Microgrids and Local Energy Balance

A microgrid is a small power system that can run on its own: a village, a hospital campus, a military base, an island. It has its own generators, its own storage and its own loads, and it can either connect to the main grid or operate as an island when the main grid is absent or fails. The appeal is resilience and independence. The challenge is balance. In a large grid, thousands of generators and millions of loads average each other out. In a microgrid, every kilowatt has to be accounted for at every moment. If the evening peak arrives and the sun has set, the energy must already be in storage or come from a fuel-driven generator. Sizing such a system means matching load profiles, generation profiles and storage capacity over the hours of the day and the seasons of the year.

The Established Approaches

The microgrid concept was formalised in the early 2000s as a cluster of loads and small sources operating as a single controllable system, able to disconnect from the main grid and keep running [1]. Its operating principles are well established [2].

The energy balance. At every moment, generation plus storage discharge must equal load plus losses plus storage charge. Over a full day, the energy generated must cover the energy consumed plus losses. The power balance sets the size of inverters and generators; the energy balance sets the size of storage.

Control in islanded mode. Without the main grid as reference, the microgrid’s inverters must set voltage and frequency themselves. Droop control lets parallel inverters share load without fast communication by adjusting frequency and voltage slightly with output [3].

Typical generation mix. Most off-grid microgrids today combine solar photovoltaics with batteries and a diesel generator for the hours and days that solar and storage cannot cover. Solar provides cheap energy in daylight; batteries shift it into the evening; diesel covers long cloudy periods, at the cost of fuel logistics and emissions.

These methods are proven in many installations. Their weak point is the mismatch between when weather-driven sources produce and when people consume. Much of a microgrid’s storage and backup capacity exists only to bridge that gap.

The Open Question

Established microgrid design starts from variable sources and adds storage and fuel-based backup until the energy balance holds in the worst week of the year. What is not yet part of standard design is generation that runs continuously, independent of daylight and weather, from ambient inputs converted on site. With such sources, the base load could be covered around the clock, and storage would only need to shift the difference between the base and the peak. Sizing that kind of decentralized energy cell requires a model of each continuous unit that can be checked against conservation of energy.

A Generalized Approach

A generalized approach to this problem is represented by the Schubart Master Formula, which gives the output of each continuous unit from its inputs:

P(t) = η · ∫_V Φ_eff(r, t) · σ_eff(E) dV

The derivation, notation and open questions are published at pt-schubart.com. The model is introduced in Mathematical Models for Multi-Source Energy Harvesting.

Definitions

P(t): instantaneous electrical output, the usable electrical power delivered at time t. Unit: watt (W). Energy over an interval is the time integral of P(t), in joules or watt-hours.
η: total conversion efficiency, dimensionless, between 0 and 1. It combines coupling, resonance, rectification and electrical losses.
Φ_eff(r, t): effective ambient flux density at position r and time t, the local strength of all ambient inputs acting on the material.
σ_eff(E): energy-dependent effective interaction cross-section, how strongly the material couples to an input of energy E.
∫_V: volume integral over the active material.

Together, Φ_eff · σ_eff is the coupled power per unit volume (W/m³), so the integral over V yields watts. How that product divides between the two factors depends on the input channel, so no separate units are assigned to them here.

Assumptions and Boundaries

The Schubart Master Formula is a proposed generalized framework, not an established equation of physics. Its assumptions and limits:

Each unit is an open, non-equilibrium system; its output is bounded by coupled input power: P_out ≤ ΣP_in.
Unit output must be verified by measurement before it is used for sizing; the example below uses illustrative values only.
Inverter, cable and storage losses are outside the formula and must be added to the balance.
The microgrid can stay connected to the main grid, which remains available as connection, reserve and balancing layer.

Worked Example

All values below are illustrative values, chosen to show the method. They are not measured results.

The load. A village has a base load of 15 kW for 18 hours a day and an evening peak of 50 kW for 6 hours. Daily consumption is 15 kW × 18 h + 50 kW × 6 h = 270 kWh + 300 kWh = 570 kWh.

Continuous generation. Six continuous units, each with an illustrative output of 4 kW, deliver 24 kW around the clock, or 576 kWh per day. Daily energy balance: 576 kWh ≥ 570 kWh.

Storage. Outside the peak, the units produce 9 kW more than the base load for 18 hours: 162 kWh of surplus. During the peak, they fall 26 kW short for 6 hours: 156 kWh of deficit. A battery of about 160 kWh usable capacity, plus a margin for losses, closes the daily cycle.

Comparison. A solar-based design for the same village would need enough panels to produce 570 kWh in daylight hours alone, and storage for the entire night, typically several hundred kilowatt-hours, plus a diesel generator for cloudy periods. Continuous generation shrinks the storage to what the peak requires.

Where It Leads

A microgrid built this way is a decentralized energy cell: one level in the seven-step architecture from sensors to hybrid infrastructure described in From Sensor to Grid. How many such cells add up at regional scale is the subject of Distributed Generation: Mathematical Models for Many Small Sources, and the scaling from material to grid is set out in From Microscopic Conversion to Grid Scale.

References

R. H. Lasseter, “MicroGrids,” Proceedings of the IEEE Power Engineering Society Winter Meeting 1, 305–308 (2002).
N. Hatziargyriou, H. Asano, R. Iravani and C. Marnay, “Microgrids,” IEEE Power and Energy Magazine 5(4), 78–94 (2007).
M. C. Chandorkar, D. M. Divan and R. Adapa, “Control of parallel connected inverters in standalone AC supply systems,” IEEE Transactions on Industry Applications 29, 136–143 (1993).
T. Ackermann, G. Andersson and L. Söder, “Distributed generation: a definition,” Electric Power Systems Research 57, 195–204 (2001).
Schubart Master Formula: derivation, notation and open questions, pt-schubart.com.

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