diff --git a/flexmeasures/data/models/planning/tests/test_solver.py b/flexmeasures/data/models/planning/tests/test_solver.py
index fb563f020c..f71c9ac060 100644
--- a/flexmeasures/data/models/planning/tests/test_solver.py
+++ b/flexmeasures/data/models/planning/tests/test_solver.py
@@ -2986,3 +2986,95 @@ def run_sequential_scheduler():
     assert total_cost_all_devices == sum(
         expected_cost[1] for expected_cost in expected_costs
     ), "Total cost mismatch."
+
+
+@pytest.mark.parametrize(
+    ("capacity_in_w"),
+    [
+        10,
+        1.1,
+        1.01,
+        1.001,
+        1.0001,
+        1.00001,
+        1.000001,
+        1.0000001,
+        1.00000001,
+        1.000000001,
+        1.0000000001,
+        1.00000000001,
+        1.000000000001,
+        1.0000000000001,
+        1.00000000000001,
+        1.000000000000001,
+        pytest.param(
+            1.0000000000000001, marks=pytest.mark.xfail(reason="some rounding bug")
+        ),
+        pytest.param(
+            1.00000000000000001, marks=pytest.mark.xfail(reason="some rounding bug")
+        ),
+        pytest.param(
+            1.000000000000000001, marks=pytest.mark.xfail(reason="some rounding bug")
+        ),
+        pytest.param(
+            1.0000000000000000001, marks=pytest.mark.xfail(reason="some rounding bug")
+        ),
+        pytest.param(1, marks=pytest.mark.xfail(reason="some rounding bug")),
+    ],
+)
+def test_battery_kW(
+    add_battery_assets,
+    db,
+    capacity_in_w,
+):
+    """
+    Scheduling a very small battery of 1Wh (1e-6 MWh) with a power capacity of 1W with a constant
+    usage of 1W.
+    """
+
+    epex_da, battery = get_sensors_from_db(
+        db,
+        add_battery_assets,
+        battery_name="Test battery",
+        power_sensor_name="power (kW)",
+    )
+    tz = pytz.timezone("Europe/Amsterdam")
+    start = tz.localize(datetime(2015, 1, 1))
+    end = tz.localize(datetime(2015, 1, 2))
+    resolution = timedelta(minutes=15)
+    soc_max = capacity_in_w  # assumes 1 hour to fully charge
+    soc_usage_in_w = capacity_in_w
+    device_capacity_in_w = capacity_in_w
+
+    scheduler: Scheduler = StorageScheduler(
+        battery,
+        start,
+        end,
+        resolution,
+        flex_model={
+            "soc-at-start": "0 kWh",
+            "soc-max": f"{soc_max} Wh",
+            "soc-min": "0.0 kWh",
+            "soc-usage": [f"{soc_usage_in_w} W"],
+            "charging-efficiency": 1,
+            "discharging-efficiency": 1,
+            "storage-efficiency": 1,
+            "power-capacity": f"{device_capacity_in_w} W",
+            "consumption-capacity": f"{device_capacity_in_w} W",
+            "production-capacity": f"{device_capacity_in_w} W",
+        },
+        flex_context={
+            "site-power-capacity": "1 MW",
+            "consumption-price": "100 EUR/MWh",
+            "production-price": "90 EUR/MWh",
+        },
+    )
+    schedule = scheduler.compute()
+
+    # Check if constraints were met
+    soc_usage = (
+        pd.Series(soc_usage_in_w / 1000, index=schedule.index)
+        * resolution
+        / timedelta(hours=1)
+    )
+    check_constraints(battery, schedule, 0, soc_usage=soc_usage)
diff --git a/flexmeasures/data/models/planning/tests/utils.py b/flexmeasures/data/models/planning/tests/utils.py
index 9c270dd9ec..c16ae4e4c8 100644
--- a/flexmeasures/data/models/planning/tests/utils.py
+++ b/flexmeasures/data/models/planning/tests/utils.py
@@ -10,6 +10,7 @@ def check_constraints(
     sensor: Sensor,
     schedule: pd.Series,
     soc_at_start: float,
+    soc_usage: pd.Series | float = 0,
     roundtrip_efficiency: float = 1,
     storage_efficiency: float = 1,
     tolerance: float = 0.00001,
@@ -22,6 +23,8 @@ def check_constraints(
         storage_efficiency=storage_efficiency,
         decimal_precision=6,
     )
+    soc_usage = soc_usage.reindex(soc_schedule.index).shift(1).fillna(0).cumsum()
+    soc_schedule -= soc_usage  # todo: move into integrate_time_series
     with pd.option_context("display.max_rows", None, "display.max_columns", 3):
         print(soc_schedule)
     capacity = sensor.get_attribute(
@@ -30,9 +33,10 @@ def check_constraints(
     )
     assert min(schedule.values) >= capacity * -1 - tolerance
     assert max(schedule.values) <= capacity + tolerance
+    # breakpoint()
     for soc in soc_schedule.values:
-        assert soc >= sensor.get_attribute("min_soc_in_mwh")
-        assert soc <= sensor.get_attribute("max_soc_in_mwh")
+        assert soc >= sensor.get_attribute("min_soc_in_mwh") - tolerance
+        assert soc <= sensor.get_attribute("max_soc_in_mwh") + tolerance
     return soc_schedule