energy accounting

HSEM Energy Accounting — Mathematical Reference

This document defines the physical energy flow model used by the HSEM planner. Every equation here corresponds to a constraint enforced in the SoC simulation, the cost function, and the MILP solver.

---

Per-slot energy balance

For every planning slot, energy balance must hold:

$$ net\_load[t] = house\_load[t] + ev\_planned\_load[t] - \mathrm{pv}[t] $$

When EV integration is disabled, $ev\_planned\_load[t] = 0$.

Positive $net\_load[t]$ means the house needs energy (import or battery discharge). Negative $net\_load[t]$ means there is surplus energy (export or battery charge).

---

Grid and battery flows

Source-sink separation

The battery and grid flows must satisfy:

$$ house\_load[t] = pv\_to\_house[t] + battery\_to\_house[t] + grid\_to\_house[t] $$

$$ grid\_import[t] = grid\_to\_house[t] + grid\_to\_battery[t] + ev\_grid\_import[t] $$

PV production split

$$ \mathrm{pv}[t] = pv\_to\_house[t] + pv\_to\_ev[t] + pv\_to\_battery[t] + pv\_exported[t] + pv\_curtailed[t] $$

Battery charge

$$ battery\_charge\_stored[t] = (pv\_to\_battery[t] + grid\_to\_battery[t]) \cdot \eta_{chg} $$

Where $\eta_{chg} = charge\_efficiency\_pct / 100$.

Grid import for charging

$$ grid\_to\_battery[t] = battery\_charge\_stored[t] / \eta_{chg} $$

Key invariant: The cost function prices $grid\_to\_battery[t]$, not $battery\_charge\_stored[t]$. This ensures conversion losses are included in the import cost.

Battery discharge

$$ usable\_discharge[t] = battery\_removed[t] \cdot \eta_{dis} $$

Where $\eta_{dis} = discharge\_efficiency\_pct / 100$.

The battery energy removed to supply a target house load:

$$ battery\_removed[t] = house\_load\_from\_battery[t] / \eta_{dis} $$

---

SoC forward simulation

For each slot:

$$ soc\_after\_kwh[t] = soc\_before\_kwh[t] + charge\_stored[t] - battery\_removed[t] $$

SoC bounds

$$ soc\_after\_kwh[t] \in [min\_soc\_kwh, max\_soc\_kwh] $$

Where:

$$ min\_soc\_kwh = rated\_kwh \cdot \frac{end\_of\_discharge\_soc\_pct}{100} $$

$$ max\_soc\_kwh = rated\_kwh \cdot \frac{battery\_max\_soc\_pct}{100} $$

$$ usable\_kwh = max\_soc\_kwh - min\_soc\_kwh $$

Power limits (per-slot energy caps)

$$ charge\_stored[t] \leq max\_charge\_per\_slot = \frac{max\_charge\_power\_w}{1000} \cdot \frac{interval\_minutes}{60} $$

$$ battery\_removed[t] \leq max\_discharge\_per\_slot = \frac{max\_discharge\_power\_w}{1000} \cdot \frac{interval\_minutes}{60} $$

When $max\_discharge\_power\_w$ is None (unlimited), the per-slot cap is relaxed to $usable\_kwh$.

---

EV charger energy model

AC appliance model

The EV charger draws from the AC bus — it never draws from the house battery:

$$ ev\_ac\_load[t] = \frac{ev\_battery\_charged[t]}{charger\_efficiency} $$

Where $charger\_efficiency = charger\_efficiency\_pct / 100$.

Three-field EV load model

Field	Formula	Meaning
ev_planned_load_kwh	Sum of EV AC loads NOT in house load	Added to net consumption
ev_accounted_load_kwh	Sum of EV AC loads already in house load	NOT added to net consumption
ev_total_planned_load_kwh	ev_planned + ev_accounted	Total EV activity (diagnostics)

Net surplus filtering

The EV planner selects slots using net consumption after house load:

$$ slot\_net\_surplus[t] = \max(-estimated\_net\_consumption[t], 0) $$

Where $estimated\_net\_consumption[t] = house\_load[t] - \mathrm{pv}[t]$.

Historical note (PR #397, #406)

Before PR #406, the EV planner incorrectly used estimated_net_consumption which was 0.0 at EV planning time (net consumption had not been populated yet). This meant every slot appeared to have zero surplus, so the EV was always scheduled as grid-import.

PR #397 fixed this by deriving surplus directly from raw PV and house load fields. PR #406 improved it further by running populate_net_consumption before EV planning so that PV confidence decay is automatically applied.

---

Round-trip efficiency

$$ \eta_{roundtrip} = \eta_{chg} \cdot \eta_{dis} $$

$$ roundtrip\_loss = 1 - \eta_{roundtrip} $$

Example

With 97 % charge and 97 % discharge efficiency:

$$ \eta_{roundtrip} = 0.97 \cdot 0.97 = 0.9409 $$

$$ \mathrm{loss} = 1 - 0.9409 = 0.0591 \mathrm{ (5.91 %)} $$

Charging 10 kWh from the grid:

• Grid draws: $10 / 0.97 = 10.31$ kWh
• Battery stores: 10 kWh
• Discharging 10 kWh battery energy:
• House receives: $10 \cdot 0.97 = 9.7$ kWh
• Net round-trip: 10.31 kWh grid → 9.7 kWh house = 5.91 % loss

---

PV confidence decay

For multi-day horizons, PV estimates are discounted:

Day offset	Decay factor
0 (today)	1.00
1 (tomorrow)	0.90
2 (day after)	0.80

$$ pv\_decayed[t] = pv\_raw[t] \cdot decay\_factor[day_offset] $$

Prices are not decayed because spot-market prices are typically firm by mid-day.

---

Energy unit conventions

Conversion	Formula
W → kW	$\mathrm{kW} = \mathrm{W} / 1000$
Wh → kWh	$\mathrm{kWh} = \mathrm{Wh} / 1000$
Power → energy	$\mathrm{kWh} = \mathrm{kW} \times \mathrm{hours}$
Accumulated energy	$\mathrm{kWh} = power\_W \times \frac{elapsed\_seconds}{3600} / 1000$

All internal planner calculations use kWh for energy and kW for power. Power limits from Huawei Solar are received in Watts and converted at the planner boundary.