Dissipative Energy Flow (DEF) Overview

• DEF is an energy accounting method that distinguishes between energy injection, storage, and dissipation in systems ranging from power networks to quantum and ferromagnetic channels.
• In power systems, DEF employs quadratic supply-rate transformations to pinpoint oscillatory energy sources and sinks, aiding in accurate disturbance localization.
• Extensions to FACTS devices, turbulent flows, and Floquet quantum systems show DEF’s practical role in revealing energy pathways and guiding control strategies.

Dissipative Energy Flow (DEF) is a context-dependent term for formulations that track how supplied or oscillatory energy is partitioned among transfer pathways, storage-like contributions, and dissipative sinks. In electric-power systems, DEF most commonly denotes the dissipating energy flow method for oscillation-source localization, built from voltage–current measurements and interpreted through passivity and quadratic supply rates (Chevalier et al., 2019). In related but nonidentical settings, a DEF viewpoint appears as global energy-flux accounting in controlled wall turbulence (Gatti et al., 2018) and as the steady-state energy current from a periodically driven quantum system into a bath (Langemeyer et al., 2014). By contrast, in spin-hydrodynamic ferromagnetics the same acronym usually denotes dissipative exchange flow, not dissipative energy flow (Estiphanos et al., 2024).

1. Terminological scope and conceptual core

Across the literatures represented here, DEF is not a single universal formalism. Its common structure is an energy accounting in which directionality matters: a component, mode, or pathway is characterized by whether it injects energy, absorbs it, or redistributes it. In the power-systems literature, that directional interpretation is explicit: DEF is used to determine whether a network element is acting as a source or a sink of oscillatory energy, and in modern extensions it is also used to infer propagation paths across branches and meshed networks (Chatterjee et al., 15 Aug 2025).

The strongest canonical meaning is therefore the power-system one. There, DEF is an online or measurement-oriented diagnostic for sustained oscillations, especially low-frequency forced oscillations and, more recently, sub- and super-synchronous control interactions. Its central observable is a cumulative energy-like quantity whose slope indicates whether energy is being injected into the oscillatory network or extracted from it (Chevalier et al., 2019).

Other uses are structurally analogous but physically distinct. In fully developed turbulent channel flow under control, the relevant object is a global partition of supplied power into mean-flow transport, transfer to turbulence, and viscous dissipation; the 2018 channel-flow study is highly relevant to a Dissipative Energy Flow viewpoint, even though it does not use the term “DEF” (Gatti et al., 2018). In periodically driven open quantum systems, the analogous quantity is the steady-state energy dissipation rate into the bath, resolved into Floquet transition channels (Langemeyer et al., 2014). These usages share an emphasis on directed energy throughput, but not a common notation or application domain.

2. Power-system DEF as a quadratic energy and passivity construction

In the low-frequency power-system formulation, the DEF quantity can be written as

$$W_{\rm DE}=\int {\rm Im}\{I^*\,{\rm d}V\} =\int \left(P\dot\theta + Q\frac{\dot V}{V}\right)\,{\rm d}t ,$$

and for small perturbations around $V_0$,

$$W_{\rm DE}\approx \int \left(P\dot\theta + Q\frac{\dot V}{V_0}\right)\,{\rm d}t .$$

This time-domain expression is the classical dissipating-energy quantity used in source-location methods (Chevalier et al., 2019).

The same construction admits a frequency-domain representation at forcing frequency $\Omega_d$. For rectangular perturbation phasors $\tilde{\bf V}(\Omega_d)$ and $\tilde{\bf I}(\Omega_d)$, the dissipating power is

$$P^\star = {\rm Re}\{\tilde{\bf V}(\Omega_d)^\dagger {\bf M}\tilde{\bf I}(\Omega_d)\}, \qquad {\bf M}= \begin{bmatrix} 0 & j\ -j & 0 \end{bmatrix}.$$

Equivalent formulations use the quadratic matrix

${\mathcal Q}_b= \begin{bmatrix} 0 & -\frac{1}{j\Omega}\[4pt] \frac{1}{j\Omega} & 0 \end{bmatrix},$

applied bus by bus as a block-diagonal transformation (Chevalier et al., 2019).

A central theoretical result is that DEF is not merely heuristic energy tracing. It is a particular quadratic supply-rate or passivity transformation applied to incremental voltage–current perturbations. For a component with admittance relation $\tilde{\bf I}=\mathcal Y \tilde{\bf V}$, passivity under the DEF transformation is equivalent to

$${\bf M}\mathcal Y + ({\bf M}\mathcal Y)^\dagger \succeq 0 .$$

At the network level, the transformed energy balance follows from Tellegen-type conservation: if non-source elements are passive or lossless under the transformation, the source must appear as the non-passive injector (Chevalier et al., 2019).

The same framework also identifies a fundamental limitation. There exists no passivity transformation, i.e. no quadratic energy function, which can simultaneously render all components of a lossy classical power system passive. Resistive transmission elements are the principal obstruction. This leads directly to the characteristic DEF failure mode in lossy networks: lines and resistive loads can appear as positive contributors to dissipating energy, creating false positives or masking the true source (Chevalier et al., 2019).

That limitation motivates a simulation-free screening criterion based on the dynamic Ward equivalent seen from a candidate source bus. If the eigenvalues of the transformed source-bus equivalent are both positive, DEF is predicted to succeed; if both are negative, it is predicted to fail; if they are mixed-sign, DEF is predicted to be unreliable. This eigenvalue test is one of the clearest outcomes of the passivity interpretation because it converts an empirical method into a frequency- and topology-dependent diagnostic (Chevalier et al., 2019).

3. Extensions to FACTS devices and multi-frequency SSCI diagnosis

Extension of DEF to controller-dominated networks requires explicit treatment of path dependence and frequency selectivity. For FACTS devices, the key question is whether the device’s transient-energy expression contains a path-dependent term. If it does, the device can behave as a source or sink of oscillation energy under DEF; if it does not, the device is DEF-neutral (Chatterjee et al., 2021).

For a TCSC modeled as a variable series susceptance $V_0$0, the device energy decomposes into a path-independent part and a dissipating-energy-related part,

$V_0$1

If $V_0$2, or if $V_0$3 is an explicit algebraic function of $V_0$4, the expression remains path-independent and the TCSC is neither a source nor a sink of oscillation energy. If instead $V_0$5 is generated by a dynamic controller,

$V_0$6

then $V_0$7 becomes path-dependent and the device can switch between sink and source behavior depending on controller tuning; in the reported case study, reversing the sign of $V_0$8 reverses the DEF sign (Chatterjee et al., 2021).

For a lossless STATCOM, the dissipating-energy-related term is

$V_0$9

Under constant current control, this vanishes and the STATCOM is DEF-neutral. Under reactive-power control with $$W_{\rm DE}\approx \int \left(P\dot\theta + Q\frac{\dot V}{V_0}\right)\,{\rm d}t .$$0 droop and a PI current-reference generator, $$W_{\rm DE}\approx \int \left(P\dot\theta + Q\frac{\dot V}{V_0}\right)\,{\rm d}t .$$1 becomes path-dependent; tuning the droop coefficient $$W_{\rm DE}\approx \int \left(P\dot\theta + Q\frac{\dot V}{V_0}\right)\,{\rm d}t .$$2 can transform the STATCOM from sink to neutral to source (Chatterjee et al., 2021).

In inverter-rich systems, the extension to SSCI diagnosis is mode-specific rather than device-specific. The modern formulation starts from the same DEF quantity,

$$W_{\rm DE}\approx \int \left(P\dot\theta + Q\frac{\dot V}{V_0}\right)\,{\rm d}t .$$3

but rewrites it in the synchronous $$W_{\rm DE}\approx \int \left(P\dot\theta + Q\frac{\dot V}{V_0}\right)\,{\rm d}t .$$4 frame as

$$W_{\rm DE}\approx \int \left(P\dot\theta + Q\frac{\dot V}{V_0}\right)\,{\rm d}t .$$5

Three-phase voltage and current measurements are transformed to $$W_{\rm DE}\approx \int \left(P\dot\theta + Q\frac{\dot V}{V_0}\right)\,{\rm d}t .$$6, dominant oscillatory modes are identified spectrally, and each mode is isolated by bandpass filtering. The implemented discrete formula is

$$W_{\rm DE}\approx \int \left(P\dot\theta + Q\frac{\dot V}{V_0}\right)\,{\rm d}t .$$7

In this node-injection convention, $$W_{\rm DE}\approx \int \left(P\dot\theta + Q\frac{\dot V}{V_0}\right)\,{\rm d}t .$$8 indicates an oscillation source and $$W_{\rm DE}\approx \int \left(P\dot\theta + Q\frac{\dot V}{V_0}\right)\,{\rm d}t .$$9 indicates a sink (Chatterjee et al., 15 Aug 2025).

The principal methodological advance is frequency selectivity. In the EMT case study, the same synchronous generator is a source at $\Omega_d$0 Hz and a sink at $\Omega_d$1 Hz, while a wind generator is the sole source at $\Omega_d$2 Hz. This result rules out any frequency-independent notion of “the” source role of a device and shows that broadband DEF can obscure source–sink attribution when multiple sidebands coexist (Chatterjee et al., 15 Aug 2025).

4. DEF-style global power accounting in wall turbulence

A rigorous DEF-like framework appears in turbulent channel flow under Constant Power Input (CPI). There the total supplied power per unit wetted area is fixed, so that

$\Omega_d$3

where $\Omega_d$4 is pumping power, $\Omega_d$5 control power, $\Omega_d$6 mean dissipation, and $\Omega_d$7 turbulent dissipation (Gatti et al., 2018).

The central contribution is an extended Reynolds decomposition,

$\Omega_d$8

where $\Omega_d$9 is the laminar parabolic profile with the same bulk velocity as the actual flow, and $\tilde{\bf V}(\Omega_d)$0 is the deviation from that laminar reference. This yields an exact dissipation split

$\tilde{\bf V}(\Omega_d)$1

with vanishing cross term, and a production split

$\tilde{\bf V}(\Omega_d)$2

The reduced mean-kinetic-energy balance becomes

$\tilde{\bf V}(\Omega_d)$3

This identifies $\tilde{\bf V}(\Omega_d)$4 as the dissipation tied to the laminar profile that carries useful bulk transport, and $\tilde{\bf V}(\Omega_d)$5 as the part of pumping power diverted into turbulence (Gatti et al., 2018).

Two wall-normal Reynolds-stress integrals control the global picture: $\tilde{\bf V}(\Omega_d)$6 At fixed total power, $\tilde{\bf V}(\Omega_d)$7 determines the bulk flow rate $\tilde{\bf V}(\Omega_d)$8, while $\tilde{\bf V}(\Omega_d)$9 determines how losses split between deviation dissipation $\tilde{\bf I}(\Omega_d)$0 and turbulent dissipation $\tilde{\bf I}(\Omega_d)$1. The key caution is that successful drag-reducing control does not imply a universal decrease in either $\tilde{\bf I}(\Omega_d)$2 or $\tilde{\bf I}(\Omega_d)$3. In the DNS results, opposition control increases $\tilde{\bf I}(\Omega_d)$4 and decreases $\tilde{\bf I}(\Omega_d)$5, whereas oscillating-wall control decreases $\tilde{\bf I}(\Omega_d)$6 and increases $\tilde{\bf I}(\Omega_d)$7 (Gatti et al., 2018).

The control-relevant waste measure is therefore not $\tilde{\bf I}(\Omega_d)$8 alone and not $\tilde{\bf I}(\Omega_d)$9 alone, but

$$P^\star = {\rm Re}\{\tilde{\bf V}(\Omega_d)^\dagger {\bf M}\tilde{\bf I}(\Omega_d)\}, \qquad {\bf M}= \begin{bmatrix} 0 & j\ -j & 0 \end{bmatrix}.$$0

described as the fraction of total power not used to produce flow rate. This suggests a DEF interpretation in which the physically meaningful target is minimization of turbulence-induced waste relative to the laminar transport baseline, rather than minimization of a single dissipation channel (Gatti et al., 2018).

5. Quantum DEF as bath-directed steady-state energy current

In periodically driven open quantum systems, dissipative energy flow is the steady-state energy current from the driven system into its heat bath. For a system with periodic Hamiltonian $$P^\star = {\rm Re}\{\tilde{\bf V}(\Omega_d)^\dagger {\bf M}\tilde{\bf I}(\Omega_d)\}, \qquad {\bf M}= \begin{bmatrix} 0 & j\ -j & 0 \end{bmatrix}.$$1, Floquet states $$P^\star = {\rm Re}\{\tilde{\bf V}(\Omega_d)^\dagger {\bf M}\tilde{\bf I}(\Omega_d)\}, \qquad {\bf M}= \begin{bmatrix} 0 & j\ -j & 0 \end{bmatrix}.$$2, and bath-induced partial transition rates

$$P^\star = {\rm Re}\{\tilde{\bf V}(\Omega_d)^\dagger {\bf M}\tilde{\bf I}(\Omega_d)\}, \qquad {\bf M}= \begin{bmatrix} 0 & j\ -j & 0 \end{bmatrix}.$$3

the total dissipation rate is

$$P^\star = {\rm Re}\{\tilde{\bf V}(\Omega_d)^\dagger {\bf M}\tilde{\bf I}(\Omega_d)\}, \qquad {\bf M}= \begin{bmatrix} 0 & j\ -j & 0 \end{bmatrix}.$$4

Positive $$P^\star = {\rm Re}\{\tilde{\bf V}(\Omega_d)^\dagger {\bf M}\tilde{\bf I}(\Omega_d)\}, \qquad {\bf M}= \begin{bmatrix} 0 & j\ -j & 0 \end{bmatrix}.$$5 means net dissipation into the bath; in the periodic steady state this equals the average power supplied by the drive (Langemeyer et al., 2014).

The distinctive feature of the Floquet setting is the decomposition into genuine transitions and pseudo-transitions. Genuine transitions change the Floquet label, $$P^\star = {\rm Re}\{\tilde{\bf V}(\Omega_d)^\dagger {\bf M}\tilde{\bf I}(\Omega_d)\}, \qquad {\bf M}= \begin{bmatrix} 0 & j\ -j & 0 \end{bmatrix}.$$6 with $$P^\star = {\rm Re}\{\tilde{\bf V}(\Omega_d)^\dagger {\bf M}\tilde{\bf I}(\Omega_d)\}, \qquad {\bf M}= \begin{bmatrix} 0 & j\ -j & 0 \end{bmatrix}.$$7. Pseudo-transitions preserve the Floquet state, $$P^\star = {\rm Re}\{\tilde{\bf V}(\Omega_d)^\dagger {\bf M}\tilde{\bf I}(\Omega_d)\}, \qquad {\bf M}= \begin{bmatrix} 0 & j\ -j & 0 \end{bmatrix}.$$8, while exchanging $$P^\star = {\rm Re}\{\tilde{\bf V}(\Omega_d)^\dagger {\bf M}\tilde{\bf I}(\Omega_d)\}, \qquad {\bf M}= \begin{bmatrix} 0 & j\ -j & 0 \end{bmatrix}.$$9 with the drive–bath channel. Their contribution is

${\mathcal Q}_b= \begin{bmatrix} 0 & -\frac{1}{j\Omega}\[4pt] \frac{1}{j\Omega} & 0 \end{bmatrix},$0

This establishes that steady dissipation need not be tied to changes in Floquet-state occupation: a system can dissipate energy continuously even when its occupation probabilities are stationary (Langemeyer et al., 2014).

The analytically solvable examples sharpen that point. For the linearly forced harmonic oscillator, ${\mathcal Q}_b= \begin{bmatrix} 0 & -\frac{1}{j\Omega}\[4pt] \frac{1}{j\Omega} & 0 \end{bmatrix},$1, so the entire dissipative energy flow is carried by pseudo-transitions. For the circularly driven two-level system, both genuine and pseudo-transitions contribute, and at zero bath temperature the system does not necessarily relax to a single Floquet state. This is a major departure from equilibrium intuition and shows that DEF in driven quantum thermodynamics is controlled by ${\mathcal Q}_b= \begin{bmatrix} 0 & -\frac{1}{j\Omega}\[4pt] \frac{1}{j\Omega} & 0 \end{bmatrix},$2-resolved channels rather than by populations alone (Langemeyer et al., 2014).

6. Terminological divergence: DEF as dissipative exchange flow

In one-dimensional ferromagnetic channels, DEF refers to dissipative exchange flow, not dissipative energy flow. The governing model is the dimensionless Landau–Lifshitz equation

${\mathcal Q}_b= \begin{bmatrix} 0 & -\frac{1}{j\Omega}\[4pt] \frac{1}{j\Omega} & 0 \end{bmatrix},$3

with hydrodynamic variables

${\mathcal Q}_b= \begin{bmatrix} 0 & -\frac{1}{j\Omega}\[4pt] \frac{1}{j\Omega} & 0 \end{bmatrix},$4

In this literature, ${\mathcal Q}_b= \begin{bmatrix} 0 & -\frac{1}{j\Omega}\[4pt] \frac{1}{j\Omega} & 0 \end{bmatrix},$5 is a phase gradient and spin-current analog, and ${\mathcal Q}_b= \begin{bmatrix} 0 & -\frac{1}{j\Omega}\[4pt] \frac{1}{j\Omega} & 0 \end{bmatrix},$6 is the natural hydrodynamic flux term (Estiphanos et al., 2024).

The low-injection limit yields the linear dissipative exchange flow or spin-superfluid solution,

${\mathcal Q}_b= \begin{bmatrix} 0 & -\frac{1}{j\Omega}\[4pt] \frac{1}{j\Omega} & 0 \end{bmatrix},$7

At stronger injection, a contact soliton forms near the injector, producing a contact-soliton dissipative exchange flow. In the moderate finite-width regime studied numerically, two metastable solitons can coexist within an injection region; when two such regions interact, the DEF between them induces a steady-state cycle of soliton ejection and nucleation whose handedness depends on the relative signs of the spin injections (Estiphanos et al., 2024).

This usage is conceptually distinct from dissipative energy-flow diagnostics in power systems, turbulence, or quantum thermodynamics. The shared acronym does not imply a shared theory. The ferromagnetic DEF is a spin-transport state, whereas the other DEF formulations are energy-accounting constructs concerned with source–sink attribution, transport efficiency, or dissipation rates (Estiphanos et al., 2024).

7. Unifying themes and persistent limitations

Despite domain differences, several themes recur. First, DEF is most informative when it resolves direction rather than merely magnitude. Power-system DEF locates sources, sinks, and branchwise propagation; turbulent-channel DEF-like accounting distinguishes useful transport-supporting dissipation from turbulence-induced waste; quantum DEF resolves bath-directed flow by transition channel; and the ferromagnetic usage resolves spin-current transport and soliton motion (Chatterjee et al., 15 Aug 2025).

Second, scalar dissipation measures are often insufficient. In controlled wall turbulence, neither ${\mathcal Q}_b= \begin{bmatrix} 0 & -\frac{1}{j\Omega}\[4pt] \frac{1}{j\Omega} & 0 \end{bmatrix},$8 nor ${\mathcal Q}_b= \begin{bmatrix} 0 & -\frac{1}{j\Omega}\[4pt] \frac{1}{j\Omega} & 0 \end{bmatrix},$9 alone is a universal control objective (Gatti et al., 2018). In Floquet thermodynamics, occupations alone do not determine the dissipation rate because pseudo-transitions matter (Langemeyer et al., 2014). In power systems, positive DEF on a device does not necessarily identify the initiating disturbance source, especially in lossy or controller-dominated networks (Chevalier et al., 2019).

Third, sign interpretation depends on the bookkeeping convention. Node-injection SSCI formulations define positive DEF slope as outward oscillatory-energy injection by the element (Chatterjee et al., 15 Aug 2025). Device-centric FACTS derivations define $\tilde{\bf I}=\mathcal Y \tilde{\bf V}$0 into the device, so source behavior corresponds to $\tilde{\bf I}=\mathcal Y \tilde{\bf V}$1 and sink behavior to $\tilde{\bf I}=\mathcal Y \tilde{\bf V}$2 (Chatterjee et al., 2021). Any comparison across studies therefore requires careful attention to reference direction and variable definition.

A plausible implication is that DEF is best understood not as a single metric, but as a family of energy-flow formalisms whose value lies in exposing the topology of energetic causation: which pathways support useful transport, which pathways dissipate energy, and which components inject or absorb oscillatory content. Within each field, its practical reliability depends on how completely the chosen formulation captures the physically relevant channels.