milp optimization

HSEM MILP Optimization

The MILP solver (planner/milp_optimizer.py) finds the globally optimal battery charge/discharge schedule using scipy's HiGHS linear programming solver. It is the primary planner — heuristic candidates are generated alongside it for benchmarking and fallback, but when scipy is available the MILP solution is preferred.

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Pipeline

flowchart TD
    A[Engine passes slots with populated prices / PV / consumption]
    B[Identify future active slot indices past vs fixed-zero]
    C[Build per-slot data arrays: p_imp, p_exp, net_load, base_load, pv_avail]
    D{EV configs provided?}
    E[Rebuild net_load without pre-computed EV planned loads]
    F[Keep fixed EV planned load in net_load]
    G[Build objective vector c_obj: import cost, export revenue, cycle cost, conversion loss, terminal-SoC credit, SoC penalties]
    H[Build equality constraints A_eq: energy balance per slot inc. EV charger load]
    I[Build inequality constraints A_ub: SoC recurrence soft bounds, mutual exclusion, cycle cost auxiliary m >= ec/ed, EV cumulative SOC, EV deadline target]
    J[Build variable bounds: ec, ed capped; pv fixed to actual surplus; penalties >= 0]
    K[linprog method = highs, timeout = 2s]
    L{Solution found?}
    M[Decode solution: ec, ed, gi, ge, pv, m, penalties, EV charging]
    N[Write recommendations to output slots BatteriesChargeGrid, BatteriesChargeSolar, BatteriesDischargeMode, ForceBatteriesDischarge]
    O[Compute penalty violation diagnostics]
    P[Return slots and diagnostics]
    Q[Return None: solver failed or no future slots]

    A --> B --> C --> D
    D -->|Yes| E --> G
    D -->|No| F --> G
    G --> H --> I --> J --> K --> L
    L -->|Yes| M --> N --> O --> P
    L -->|No| Q

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Variable layout

Base battery variables (8 × n slots)

For each slot t ∈ 0…n-1 the LP variable vector x contains eight decision variables:

Offset	Variable	Name	Description	Bounds
0	ec[t]	ec_off	Energy charged and stored in battery this slot (kWh)	[0, max_charge_per_slot]
n	ed[t]	ed_off	Energy discharged from battery this slot (kWh)	[0, max_discharge_per_slot]
2n	gi[t]	gi_off	Grid import this slot (kWh)	[0, ∞)
3n	ge[t]	ge_off	Grid export this slot (kWh)	[0, ∞)
4n	pv[t]	pv_off	PV surplus available in slot t (kWh)	[pv_avail[t], pv_avail[t]] (fixed)
5n	m[t]	m_off	Auxiliary variable ≥ max(ec[t], ed[t]) for cycle cost (kWh)	[0, ∞)
6n	s_max_pen[t]	s_max_off	SoC upper penalty — kWh by which state of charge exceeds usable_kwh	[0, ∞)
7n	s_min_pen[t]	s_min_off	SoC lower penalty — kWh by which state of charge drops below 0	[0, ∞)

The state of charge soc[t] is not an explicit variable — it is derived from the forward recurrence:

$$ soc[t] = soc[0] + \sum_{k=0}^{t} \bigl( ec[k] - ed[k] \bigr) $$

Penalty variables s_max_pen and s_min_pen prevent infeasibility when the initial SoC lies outside [0, usable_kwh]. Their objective coefficient is extremely high (max(p_imp) × 100), so they are only used when the initial state is physically out of bounds.

EV co-optimization extension

When one or more active EVs are provided, the variable vector expands to:

$$ \text{total variables} = 8n + n \cdot E + E $$

where E is the number of active EVs.

Offset	Variable	Name	Description	Bounds
8n + i·n	evN_c[t]	EV N DC-side charge per slot (kWh)	[0, evN.max_charge_per_slot]
8n + E·n + i	evN_pen	EV N deadline target slack (kWh shortfall)	[0, ∞)

The EV charger AC load entering the energy balance equation is evN_c[t] / charger_efficiency.

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Objective function

$$ \begin{aligned} \mathrm{minimise} \quad \sum_{t} \delta_t \cdot \bigg[ & p_{\mathrm{imp}}[t] \cdot gi[t] && \text{grid import cost} \\ - & p_{\mathrm{exp}}[t] \cdot ge[t] && \text{export revenue} \\ + & \alpha \cdot m[t] && \text{battery cycle cost (depreciation)} \\ + & \epsilon_{\mathrm{chg}} \cdot p_{\mathrm{imp}}[t] \cdot ec[t] && \text{charge-side conversion loss cost} \\ + & \epsilon_{\mathrm{dis}} \cdot p_{\mathrm{imp}}[t] \cdot ed[t] && \text{discharge-side conversion loss cost} \\ - & \gamma \cdot \bigl( ec[t] - ed[t] \bigr) && \text{terminal-SoC replacement credit} \\ + & p_{\mathrm{soc}} \cdot \bigl( \mathrm{s_max_pen}[t] + \mathrm{s_min_pen}[t] \bigr) && \text{SoC soft-constraint penalties} \bigg] \\ \end{aligned} $$

Plus EV deadline penalties (undiscounted — deadline is a hard commitment):

$$ \sum_{v=1}^{E} p_{\mathrm{ev_pen}}^{(v)} \cdot \mathrm{ev_pen}_v $$

Where:

Symbol	Description
$\delta_t$	Time discount per slot: $\delta_t = r^{\Delta t}$ where $\Delta t$ is hours from now
$p_{\mathrm{imp}}[t]$	Grid import price (currency/kWh)
$p_{\mathrm{exp}}[t]$	Grid export price (currency/kWh), clamped to 0 when below min_export_price
$\alpha$	Battery cycle cost per kWh: $\alpha = \frac{P \cdot L_{pct}/100}{2 \cdot N \cdot C_u}$
$\epsilon_{\mathrm{chg}}$	Charge-side loss fraction: $\epsilon_{\mathrm{chg}} = 1 - \eta_{\mathrm{chg}}$
$\epsilon_{\mathrm{dis}}$	Discharge-side loss fraction: $\epsilon_{\mathrm{dis}} = 1 - \eta_{\mathrm{dis}}$
$\gamma$	Terminal-SoC replacement price (currency/kWh), from the engine
$p_{\mathrm{soc}}$	SoC penalty cost: $\max(p_{\mathrm{imp}}) \times 100$
$p_{\mathrm{ev_pen}}^{(v)}$	EV deadline penalty for EV v: $\max(p_{\mathrm{imp}}) \cdot \mathrm{capacity} \cdot 100$

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Constraints

Equality constraint — energy balance per slot

For each slot $t$:

$$ gi[t] + pv[t] + ed[t] \cdot \eta_{\mathrm{dis}} = \mathrm{base_load}[t] + \frac{ec[t]}{\eta_{\mathrm{chg}}} + ge[t] + \sum_{v=1}^{E} \frac{\mathrm{ev_c}_v[t]}{\eta_{\mathrm{charger}}^{(v)}} $$

• base_load[t] = $\max(\mathrm{net_load}[t], 0)$ — demand the grid/battery must satisfy (kWh)
• net_load[t] = avg_house_consumption[t] - solcast_pv_estimate[t] (when EV co-optimisation active)
• pv_avail[t] = $\max(-\mathrm{net_load}[t], 0)$ — PV surplus fixed to the pv[t] variable bounds
• EV charger efficiency re-scales DC-side charge to AC grid/PV load

Inequality constraints

SoC upper bound (soft):

$$ \sum_{k=0}^{t} \bigl( ec[k] - ed[k] \bigr) - \mathrm{s_max_pen}[t] \leq C_u - soc_0 $$

SoC lower bound (soft):

$$ -\sum_{k=0}^{t} \bigl( ec[k] - ed[k] \bigr) - \mathrm{s_min_pen}[t] \leq soc_0 $$

Mutual exclusion — no simultaneous charge + discharge:

$$ \frac{ec[t]}{\mathrm{max_charge}} + \frac{ed[t]}{\mathrm{max_discharge}} \leq 1 $$

Cycle cost auxiliary — forcing $m[t] \geq ec[t]$ and $m[t] \geq ed[t]$:

$$ -m[t] + ec[t] \leq 0 $$ $$ -m[t] + ed[t] \leq 0 $$

EV cumulative SoC upper bound (per EV v):

$$ \sum_{k=0}^{t} \mathrm{ev_c}_v[k] \leq \mathrm{capacity}_v - \mathrm{initial_soc}_v $$

EV deadline target (soft, per EV v):

$$ \mathrm{initial_soc}_v + \sum_{k=0}^{D_v} \mathrm{ev_c}_v[k] + \mathrm{ev_pen}_v \geq \mathrm{target}_v $$

Where $D_v$ is the deadline slot index for EV v.

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Post-processing

After solving, the solution is decoded into slot recommendations:

flowchart TD
    A[LP solution: ec, ed, gi, ge]
    B{ec > threshold AND ed > threshold?}
    C[Numerical tolerance conflict: resolve by net profit]
    D{Round-trip profitable?}
    E[Keep ec, zero ed]
    F[Zero both]
    G{ec > threshold?}
    H{PV surplus available?}
    I[BatteriesChargeSolar]
    J[BatteriesChargeGrid]
    K{ed > threshold?}
    L{Grid export > 0 AND price >= min_export_price?}
    M[ForceBatteriesDischarge]
    N[BatteriesDischargeMode]
    O[Write EV charge decisions]
    P[Recompute estimated_net_consumption and estimated_cost per slot]
    Q[Compute penalty violation diagnostics]

    A --> B
    B -->|Yes| C --> D
    D -->|Yes| E --> G
    D -->|No| F --> G
    B -->|No| G
    G -->|Yes| H
    H -->|Yes| I --> O
    H -->|No| J --> O
    G -->|No| K
    K -->|Yes| L
    L -->|Yes| M --> O
    L -->|No| N --> O
    K -->|No| O
    O --> P --> Q

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EV charging fields written to slots

Field	Source
ev_planned_load_kwh	AC load added when base_load_includes_ev is False
ev_accounted_load_kwh	AC load when already captured in house consumption
ev_total_planned_load_kwh	Total AC load (sum of planned + accounted)
ev_charger_calculated_power	Target AC power (W) for primary EV
ev_second_charger_calculated_power	Target AC power (W) for second EV

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Assumptions

• Linear relaxation: Binary charge/discharge flags are relaxed to continuous because the mutual-exclusion constraint and per-slot power caps already prevent simultaneous charge + discharge in the optimal solution.
• Deterministic inputs: All forecasts (prices, PV, load) are treated as known with certainty — no stochastic programming.
• Cycle cost proxy: The m[t] = max(ec[t], ed[t]) formulation counts the larger of charge or discharge per slot, matching the 2× denominator in the cycle cost formula.
• Time discount: The objective uses exponential discounting with time_discount_rate^hours_ahead to match the selector's discounted score.
• Export price clamping: Negative export prices and prices below min_export_price are clamped to 0 before solving, reflecting physical inverter behaviour.
• Terminal-SoC credit: Undiscounted in the objective, matching the cost function's terminal_soc_value.

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Solver configuration

Parameter	Value	Rationale
Method	highs	scipy's HiGHS is the only supported LP method
Timeout	2.0 s	Covers 192-slot (768+ variable) problems where preprocessing reaches 200-400 ms
pv[t] bounds	(pv_avail[t], pv_avail[t])	Fixed — PV surplus is not chosen by the LP

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Fallback

If scipy is unavailable, usable_kwh ≤ 0, or the solver fails (crash, timeout, or non-success status), solve_milp() returns None. The engine silently drops the MILP candidate and the heuristic candidates compete as normal. Pickup is be measured via the hsem_plan_origin metric: milp when the LP succeeds, rule_based otherwise.

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Related

• HSEM Planner Specification
• Cost Function Math
• Energy Accounting
• Candidate Generation