Stochastic Programming¶

The stochastic extension enables robust capacity expansion planning under uncertainty by considering multiple future scenarios simultaneously. It is implemented in master_problem.jl via create_stochastic_master_problem().

1. Overview¶

The deterministic Master Problem assumes perfect knowledge of future conditions -- fuel prices, demand growth, technology costs, and renewable resource availability are all known with certainty. In practice, these parameters are subject to significant uncertainty over multi-decade planning horizons. The stochastic formulation [9], [10] addresses this by considering multiple scenarios, each representing a plausible future, and optimizing the expected cost across all scenarios.

The key insight of stochastic programming is the distinction between decisions that must be made before uncertainty is resolved (here-and-now decisions) and decisions that can adapt to realized conditions (wait-and-see decisions):

First-stage (here-and-now): Investment decisions $I^{gen}_{y,g,n}$, $I^{bat,P}_{y,b,n}$, $I^{bat,E}_{y,b,n}$, $I^{tr}_{y,i,j}$ are made once and must be the same across all scenarios. These represent infrastructure commitments that cannot be changed after construction.
Second-stage (wait-and-see): Operational dispatch decisions can differ by scenario. Each scenario has its own set of representative day dispatch variables reflecting how the system would operate under that particular realization of uncertainty.

2. Scenario Tree Structure¶

2.1 Scenario Definition¶

Each scenario $\omega \in \Omega$ is defined by a probability weight and a set of parameter multipliers:

\[ \Omega = \{(\omega_1, \pi_1), (\omega_2, \pi_2), \ldots, (\omega_S, \pi_S)\} \quad \text{where } \sum_{s=1}^{S} \pi_s = 1 \tag{STOCH-1} \]

The probability weights $\pi_s$ represent the decision-maker's belief about the likelihood of each scenario. They must be non-negative and sum to one, forming a valid probability distribution.

2.2 Scenario Multipliers¶

The ScenarioMultipliers struct defines multiplicative factors that are applied to the base-case parameters to create each scenario's parameter set. This approach is parsimonious: rather than specifying complete parameter sets for each scenario, only the deviations from the base case are specified.

Multiplier	Default	Description	Affected Parameters
`demand_multiplier`	1.0	Scales electrical demand	$D_{n,t}$
`fuel_cost_multiplier`	1.0	Scales all fuel costs	$c^{fuel}_{g,n}$
`renewable_availability_multiplier`	1.0	Scales RE capacity factors	$\alpha_{g,t,n}$
`investment_cost_multiplier`	1.0	Scales generation investment costs	$c^{inv}_{g,n}$
`battery_cost_multiplier`	1.0	Scales storage investment costs	$c^{inv,P}_{b,n}$, $c^{inv,E}_{b,n}$
`co2_cost_multiplier`	1.0	Scales CO2 emission penalty	$C^{CO2}$
`discount_rate_multiplier`	1.0	Scales discount rate	$r$
`demand_growth_multiplier`	1.0	Scales annual demand growth rate	$\gamma$
`ev_adoption_multiplier`	1.0	Scales EV fleet growth rate	$r_{EV}$
`technology_improvement_multiplier`	1.0	Scales efficiency improvement	Various

Multipliers are applied by apply_scenario_multipliers(), which creates modified copies of the generator and battery configurations for each scenario. A multiplier of 1.0 means no change from the base case; values greater than 1.0 increase the parameter, and values less than 1.0 decrease it.

2.3 Scenario Construction¶

The effective parameter for scenario $\omega_s$ is computed as:

\[ \theta_s = \theta^{base} \times m_s \]

where $\theta^{base}$ is the base-case parameter value and $m_s$ is the corresponding multiplier. For example, if the base fuel cost is 50 $/MWh and the fuel cost multiplier for a "high fuel price" scenario is 1.5, the effective fuel cost in that scenario is 75 $/MWh.

3. Two-Stage Formulation¶

3.1 Mathematical Structure¶

The stochastic Master Problem has the following two-stage structure:

First stage (deterministic, common to all scenarios):

\[ \min_{I} \; C^{1st}(I) + \mathbb{E}_\omega\left[Q(I, \omega)\right] \]

subject to investment constraints (INV-1 through INV-3, BUD-1, TXN-1) that are scenario-independent.

Second stage (recourse, scenario-dependent):

For each scenario $\omega_s$, the recourse function $Q(I, \omega_s)$ is the optimal operational cost given investment decisions $I$ and scenario parameters $\omega_s$:

\[ Q(I, \omega_s) = \min_{P,E,L,\ldots} \; C^{op}_s(P, E, L, \ldots) \]

subject to operational constraints (power balance, generator limits, battery dynamics, etc.) parameterized by scenario $\omega_s$ and linked to investment decisions through cumulative capacity expressions.

3.2 Non-Anticipativity¶

The non-anticipativity constraint is enforced implicitly: there is only one copy of the first-stage investment variables shared across all scenarios. This is equivalent to the explicit constraint:

\[ I^{gen,(\omega_1)}_{y,g,n} = I^{gen,(\omega_2)}_{y,g,n} = \ldots = I^{gen,(\omega_S)}_{y,g,n} \qquad \forall y, g, n \]

and similarly for battery and transmission investment variables.

In the JuMP implementation, this is achieved by using a single set of investment variables that appear in the constraints of all scenarios. No additional constraints are needed.

Second-stage variables (representative day operational variables) are scenario-specific, each with its own index:

\[ P^{day,(\omega_s)}_{g,n,t} \text{ varies by scenario } s \quad \text{(separate variable set per scenario)} \]

4. Objective Function¶

4.1 Stochastic Objective¶

The stochastic objective minimizes the sum of deterministic first-stage investment costs and expected second-stage operational costs:

\[ \min \; Z^{stoch} = \underbrace{\sum_{y=1}^{Y} \frac{C^{inv}_y}{(1+r)^{y-1}}}_{\text{First stage (deterministic)}} + \underbrace{\sum_{s=1}^{S} \pi_s \sum_{y=1}^{Y} \frac{C^{op}_{y,s}}{(1+r_s)^{y-1}}}_{\text{Second stage (expected cost)}} + C^{slack} \tag{STOCH-OBJ} \]

Built by build_stochastic_objective!().

4.2 First-Stage Investment Cost¶

Investment costs may be modified by scenario-specific multipliers. However, since investment decisions are common across scenarios, the investment cost in the objective uses either the base-case costs (unscaled) or the expected investment cost:

\[ C^{inv}_y = \sum_{g,n} c^{inv}_{g,n} \cdot I^{gen}_{y,g,n} + \sum_{b,n} \left( c^{inv,P}_{b,n} \cdot I^{bat,P}_{y,b,n} + c^{inv,E}_{b,n} \cdot I^{bat,E}_{y,b,n} \right) + \sum_{(i,j)} c^{tr}_i \cdot I^{tr}_{y,i,j} \]

4.3 Second-Stage Operational Cost¶

For each scenario $s$, the operational cost is computed from representative day dispatch, scaled to annual:

\[ C^{op}_{y,s} = \frac{365}{|\mathcal{D}_y|} \sum_{d \in \mathcal{D}_y} C^{day}_{y,d,s} \]

where $C^{day}_{y,d,s}$ is the daily operational cost under scenario $s$, computed using the scenario-modified fuel costs, maintenance costs, and penalty coefficients.

4.4 Scenario-Adjusted Discount Rate¶

Each scenario can have a different effective discount rate:

\[ r_s = r^{base} \times m^{discount}_s \]

where $m^{discount}_s$ is the discount_rate_multiplier for scenario $s$. This allows modeling scenarios with different capital market conditions or risk premia.

4.5 Expected Cost Calculation¶

The expected total cost is a weighted sum over scenarios:

\[ \mathbb{E}_\omega[Z] = \sum_{s=1}^{S} \pi_s \cdot Z_s \]

For example, with three scenarios (high demand, base case, green transition) with probabilities (0.3, 0.5, 0.2):

\[ \mathbb{E}[Z] = 0.3 \cdot Z_{high} + 0.5 \cdot Z_{base} + 0.2 \cdot Z_{green} \]

The optimal investment plan minimizes this expected cost, hedging against all three futures simultaneously.

4.6 Slack Penalties¶

Slack penalties are applied per scenario with probability weighting:

\[ C^{slack} = \sum_{s=1}^{S} \pi_s \sum_{y=1}^{Y} \left[ c^{slack} \cdot s^{re}_{y,s} + c^{slack} \cdot s^{bud}_{y} + \sum_n c^{slack} \cdot s^{cap}_{y,n,s} \right] \]

The budget slack $s^{bud}_y$ is scenario-independent (since investment is common), while RE target and capacity adequacy slacks are scenario-dependent.

5. Constraint Structure¶

5.1 Scenario-Independent Constraints¶

The following constraints apply identically across all scenarios:

Constraint Family	Reference	Description
INV-1 to INV-3	Capacity Expansion	Cumulative investment limits
BUD-1	Capacity Expansion	Annual investment budget
TXN-1	Capacity Expansion	Transmission symmetry

5.2 Scenario-Dependent Constraints¶

For each scenario $s$, a separate set of representative day operational constraints is created with scenario-modified parameters:

Constraint	Modification
Power balance (PB-1)	Demand scaled by `demand_multiplier` and `demand_growth_multiplier`
Generator capacity (GEN-1)	RE availability scaled by `renewable_availability_multiplier`
Operational cost (OBJ-5)	Fuel costs scaled by `fuel_cost_multiplier`
CO2 emissions (CO2-1)	CO2 penalty scaled by `co2_cost_multiplier`
RE target (RE-DAY-1)	Target may be adjusted based on scenario parameters

The cumulative capacity expressions (CUM-1 through CUM-4) link the common investment variables to each scenario's operational constraints, ensuring that the same physical infrastructure serves all scenarios.

6. Illustrative Example¶

Consider a three-scenario planning problem for a small island grid:

Scenario	Probability	Key Assumptions
High Demand	0.3	20% higher demand, 10% higher fuel costs
Base Case	0.5	No modifications
Green Transition	0.2	15% better RE resources, 20% cheaper RE investment, 30% cheaper batteries

Configuration:

master_problem:
  stochastic: true

systems:
  island_grid:
    stochastic_scenarios:
      high_demand:
        probability: 0.3
        multipliers:
          demand_multiplier: 1.2
          fuel_cost_multiplier: 1.1

      base_case:
        probability: 0.5
        multipliers:
          demand_multiplier: 1.0

      green_transition:
        probability: 0.2
        multipliers:
          renewable_availability_multiplier: 1.15
          investment_cost_multiplier: 0.8
          battery_cost_multiplier: 0.7

Interpretation: The optimizer will select investments that perform reasonably well across all three futures. If the "green transition" scenario strongly favors solar + storage, but the "high demand" scenario requires firm capacity, the stochastic solution will include some firm generation capacity that the purely green-transition deterministic solution might omit.

7. Computational Considerations¶

7.1 Problem Size¶

The stochastic formulation scales the second-stage problem linearly with the number of scenarios:

Component	Deterministic	Stochastic ($S$ scenarios)
Investment variables	$V_1$	$V_1$ (shared)
Operational variables	$V_2$	$S \times V_2$
Investment constraints	$C_1$	$C_1$ (shared)
Operational constraints	$C_2$	$S \times C_2$
Total LP size	$V_1 + V_2$, $C_1 + C_2$	$V_1 + S \cdot V_2$, $C_1 + S \cdot C_2$

For a typical Master Problem with 5 representative days, 25 years, and 3 nodes, the deterministic problem might have ~100,000 variables. With 3 scenarios, the stochastic problem grows to ~250,000 variables (investment variables shared, operational variables tripled).

7.2 Solution Time¶

Solution time scales roughly linearly with the number of scenarios for LP problems (simplex or barrier methods). For practical planning studies:

2--3 scenarios: Minimal overhead, suitable for sensitivity analysis
5--10 scenarios: Moderate computation, captures key uncertainties
20+ scenarios: Significant computation; consider scenario reduction techniques

All scenarios share the same representative day selection (from the base-case demand profile). This simplifies the implementation and ensures that the same temporal structure is used across scenarios, differing only in parameter values.

When TSAM is enabled (use_tsam: true), TSAM periods and weights are computed once from the base case and shared across all scenarios. Only cost and demand multipliers differ between scenarios.

8. Implementation Notes¶

Partial Implementation

The stochastic framework is structurally complete but has limited testing. For production use, verify scenario results carefully and consider starting with 2--3 scenarios before scaling up.

8.1 Implementation Details¶

The stochastic Master Problem creates separate representative day variables per scenario via create_day_operational_vars!()
Scenario multipliers are applied to the base input before building each scenario's constraints via apply_scenario_multipliers()
The solver sees a single large LP with shared investment variables and per-scenario operational blocks
Solution extraction returns per-scenario operational results alongside the common investment plan

8.2 Function Call Graph¶

create_stochastic_master_problem(input)
  |-- build_master_variables!(model, input)         # Shared investment variables
  |-- add_investment_constraints!()                  # Shared constraints
  |-- add_budget_constraints!()                      # Shared
  |-- add_transmission_symmetry_constraints!()       # Shared
  |-- for each scenario s:
  |     |-- apply_scenario_multipliers(input, s)     # Create scenario params
  |     |-- add_representative_days_validation!(s)   # Per-scenario dispatch
  |     |-- calculate_target_ratios(input_s)         # Per-scenario RE targets
  |     +-- add_capacity_adequacy_constraints!(s)    # Per-scenario adequacy
  +-- build_stochastic_objective!()                  # Expected cost objective

9. Relationship to Other Formulations¶

The stochastic extension is compatible with all other ESFEX features:

Feature	Compatibility	Notes
MGA/SPORES	Sequential	Run MGA after stochastic solve to explore near-optimal alternatives
DC Power Flow	Full	Per-scenario transmission constraints
Primary Energy	Full	Per-scenario fuel supply constraints
Multi-System	Full	Per-scenario inter-system coupling
TSAM	Full	Shared periods, per-scenario costs
NPV Iteration	Full	Applied after stochastic solve

References¶

The two-stage stochastic programming formulation follows the framework of Birge and Louveaux [9]. Decision-making under uncertainty in electricity markets is treated by Conejo et al. [10]. The interaction between stochastic capacity expansion and unit commitment constraints is analyzed by Schwele et al. [35]. The EVPI and VSS metrics for quantifying the value of stochastic solutions are defined in Birge and Louveaux [9] (Ch. 4).

See the full bibliography for complete citation details.

Component	Deterministic	Stochastic (\(S\) scenarios)
Investment variables	\(V_1\)	\(V_1\) (shared)
Operational variables	\(V_2\)	\(S \times V_2\)
Investment constraints	\(C_1\)	\(C_1\) (shared)
Operational constraints	\(C_2\)	\(S \times C_2\)
Total LP size	\(V_1 + V_2\), \(C_1 + C_2\)	\(V_1 + S \cdot V_2\), \(C_1 + S \cdot C_2\)

Multiplier	Default	Description	Affected Parameters
`demand_multiplier`	1.0	Scales electrical demand	\(D_{n,t}\)
`fuel_cost_multiplier`	1.0	Scales all fuel costs	\(c^{fuel}_{g,n}\)
`renewable_availability_multiplier`	1.0	Scales RE capacity factors	\(\alpha_{g,t,n}\)
`investment_cost_multiplier`	1.0	Scales generation investment costs	\(c^{inv}_{g,n}\)
`battery_cost_multiplier`	1.0	Scales storage investment costs	\(c^{inv,P}_{b,n}\), \(c^{inv,E}_{b,n}\)
`co2_cost_multiplier`	1.0	Scales CO2 emission penalty	\(C^{CO2}\)
`discount_rate_multiplier`	1.0	Scales discount rate	\(r\)
`demand_growth_multiplier`	1.0	Scales annual demand growth rate	\(\gamma\)
`ev_adoption_multiplier`	1.0	Scales EV fleet growth rate	\(r_{EV}\)
`technology_improvement_multiplier`	1.0	Scales efficiency improvement	Various