Dynamic Energy

In a system dynamics model, a stock that changes is said to have kinetic energy. The faster the stock changes, the more kinetic energy it has. This energy can be termed dynamic energy as it is not mechanical but connected with how a stock changes its value. Figure 1 shows the graph of a stock called Infected and its kinetic energy. The kinetic energy is highest when the stock is changing the most. It is half the net flow squared.

Figure 1: Infected stock and its kinetic energy

Work Done

In the Newtonian Interpretative Framework, feedback loops act like forces. Forces do “work” on objects. In system dynamics, feedback loops do “work” on stocks changing their kinetic energy. For example, in the SI model (Figure 2), energy flows into the infected stock through R1, the reinforcing loop. The balance loop, B1, removes energy. Energy removal is delayed compared with injection, hence the rise in kinetic energy of Infected (Figure 1)1.

Figure 2: SI model with energy flows. Loop R injects energy. Loop B removes it.

Energy is conserved in each stock flow subsystem. In the SI model, the work done by the two loops on Infected is balanced by its kinetic energy. Figure 3 shows the energy injected by R1 rises first. Initially, most of this energy enters the kinetic energy (displayed negative, “- change in KE”, to indicate it is output). Later, more energy is removed by B1 so that kinetic energy falls to zero and the system stabilises.

Figure 3: Work done by feedback loops on infected compared with its kinetic energy (displayed as negative).

First-order reinforcing loops are energy sources. First-order balancing loops are energy sinks.


The power of a loop is the rate of change of energy. The powers of the two loops on the SI model are shown in Figure 4. The energy sink, B1, lags the energy source, R1. The excess enters the infected’s kinetic energy. When the two powers are numerically equal, the infected curve is at its inflexion point (Figure 1). The loop with the greater power is the dominant loop. Thus, the inflexion point is where loop dominance changes. This dominance analysis is the same as the loop impact method, as power is proportional to loop impact.

Figure 4: The two loops’ power (rate of change of energy) compared.

Second-Order Balancing Loop

Second-order balancing loops can be thought of as energy exchange between two stock-flow systems. The Workforce-Inventory model has a single second-order balancing loop, B2 (Figure 5). The system is driven by the exogenous demand, D1, a stop increase in demand at time 5, and the delayed expected demand, E1. The stocks oscillate once demand increases and eventually stabilise with the workforce at a higher equilibrium (Figure 6)2.

Figure 5: Workforce-Inventory system with a second-order balancing loop, B2, exchanging energy between stocks.
Figure 6: Damped oscillations in the Workforce-Inventory system.

D1 injects energy into Inventory, which is transferred by B2 into Workforce (Figure 5). E1 also injects energy into Workforce. All the energy in the workforce stock (in the form of kinetic energy) is removed by B1.

Figure 7 shows the work done by the loops on the system (top) and the two main stocks (bottom). Work done is an accumulation. Here, it has been measured from time 0 until equilibrium is reached. 1.56 units of energy entered Inventory from D1 (bottom of diagram)3. This energy is removed from Inventory and deposited in Workforce by B2, 1.56 units. E1 injects 0.59 units into Workforce directly. The sum of these two 1.56 + 0.59 = 2.15 units has been removed by B1, hence stability. 36.4% of the energy is input through D1, and 13.6% through E1. It is all removed by B1, 50% (top diagram).

Figure 7: Work done by loops on the Workforce-Inventory system (top) and the stocks (bottom).

Second-order balancing loops exchange energy between stock-flow systems in a conserved manner. Figure 8 displays the energy flows on the stock-flow diagram, with the green double-headed arrow indicating exchange by the second-order balancing loop B2.

Figure 8:


Feedback loops and exogenous influences either inject energy into a stock-flow system, remove energy or exchange energy.

  • First-order reinforcing loops inject energy.
  • First-order balancing loops remove energy.
  • Second-order balancing loops exchange energy in a conserved fashion.
  • Second-order reinforcing loops inject energy – exchange is not conserved.
  • All higher-order loops inject energy.
  • Exogenous effects inject energy, unless they fade or are removed.

The only stabilising structure is the first-order balancing loop, whose presence is necessary for stability and equilibrium.

References and Notes

  1. The SI model is discussed in Hayward J. & Roach P.A. (2023). A Comparison of Loop Dominance Methods: Measures and Meaning. Presented at the 41st International Conference of the System Dynamics Society, Chicago, July 2023. ↩︎
  2. The Workforce-Inventory model is discussed in 1. above and in the journal paper: Hayward J. & Roach P.A. (2024). A Comparison of Loop Dominance Methods: Measures and Meaning. System Dynamics Review, in press. https://doi.org/10.1002/sdr.1757 Open Access. ↩︎
  3. The units of energy here are (workers per month)^2. The energy units on Inventory have been harmonised with those of Workforce. ↩︎