisations as in (Konstantelos and Strbac, 2015) and (Munoz et al., 2014). In a similar vein, suitable constraints with respect to spectral risk measures such as the Conditional Value-at-Risk, as in (Rockafellar and Uryasev, 2000), can be included to render the planner risk-averse. Robust decision methods in the context of system planning can refer to two variants; uncertainty intervals and utilisation of the regret concept. The former guarantees optimal performance given a deterministic description of the uncertainty state space (Bertsimas et al., 2011). Naturally, such a formulation lends itself mostly to static descriptions of uncertainty and cannot take advantage of its inter-temporal resolution structure which is an important characteristic of dynamic system planning (Epstein et al., 1980). The latter identifies the optimal planning strategy so as to 4

minimize a planner’s worst-case regret under all considered realizations (Gorenstin et al., 1993); the regret related to a particular scenario is the difference between the realized system cost and the system cost that would have been incurred under the assumption of perfect foresight. The concept of regret has been successfully used in the past in the context of transmission planning problems (van der Weijde et al., 2012; Chen et al., 2014). For example, authors in (van der Weijde et al., 2012) develop a minimax regret transmission planning model to examine the cost of disregarding uncertainty associated with future economic, technology and regulatory developments in the UK. The attractiveness of adopting a ‘minimax regret’ decision criterion lies in its ex-post decision-making nature; decisions are based on consequences of scenario occurrences. Probabilistic models, on the other hand, identify optimal decisions without considering the consequences after uncertainty has been resolved (Miranda and Proenca, 1998). The former philosophy is very much in line with existing regulatory frameworks which are based on ex-post performance evaluation against ex-ante-defined targets. Note that by definition, regret is defined in terms of the optimal scenario-specific solutions obtained under perfect foresight 𝜔𝑠 . It follows that the perfect information problem for each scenario must be solved before proceeding with the MILP min-max regret formulation shown below. 𝐦𝐢𝐧{𝑟} 𝑟 ≥ 𝑟𝑠 𝑟𝑠 = ∑

𝐼 (𝑟𝑒𝐼 𝑐𝑚

+

𝑂) 𝑟𝑒𝑂 𝑐𝑚 −

𝜔𝑠

,∀𝑠 ∈ Ω𝑆

(16) (17)

,∀𝑠 ∈ Ω𝑆

(18)

,{∀𝑚: 𝑒(𝑚) = 1, 𝑚 ≠ 1}, 𝑙 ∈ Ω𝐿

(19)

∀𝑚∈𝑠

𝐹1,𝑙 = 𝐹𝑚,𝑙

Subject to constraints (2)-(15) applied to all scenario trees nodes 𝑚 ∈ Ω𝑀 . The objective function (16) minimizes the maximum regret 𝑟. Constraint (17) ensures that 𝑟 is equal or greater to the regret of each scenario 𝑟𝑠 . The combination of constraints (16) and (17) applies the min-max operator across all scenarios; the worst case realization does not have to be defined a priori but is a result of the optimization. Equation (18) computes the regret for each scenario by aggregating the investment and operation costs across all scenario tree nodes that belong to scenario 𝑠 and subtracting the optimal cost 𝜔𝑠 obtained from the deterministic problem associated to scenario 𝑠. Equation (19) imposes non-anticipativity; decisions taken in the first epoch are common across all scenarios. Note that although for simplicity solely first-stage branching is considered, the presented optimization framework employs a node-variable formulation that could be extended in a straightforward manner to scenario trees of arbitrary size and shape.

Complementary references Bertsimas, D., Brown, D. B., & Caramanis, C. (2011). Theory and applications of robust optimization. SIAM review, 53(3), 464-501. Binato, S., Pereira, M. V. F., & Granville, S. (2001). A new Benders decomposition approach to solve power transmission network design problems. IEEE Transactions on Power Systems, 16(2), 235-240. Chen, B., Wang, J., Wang, L., He, Y., & Wang, Z. (2014). Robust optimization for transmission expansion planning: Minimax cost vs. minimax regret. IEEE Transactions on Power Systems, 29(6), 3069-3077. Epstein, L. G. (1980). Decision making and the temporal resolution of uncertainty. International economic review, 269-283. 5

FICO Xpress (2009). Xpress-optimizer http://www.fico.com/en/node/8140?file=5097.

reference

manual.

[Online],

Available:

Gorenstin, B. G., Campodonico, N. M., Costa, J. P., & Pereira, M. V. F. (1993). Power system expansion planning under uncertainty. IEEE Transactions on Power Systems, 8(1), 129-136. National Grid (2017). [Online], Available: http://investors.nationalgrid.com/about-us/our-markets/ukprofile.aspx. Rockafellar, R. T., & Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of risk, 2, 21-42.

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