Distributed Generation: Mathematical Models for Many Small Sources

Power systems were built around a few large generators. Today they connect millions of small ones: rooftop solar, small wind turbines, combined heat and power units, batteries and electric vehicles. Each of these sources is too small to matter on its own, and each behaves differently over the day and the year. Together they can rival a large power plant, or leave a gap when they all fall short at the same moment. Planning such a system means answering questions that do not arise for a single plant. How much firm capacity do ten thousand small generators provide? How much does their combined output vary? How much reserve is still needed? The answers depend less on the size of each unit than on how their outputs are correlated in time.

The Established Approaches

Distributed generation is defined by its connection to the distribution network close to consumers, rather than by technology or size [1]. Several tools describe what many small units add up to.

Aggregation statistics. For N units with mean output μ and standard deviation s, the mean of the total is N·μ. If the units vary independently, the standard deviation of the total grows only with √N, so the relative variability falls as 1/√N. If they are fully correlated, as rooftop solar panels in one region largely are, the standard deviation grows with N and aggregation brings no smoothing.

Capacity credit. The effective load-carrying capability measures how much additional load a generator allows the system to serve at the same reliability level [2]. Firm, dispatchable plants have a high capacity credit; weather-driven sources have a low one that falls further as their share grows.

Capacity factor. The ratio of average output to rated output captures how much of the time a source effectively delivers. Typical values for solar photovoltaics lie well below 30 per cent, depending on location.

Virtual power plants. Many small units can be coordinated through communication and control so that they act on the market and in system operation like one larger plant [3].

These models handle the sources that exist today well. Their central assumption is that small generators are variable and often correlated, and that their contribution must be firmed up by storage, reserve plants or the grid.

The Open Question

The established models describe variable, weather-driven sources. They do not cover sources that convert continuously available ambient energy, whose output is not tied to sunlight or wind and whose fluctuations between units are largely independent. For such sources, aggregation should smooth the total far more than for solar, and capacity credit should behave differently. To model this, the output of each unit must be expressed in terms of its own inputs and efficiency, so that the system-level statistics can be built from physically grounded unit models rather than from weather data.

A Generalized Approach

A generalized approach to this problem is represented by the Schubart Master Formula, which provides the unit model from which the system total is built:

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

For N units, the system output is the sum of the unit outputs, P_total(t) = Σ P_i(t). The derivation, notation and open questions are published at pt-schubart.com. The unit 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.
The smoothing effect depends on how independent the units’ fluctuations really are. This must be measured, not assumed.
Network losses, inverter losses and local voltage limits are outside the formula and are added at system level.
Distributed units work alongside the grid. The grid remains the connection, reserve and balancing layer.

Worked Example

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

Take 1,000 units, each with an average output of 1 kW and a standard deviation of 0.1 kW over short time scales.

Independent fluctuations: total mean 1,000 kW = 1 MW; standard deviation 0.1 kW × √1,000 ≈ 3.2 kW, or about 0.3 per cent of the total.
Fully correlated fluctuations: total mean 1 MW; standard deviation 0.1 kW × 1,000 = 100 kW, or 10 per cent of the total.

The same units, with the same individual behaviour, give a total that is either nearly flat or noticeably variable, depending only on correlation. For continuously operating units whose inputs are local, the first case is the design goal; for solar panels on neighbouring roofs, the second case is closer to reality. The unit model P_i(t) is what allows the correlation to be traced back to physical inputs.

Where It Leads

At the scale of a village, campus or island, many units joined by a local grid form a decentralized energy cell; see Microgrids and Local Energy Balance. The full chain from material to grid is described in From Microscopic Conversion to Grid Scale and From Sensor to Grid.

References

T. Ackermann, G. Andersson and L. Söder, “Distributed generation: a definition,” Electric Power Systems Research 57, 195–204 (2001).
L. L. Garver, “Effective load carrying capability of generating units,” IEEE Transactions on Power Apparatus and Systems PAS-85, 910–919 (1966).
D. Pudjianto, C. Ramsay and G. Strbac, “Virtual power plant and system integration of distributed energy resources,” IET Renewable Power Generation 1, 10–16 (2007).
N. Hatziargyriou, H. Asano, R. Iravani and C. Marnay, “Microgrids,” IEEE Power and Energy Magazine 5(4), 78–94 (2007).
Schubart Master Formula: derivation, notation and open questions, pt-schubart.com.

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