InvestParameters

InvestParameters model investment decisions in optimization problems, enabling both binary (invest/don't invest) and continuous sizing choices with comprehensive cost modeling.

Investment Decision Types

FlixOpt supports two main types of investment decisions:

Binary Investment

Fixed-size investment creating a yes/no decision (e.g., install a 100 kW generator):

\[\label{eq:invest_binary} v_\text{invest} = s_\text{invest} \cdot \text{size}_\text{fixed} \]

With: - $v_\text{invest}$ being the resulting investment size - $s_\text{invest} \in \{0, 1\}$ being the binary investment decision - $\text{size}_\text{fixed}$ being the predefined component size

Behavior: - $s_\text{invest} = 0$: no investment ($v_\text{invest} = 0$) - $s_\text{invest} = 1$: invest at fixed size ($v_\text{invest} = \text{size}_\text{fixed}$)

Continuous Sizing

Variable-size investment with bounds (e.g., battery capacity from 10-1000 kWh):

\[\label{eq:invest_continuous} s_\text{invest} \cdot \text{size}_\text{min} \leq v_\text{invest} \leq s_\text{invest} \cdot \text{size}_\text{max} \]

With: - $v_\text{invest}$ being the investment size variable (continuous) - $s_\text{invest} \in \{0, 1\}$ being the binary investment decision - $\text{size}_\text{min}$ being the minimum investment size (if investing) - $\text{size}_\text{max}$ being the maximum investment size

Behavior: - $s_\text{invest} = 0$: no investment ($v_\text{invest} = 0$) - $s_\text{invest} = 1$: invest with size in $[\text{size}_\text{min}, \text{size}_\text{max}]$

This uses the bounds with state pattern described in Bounds and States.

Optional vs. Mandatory Investment

The mandatory parameter controls whether investment is required:

Optional Investment (mandatory=False, default): $$\label{eq:invest_optional} s_\text{invest} \in {0, 1} $$

The optimization can freely choose to invest or not.

Mandatory Investment (mandatory=True): $$\label{eq:invest_mandatory} s_\text{invest} = 1 $$

The investment must occur (useful for mandatory upgrades or replacements).

Effect Modeling

Investment effects (costs, emissions, etc.) are modeled using three components:

Fixed Effects

One-time effects incurred if investment is made, independent of size:

\[\label{eq:invest_fixed_effects} E_{e,\text{fix}} = s_\text{invest} \cdot \text{fix}_e \]

With: - $E_{e,\text{fix}}$ being the fixed contribution to effect $e$ - $\text{fix}_e$ being the fixed effect value (e.g., fixed installation cost)

Examples: - Fixed installation costs (permits, grid connection) - One-time environmental impacts (land preparation) - Fixed labor or administrative costs

Specific Effects

Effects proportional to investment size (per-unit costs):

\[\label{eq:invest_specific_effects} E_{e,\text{spec}} = v_\text{invest} \cdot \text{spec}_e \]

With: - $E_{e,\text{spec}}$ being the size-dependent contribution to effect $e$ - $\text{spec}_e$ being the specific effect value per unit size (e.g., €/kW)

Examples: - Equipment costs (€/kW) - Material requirements (kg steel/kW) - Recurring costs (€/kW/year maintenance)

Piecewise Effects

Non-linear effect relationships using piecewise linear approximations:

\[\label{eq:invest_piecewise_effects} E_{e,\text{pw}} = \sum_{k=1}^{K} \lambda_k \cdot r_{e,k} \]

Subject to: $$ v_\text{invest} = \sum_{k=1}^{K} \lambda_k \cdot v_k $$

With: - $E_{e,\text{pw}}$ being the piecewise contribution to effect $e$ - $\lambda_k$ being the piecewise lambda variables (see Piecewise) - $r_{e,k}$ being the effect rate at piece $k$ - $v_k$ being the size points defining the pieces

Use cases: - Economies of scale (bulk discounts) - Technology learning curves - Threshold effects (capacity tiers with different costs)

See Piecewise for detailed mathematical formulation.

Retirement Effects

Effects incurred if investment is NOT made (when retiring/not replacing existing equipment):

\[\label{eq:invest_retirement_effects} E_{e,\text{retirement}} = (1 - s_\text{invest}) \cdot \text{retirement}_e \]

With: - $E_{e,\text{retirement}}$ being the retirement contribution to effect $e$ - $\text{retirement}_e$ being the retirement effect value

Behavior: - $s_\text{invest} = 0$: retirement effects are incurred - $s_\text{invest} = 1$: no retirement effects

Examples: - Demolition or disposal costs - Decommissioning expenses - Contractual penalties for not investing - Opportunity costs or lost revenues

Total Investment Effects

The total contribution to effect $e$ from an investment is:

\[\label{eq:invest_total_effects} E_{e,\text{invest}} = E_{e,\text{fix}} + E_{e,\text{spec}} + E_{e,\text{pw}} + E_{e,\text{retirement}} \]

Effects integrate into the overall system effects as described in Effects, Penalty & Objective.

Integration with Components

Investment parameters modify component sizing:

Without Investment

Component size is a fixed parameter: $$ \text{size} = \text{size}_\text{nominal} $$

With Investment

Component size becomes a variable: $$ \text{size} = v_\text{invest} $$

This size variable then appears in component constraints. For example, flow rate bounds become:

\[ v_\text{invest} \cdot \text{rel}_\text{lower} \leq p(t) \leq v_\text{invest} \cdot \text{rel}_\text{upper} \]

Using the scaled bounds pattern from Bounds and States.

Cost Annualization

Important: All investment cost values must be properly weighted to match the optimization model's time horizon.

For long-term investments, costs should be annualized:

\[\label{eq:annualization} \text{cost}_\text{annual} = \frac{\text{cost}_\text{capital} \cdot r}{1 - (1 + r)^{-n}} \]

With: - $\text{cost}_\text{capital}$ being the upfront investment cost - $r$ being the discount rate - $n$ being the equipment lifetime in years

Example: €1,000,000 equipment with 20-year life and 5% discount rate $$ \text{cost}_\text{annual} = \frac{1{,}000{,}000 \cdot 0.05}{1 - (1.05)^{-20}} \approx €80{,}243/\text{year} $$

Implementation

Python Class: InvestParameters

Key Parameters: - fixed_size: For binary investments (mutually exclusive with continuous sizing) - minimum_size, maximum_size: For continuous sizing - mandatory: Whether investment is required (default: False) - effects_of_investment: Fixed effects incurred when investing (replaces deprecated fix_effects) - effects_of_investment_per_size: Per-unit effects proportional to size (replaces deprecated specific_effects) - piecewise_effects_of_investment: Non-linear effect modeling (replaces deprecated piecewise_effects) - effects_of_retirement: Effects for not investing (replaces deprecated divest_effects)

See the InvestParameters API documentation for complete parameter list and usage examples.

Used in: - Flow - Flexible capacity decisions - Storage - Storage sizing optimization - LinearConverter - Converter capacity planning - All components supporting investment decisions

Examples

Binary Investment (Solar Panels)

solar_investment = InvestParameters(
    fixed_size=100,  # 100 kW system
    mandatory=False,  # Optional investment (default)
    effects_of_investment={'cost': 25000},  # Installation costs
    effects_of_investment_per_size={'cost': 1200},  # €1200/kW
)

Continuous Sizing (Battery)

battery_investment = InvestParameters(
    minimum_size=10,  # kWh
    maximum_size=1000,
    mandatory=False,  # Optional investment (default)
    effects_of_investment={'cost': 5000},  # Grid connection
    effects_of_investment_per_size={'cost': 600},  # €600/kWh
)

With Retirement Costs (Replacement)

boiler_replacement = InvestParameters(
    minimum_size=50,  # kW
    maximum_size=200,
    mandatory=False,  # Optional investment (default)
    effects_of_investment={'cost': 15000},
    effects_of_investment_per_size={'cost': 400},
    effects_of_retirement={'cost': 8000},  # Demolition if not replaced
)

Economies of Scale (Piecewise)

battery_investment = InvestParameters(
    minimum_size=10,
    maximum_size=1000,
    piecewise_effects_of_investment=PiecewiseEffects(
        piecewise_origin=Piecewise([
            Piece(0, 100),    # Small
            Piece(100, 500),  # Medium
            Piece(500, 1000), # Large
        ]),
        piecewise_shares={
            'cost': Piecewise([
                Piece(800, 750),  # €800-750/kWh
                Piece(750, 600),  # €750-600/kWh
                Piece(600, 500),  # €600-500/kWh (bulk discount)
            ])
        },
    ),
)