Long-term system expansion using decomposition
Introduction
CERES applies Benders decomposition to solve the long-term power system expansion planning problem using a Least-Cost Planning (LCP) framework. This approach separates the investment (master) problem from the operational (subproblem) model, which covers hourly dispatch across multiple scenarios and years. The decomposition allows high-resolution operational modeling with scalable investment optimization.
1. Linking Investments and Installed Capacity
The decision to invest in new assets determines the installed capacity of generation, transmission, storage, and other elements. For generation:
$$ g_{cap,g,m} = \sum_{mm=m-LT_{g}}^{m} g_{inv,g,mm} $$
Where:
\(g_{cap,g,m}\): installed generation capacity of technology at node ,
\(g_{inv,g,m}\): new generation investments in year .
\(LT_{g}\): lifetime of generator.
The same formulation applies to transmission lines, storage capacity, and other asset classes, with respective investment variables and capacity interpretations.
2. Renewable Energy Target Constraint
To meet policy goals, CERES enforces a renewable capacity share constraint:
$$ \sum_{reg(g)} g_{cap,g,m} · GP_{g} \ge RT_{reg(g),m} $$
Where \(RT_{reg(g),m}\) is the minimum volume of RES generation and \(GP_{g}\) is the relative production of green electricity per MW.
3. Peak Demand Coverage Constraint
To ensure system adequacy, CERES imposes: $$ \sum_{reg(g)} g_{cap,g,m} · PC_{g} \ge PD_{reg(g),m} $$
Where:
\(PC_{g}\): peak contribution per MW of power plants,
\(PD_{reg(g),m}\): projected peak demand in MW.
4. Capital Expenditure (CAPEX) Calculation
The total capital expenditure is computed from investment decisions: $$ CAPEX_m = \sum_{g} g_{inv,g,m} · Cpx_{g,m} + … $$
Where \(Cpx_{g,mm}\) is the capital cost per unit of new capacity. Additional terms cover storage, lines, hydrogen, or natural gas infrastructure.
5. Fixed Operational Expenditures (OPEX)
The fixed O&M cost is calculated based on installed capacity: $$ OPEXF_m = \sum_{g,mm} g_{cap,g,mm} · Opxf_{g,mm} + … $$
Where \(Opxf_{g,mm}\) is the fixed opex cost per unit of capacity. Additional terms cover storage, lines, hydrogen, or natural gas infrastructure.
6. Variable Operational Costs via Benders Optimality Cuts
After solving the subproblem (dispatch), CERES extracts variable operational costs and generates Benders cuts:
\(OPEXV_m \ge OPEXV_{sim,m,iter-1} + \sum_{g} μ_{g,m,iter-1} · ( g_{cap,g,m} - g_{cap,g,m,iter-1} ) + ...\)
Where \(OPEXV_{sim,m,iter}\) is the simulated operating cost, \(μ_{g,m,iter}\) are dual variables from the subproblem, \(g_{cap,m,iter}\) is the generation capacity variable and \(g_{cap,g,m,iter}\) is the calculated generation capacity in a previous iteration.
7. Feasibility Cuts
When in a feasibility scenario it is not possible (e.g., due to inadequate capacity) to meet the load, feasibility cuts can be added to the master problem to exclude infeasible capacity expansion schedules:
\(\sum_{g} μ_{g,m,iter-1} · ( g_{cap,g,m} - g_{cap,g,m,iter-1} ) + ... \ge ens_{m,iter-1}\)
Where are coefficients derived from previously made feasibility scenario analysis and \(ens_{m,iter}\) reflects the resulting unserved energy.
8. Acceleration via DIA and Sampling Techniques
CERES improves computational tractability through:
DIA (Distributed Incremental Acceleration): proprietary technology to accelerate the resolution of large optimization problems,
Temporal sampling: representative months are selected and solved; the outcomes of the remaining months are estimated through robust statistical methods.
Temporal representation: though CERES allows the calculation of most accurate expansion plans, in some cases it can be useful to have a less expensive simulation that delivers results faster, in this case we allow the use of different temporal resolutions as well as typical days or full chronology.
These techniques drastically reduce computational time and costs while preserving the usefulness of long-term planning decisions.
Conclusion
By leveraging Benders decomposition and a range of modeling enhancements, CERES delivers high-speed, high-fidelity grid expansion planning. It captures investment logic, operational realities, and policy goals in an integrated and efficient framework.