Solar Export, Voltage Rise and the Iberian Blackout

0. Problem statement and conventions

In a simplified single-phase model, the load current of a domestic installation closes along the path: line conductor → load → neutral conductor → the secondary network of the distribution transformer. This is a local current path inside a shared, galvanically connected low-voltage network, not a separate isolated charge ensemble belonging to the house.

The distinction is fixed here and held to the end of the article:

\[\text{local current path} \;\neq\; \text{galvanically isolated charge ensemble}.\]

The carriers moving along that path belong to the conducting network itself. They do not arrive from the power station and they do not depart to it.

An apparent paradox follows immediately. If the current of every load closes locally — through its own branch and the shared low-voltage network — then:

1. how does the solar generation of a neighbouring house enter this circuit;
2. what exactly does the meter register, if the algebraic charge transfer across its cross-section over a period is zero;
3. how can a system of millions of local branches and current paths move to systemic collapse within tens of seconds, as happened on the Iberian Peninsula on 28 April 2025.

This article shows that all three questions are resolved by a single distinction: locality of the current balance does not mean field independence. The current balance closes locally; energy crosses boundaries; the electromagnetic regime is coupled system-wide; stability is a property of the whole coupled system, not the sum of its branches.

0.1 Sign convention

For any boundary \(\Gamma\), an outward normal is fixed. Flow directed into the installation under consideration (import) is taken as positive. Export carries a negative sign. All further signs follow from this convention and are nowhere redefined.

0.2 Definition of the boundary

The boundary \(\Gamma\) is a closed surface intersecting all conductors, field channels and material flows connecting the installation to its environment. A pair of measurement probes is not a boundary: in the presence of parasitic capacitive coupling and currents through the shield, the enclosure and the protective conductor, an incomplete boundary makes the balance impossible to close.

0.3 Time-class passport

Every measured or quantitatively compared quantity must carry a statement of its boundary, charge ensemble, time class (peak / rms / mean over a window / event quantity) and averaging window. Symbolic quantities inside derivations are not bound by this requirement: the passport is a condition for correct quantitative comparison, not for the presentation of algebra. Products of quantities of different time classes are not power. A full passport example is given in §7.4.

0.4 Notation

Reactive power is written \(Q_{\mathrm{r}}\) so as not to be confused with charge \(Q\).

1. The central fact: zero charge transfer with non-zero energy transfer

We begin not with the network but with a single cross-section of a single wire.

The local form of charge conservation:

\[\frac{\partial \rho}{\partial t} + \nabla\!\cdot\!\mathbf{J} = 0,\]

and its integral form for a volume \(V\) bounded by the surface \(\Gamma\) with an outward normal:

\[\oint_{\Gamma}\mathbf{J}\cdot d\mathbf{A} + \frac{dQ_{V}}{dt} = 0.\]

This relation holds exactly and always — in a healthy network, in a faulted network, and at the moment of a cascading disconnection.

Take the steady sinusoidal regime. The port voltage is taken as the reference quantity:

\[u(t) = \sqrt{2}\,U\cos\omega t,\]

and the current through the cross-section is

\[i(t) = \sqrt{2}\,I\cos(\omega t – \varphi).\]

Angle convention. \(\varphi\) is the angle by which the current lags the voltage. A positive \(\varphi\) corresponds to lagging current, that is to inductive consumption, and yields positive reactive power. This is the standard power-systems convention, and it is used further in §5, §9 and §11 without redefinition.

The algebraic charge transfer across the cross-section over a period \(T = 2\pi/\omega\):

\[\Delta Q = \int_{0}^{T} i(t)\,dt = \sqrt{2}\,I\int_{0}^{T}\cos(\omega t – \varphi)\,dt = 0.\]

The algebraic charge transfer across the cross-section over a full period is zero. The gross turnover of charge is of course not zero: carriers cross the section in both directions. Nevertheless energy is transferred through that same port. Expanding the instantaneous power:

\[p(t) = u(t)\,i(t) = 2UI\cos\omega t\,\cos(\omega t – \varphi),\]

and by the product-to-sum identity:

\[\boxed{\;p(t) = \underbrace{UI\cos\varphi\,\bigl(1 + \cos 2\omega t\bigr)}_{\text{unidirectional component}} \;+\; \underbrace{UI\sin\varphi\,\sin 2\omega t}_{\text{alternating component}}\;}\]

Both fundamental objects of accounting follow at once:

\[P = \langle p(t)\rangle = UI\cos\varphi, \qquad Q_{\mathrm{r}} = UI\sin\varphi, \qquad \langle Q_{\mathrm{r}}\text{-term}\rangle = 0.\]

The energy that has passed through the port over a window \(\Delta t\):

\[E_{\Gamma,\Delta t} = \int_{\Delta t} u(t)\,i(t)\,dt = \int_{\Delta t} u\,dQ.\]

The conclusion that carries the whole article. Over a period \(\int i\,dt = 0\), yet \(\int u\,i\,dt \neq 0\). An alternating circuit transfers energy precisely by oscillation.

The formulation is fixed in canonical rather than ontological form: zero algebraic charge transfer does not mean zero energy transfer. Knowing the quantity of charge that has passed is by itself insufficient to determine the energy transferred — the potential at which that transfer occurs is also required. Charge is not consumed as fuel; the energy flow of a port is determined by the joint state of the field and the current.

1.1 How small the carrier displacement is

Let us estimate how far a carrier actually moves in a domestic cable. For copper the free-electron concentration is \(n \approx 8.5\times10^{28}\ \mathrm{m^{-3}}\), the elementary charge is \(e = 1.602\times10^{-19}\) C, and the cross-section is \(A = 2.5\ \mathrm{mm^{2}} = 2.5\times10^{-6}\ \mathrm{m^{2}}\). At an rms current of \(I = 10\) A:

\[v_{d,\mathrm{rms}} = \frac{I}{n e A} = \frac{10}{8.5\times10^{28}\cdot 1.602\times10^{-19}\cdot 2.5\times10^{-6}} \approx 2.9\times10^{-4}\ \mathrm{m/s}.\]

The oscillation amplitude at \(f = 50\) Hz, \(\omega = 314\ \mathrm{s^{-1}}\):

\[x_{0} = \frac{\sqrt{2}\,v_{d,\mathrm{rms}}}{\omega} = \frac{1.41\cdot 2.9\times10^{-4}}{314} \approx 1.3\times10^{-6}\ \mathrm{m}.\]

The amplitude of the mean drift component of the electron system under these parameters is of the order of one micrometre.

This is not the trajectory of an individual electron. The microscopic motion of carriers in a metal is incomparably more complex: thermal and Fermi motion, scattering on the lattice, on defects and on phonons. The quantity \(v_{d}\) is the mean directed component of a statistical velocity distribution, and its integral gives the displacement of the centre of the drift component, not the path of a particular particle.

The estimate shows something else, and that is sufficient: macroscopic energy transfer does not require a definite set of electrons to travel from the power station to the consumer.

2. An ordinary house: the local current path and the role of the transformer

Reduce the low-voltage network to a minimum:

              DISTRIBUTION TRANSFORMER
                   secondary 230 V
                            |
              L ------------+-------------->  HOUSE
                            |                  |
                            |              [ loads ]
                            |                  |
              N ------------+------------------+
                            |
                            = (neutral earthing)

The transformer does not send carriers to the consumer. The secondary electromotive force of the transformer, together with the network impedance and the present load, forms the secondary busbar voltage, nominally

\[U \approx 230\ \frac{\text{J}}{\text{C}},\]

This is not an ideal voltage source: the actual value at the busbar depends on the load and on the line impedance — a relation to which we return in §9. The field organises the motion of carriers already present in the circuit:

\[i = \frac{dQ}{dt}.\]

At \(I = 10\) A and unity power factor:

\[P = 10\ \frac{\text{C}}{\text{s}} \times 230\ \frac{\text{J}}{\text{C}} = 2300\ \frac{\text{J}}{\text{s}} = 2.3\ \text{kW}.\]

2.1 What “the circuit is closed” actually means

The correct formulation is not “carriers travel around a loop”, but the exact form of the current balance for a chosen control volume:

\[\boxed{\;\sum_{k} i_{k}(t) + \frac{dQ_{V}}{dt} = 0.\;}\]

In the quasi-stationary regime, where charge accumulation inside the control volume can be neglected, this reduces to \(\sum_{k} i_{k} \approx 0\). The approximation is introduced here explicitly and only in this form: the article is built on exact charge discipline, and the equality \(\sum_{k} i_{k} = 0\) as a statement holding “at any instant” is not used in the text. This is a statement about the balance of currents across a cross-section, not about particle trajectories.

2.2 A caveat that must not be omitted

In a healthy installation the working current must return along the intended working conductor; the protective conductor is not intended to carry load current. However, leakage, parasitic capacitance, insulation faults and parallel conducting paths are capable of creating a current component outside the conductors enclosed by the residual-current device. The current balance always closes — but not necessarily along the conductors in which it was drawn. It is precisely on this distinction that the residual-current device is built (§6.2).

3. The field as the carrier of energy

Section §2 leaves one question open: if the carriers are local, through what physical channel does energy reach the local circuit?

The answer is given by Poynting’s theorem. The flux density of electromagnetic energy:

\[\mathbf{S}_{\mathrm{P}} = \mathbf{E}\times\mathbf{H},\]

and the power entering a volume \(V\) through the bounding surface \(\Gamma\):

\[P_{\mathrm{in}} = -\oint_{\Gamma}\mathbf{S}_{\mathrm{P}}\cdot d\mathbf{A}.\]

The differential form:

\[-\nabla\!\cdot\!\mathbf{S}_{\mathrm{P}} = \frac{\partial}{\partial t}\!\left(\frac{\varepsilon E^{2}}{2} + \frac{\mu H^{2}}{2}\right) + \mathbf{J}\cdot\mathbf{E}.\]

The right-hand side contains the rate of change of stored field energy and the Joule term \(\mathbf{J}\cdot\mathbf{E}\) — the work of the field on the carriers.

The practical consequence: the flow of electrical energy is given by the Poynting vector, not by the motion of carriers along the wire. A significant part of the flow in an ordinary circuit passes through the electromagnetic field of the space around the conductors and enters the load through its surface^[10][11]. At the same time, in conductors of finite conductivity the field also penetrates inside: there the component of the flow directed into the conductor surface provides the Joule transfer \(\mathbf{J}\cdot\mathbf{E}\)^[12]. The conductor sets the geometry and the boundary conditions; the direct work on the carriers is done by the electric component of the field.

In a transformer this is seen most clearly:

  primary ensemble Q1              secondary ensemble Q2
        |                                  |
   [ primary ]  ====== FIELD ======>  [ secondary ]

No carrier of the secondary winding can reach the primary: the windings are galvanically separated. The charge ensembles are strictly separate. What is coupled is the electromagnetic state of the two circuits — through the shared magnetic flux.

4. The appearance of photovoltaic generation

Now connect a photovoltaic inverter to the same L and N.

                DISTRIBUTION TRANSFORMER
                            |
              L ------------+---------------+------------+
                            |               |            |
                            |             HOUSE      INVERTER
                            |               |            |
              N ------------+---------------+------------+
                                                         ^
                                                    PV array

The inverter has its own direct-current circuit: PV array → MPPT and, where present, a DC/DC stage → DC link → switching bridge → output filter. But on the alternating-current side it does not form a separate line to the power station: its AC port is connected in parallel at an already existing point of common coupling and becomes one more controlled port of the same low-voltage network.

4.1 How the inverter differs from the transformer

Here a precision is required that popular accounts usually lose.

The statement “the charge ensembles are separate” is valid for the transformer and invalid for a transformerless inverter.

A qualification without which the formulation would be wrong: the absence of a transformer means the absence of guaranteed galvanic isolation, not the presence of a single fixed reference point. The topology of the DC side may be floating, switched, or temporarily decoupled by dedicated switches. The correct object of analysis is therefore the common-mode voltage, the parasitic capacitance of the array to earth and the resulting leakage current — it is precisely because of these that such inverters are required to carry leakage-current and DC-side insulation-resistance monitoring.

4.2 Grid-following: a device that follows

A grid-connected inverter in its usual design operates as a controlled current source, synchronised by a phase-locked loop to the measured grid voltage:

\[i_{\mathrm{inv}}(t) = \sqrt{2}\,I_{\mathrm{inv}}\cos\bigl(\hat{\theta}(t) + \delta\bigr), \qquad \hat{\theta} = \mathrm{PLL}\bigl[u_{\mathrm{grid}}(t)\bigr].\]

It does not set \(U\), \(f\) and \(\varphi\) — it reads them and adjusts to them. In the normal grid-connected mode such a controller presumes an external voltage and frequency reference and is not intended to form an autonomous alternating-current bus; anti-islanding protection forbids it to maintain voltage in a de-energised section. This distinction becomes decisive in §11 and §12.

5. Three states of the node

Below, \(P_{\mathrm{load}}\) is the active power of the load, \(P_{\mathrm{PV}}\) is the active power injected by the inverter, and \(P_{\mathrm{grid}}\) is the active flow across the accounting boundary (positive on import, §0.1).

Condition of validity for scalar arithmetic. Simple subtraction holds only at unity power factor on both ports and in phase coincidence. In the general case the accounting is done in complex form:

\[\underline{S}_{\mathrm{grid}} = \underline{S}_{\mathrm{load}} – \underline{S}_{\mathrm{PV}}, \qquad \underline{S} = P + jQ_{\mathrm{r}},\]

and in a three-phase network the result additionally depends on whether the accounting is per-phase or aggregated (§7.3).

5.1 State A — consumption exceeds generation

\[P_{\mathrm{load}} = 5\ \text{kW}, \quad P_{\mathrm{PV}} = 2\ \text{kW} \;\Longrightarrow\; P_{\mathrm{grid}} = +3\ \text{kW}.\]

In charge-rate terms at 230 V and \(\cos\varphi = 1\) — in rms values:

\[I_{\mathrm{rms,load}} = \frac{5000}{230} \approx 21.7\ \frac{\text{C}}{\text{s}}, \qquad I_{\mathrm{rms,PV}} = \frac{2000}{230} \approx 8.7\ \frac{\text{C}}{\text{s}}.\]

A mandatory caveat on the meaning of “C/s”. Here and below this notation denotes the rms value of the rate of alternating transfer, not a steady unidirectional charge flow: the algebraic transfer across the cross-section over a period remains zero (§1). The transformer establishes the boundary condition under which the remaining \(\approx 13.0\) C/s appears at the measurement plane, in exactly the same sense. The meter registers import.

5.2 State B — generation covers the load

\[P_{\mathrm{grid}} = 5 – 5 = 0.\]

The mean active exchange across the boundary is close to zero. This does not mean:

• absence of voltage at the boundary;
• absence of current in the conductors;
• absence of reactive exchange \(Q_{\mathrm{r}}\);
• absence of harmonic components;
• absence of non-zero instantaneous power \(p(t)\).

Exactly one quantity goes to zero — the mean \(\langle u i\rangle\) over the accounting window. All the quantities listed may remain non-zero; their vanishing does not follow from \(P_{\mathrm{grid}} = 0\), but neither is it excluded: with exact coincidence of local generation and load in \(P\), \(Q_{\mathrm{r}}\), waveform and phase, the current in the measurement plane may also turn out to be close to zero. Some of these quantities are tariffed separately.

5.3 State C — export

\[P_{\mathrm{PV}} = 7\ \text{kW}, \quad P_{\mathrm{load}} = 2\ \text{kW} \;\Longrightarrow\; P_{\mathrm{grid}} = -5\ \text{kW}.\]

                    5 kW
      HOUSE ---------------------> GRID
       ^
       | 7 kW
      PV
       |
       +-- 2 kW -> local loads

Fundamentally: the topological conduction path has not changed. The mean drift component of the carriers remains local and alternating, with an amplitude of the order of a micrometre, and the current itself reverses direction every half-period in state A as well as in state C. What changed is the sign of the mean active power \(\langle u(t)\,i(t)\rangle\) across the chosen boundary — that is, the phase relation of the port current to the voltage, not the geometry of the circuit. The formulation “the electrons went back to the power station” is incorrect for alternating current and must be excluded from technical texts.

6. What the boundary measures: two instruments, two conserved quantities

In one and the same service cross-section of a domestic installation, two fundamentally different measurement principles operate, and they rest on different conservation laws. This is perhaps the clearest illustration of two-balance discipline in the whole of power engineering.

A point of terminological precision: both instruments are installed in the measurement plane of the service entry. That plane is not a complete energy boundary in the sense of §0.2 — it intersects the conductors, but not all field and thermal channels of the installation.

   PUBLIC GRID | METER | RCD | PRIVATE INSTALLATION
                    Γ

6.1 The meter — the energy accountant of the boundary

The physical object of active-energy accounting is the time integral of the product of synchronous quantities in the measurement plane:

\[E_{\Gamma,\Delta t} = \int_{\Delta t} u(t)\,i(t)\,dt.\]

A bidirectional meter accumulates active energy in the registers of the corresponding direction according to its normatively defined measurement and aggregation algorithm. This article does not specify that algorithm: real instruments differ in internal integration intervals, in accumulation methods, in per-phase or vector summation, in configurable modes, and the settlement interval used later for tariffing may be a different temporal object altogether.

What must not be done. The instrument’s algorithm should not be conflated either with the sign of an individual instantaneous sample of \(p(t)\), or with the settlement interval subsequently applied for tariffing. For a reactive load, \(p(t)\) changes sign twice per period even when the installation remains a net consumer throughout the settlement interval; clipping every negative half-period would turn ordinary reactive exchange into fictitious export registers.

The subject of the meter is not the algebraic charge transfer (in a steady periodic regime without a direct component it is zero), but the signed integral of the product. The meter is an accountant of energy, not an accountant of electrons.

6.2 The RCD — the charge accountant of the boundary

The residual-current device measures a different quantity — the algebraic sum of the currents of all working conductors through the same cross-section:

\[i_{\Delta}(t) = \sum_{k} i_{k}(t).\]

In normal operation the algebraic sum of the instantaneous currents of all working conductors enclosed by the summing transformer is close to zero. The instrument does not need to establish the identity of carriers — it compares currents. A current closing along a path outside the enclosed conductors — for example through the protective conductor or earth — is what creates the residual current \(i_{\Delta}\). A strict zero may not be reached even in a healthy installation: interference-suppression filters, class-Y capacitors and distributed capacitance produce leakage current in the absence of any fault. As leakage current outside the enclosed conductors increases, the residual sum increases, and at \(|i_{\Delta}| > I_{\Delta n}\) the device opens the circuit. The rating \(I_{\Delta n}\) depends on the purpose of the protection: 30 mA is a typical value for additional protection of persons, not a universal rating of all RCDs.

6.3 Comparison

Logical precision is required here. Conservation of charge and conservation of energy hold independently of the completeness of our measurement model. The distinction is different: the current imbalance in the service measurement plane is directly monitored by a single differential measurement, whereas an experimentally reconstructed energy balance closes only when all energy channels and changes in stores are covered. The law of charge conservation itself, meanwhile, is set by the continuity equation independently of the residual-current device: the instrument does not measure the term \(dQ_{V}/dt\) and does not prove the full balance of a closed volume. Two instruments in one consumer unit are the practical embodiment of that distinction.

7. The metrological and the legal layers

7.1 Active, reactive, apparent

For the sinusoidal regime:

\[S = UI, \qquad P = S\cos\varphi, \qquad Q_{\mathrm{r}} = S\sin\varphi, \qquad S^{2} = P^{2} + Q_{\mathrm{r}}^{2}.\]

For the non-sinusoidal regime the last equality is invalid: a distortion component appears. The correct apparatus is defined by the standard in force, IEEE Std 1459-2025 (a revision of IEEE Std 1459-2010, published 16 May 2025), which introduces the effective apparent power \(S_{e}\), the separate accounting of fundamental and non-fundamental components and — significantly for §6.1 — a separate treatment of the measurement observation period^[3]. For a node with inverter generation this is not an academic subtlety: the current of a converter node may contain non-fundamental components as a result of switching, of control features, of filtering and of distortion in the network itself. A modern grid-connected inverter is specifically controlled so as to bring the grid current close to a sinusoid, so what is at issue is possible, not inevitable, spectral content.

7.2 Accuracy class and window

Commercial electricity meters in the EU fall under the Measuring Instruments Directive (MID) and the EN 50470 series. The part in force, EN 50470-3:2022, is titled explicitly as requirements for static meters for AC active energy of class indexes A, B and C and applies to meters of those classes in 50 and 60 Hz networks. The required class depends on the category of application: Annex MI-003 of the directive sets the permissible classes for domestic, commercial and light-industrial use and grants the Member State the right to require a higher class for certain applications^[7]. Class D is not defined by the directive.

The registration interval (frequently 15 or 30 minutes) is a market and settlement configuration, not a physical or pan-European metrological norm: it differs between countries and between categories of metering point. What matters is something else: tariffing is done on interval sums, not on instantaneous values.

7.3 Four quadrants and the method of aggregation

A four-quadrant meter may keep separate registers:

\[+A \;(\text{active import}), \quad -A \;(\text{active export}), \quad +R, \; -R \;(\text{reactive}).\]

In a three-phase installation the result depends critically on the method of aggregation. Under per-phase accounting, export on phase L1 does not offset import on L2. Under aggregated accounting (net across the three phases) it does. One and the same physical situation produces different bills. This distinction is established normatively, not physically.

7.4 An example of a full quantity passport

Two fields of this passport are filled with qualifications, and both are of principle. The boundary field is written as \(\Pi\) — a measurement plane, not the closed energy boundary \(\Gamma\) in the sense of §0.2 (see §6). The field \(\mathcal{Q}\) is canonically a charge ensemble, that is a population of carriers, and not a network as a topological object; here it points to the ensemble of carriers of the shared galvanically connected low-voltage network including the measured branch, and not to a non-existent “ensemble of the individual house” (see §0 and §8). Identifying the ensemble with the network is inadmissible in an article built on the separation of topology from a population of carriers. The example exists precisely to demonstrate that discipline and therefore has no right to violate it.

The uncertainty is deliberately not given as a number. The maximum permissible error is not assigned by accuracy class as a single value: it depends on the current range, the power factor, the temperature and other specified conditions. Any concrete number in that field must come from a certificate, not from the name of the class.

A pair of “voltage and current” without a boundary, an ensemble, a time class and a window does not describe an operating point. It describes two numbers.

7.5 The legal layer is not the physical one

The term “net metering” describes a settlement scheme, not a physical process. In Spain the arrangement is set by Real Decreto 244/2019: the simplified compensation mechanism (compensación simplificada) is an economic balance over the billing period, computed from hourly valued quantities of energy consumed from the grid and of surplus energy, whereby the value of the surplus energy may not exceed the value of the energy consumed^[8]. The physics of the boundary does not change with it — what changes is the rule for folding the registers into a bill. Conflating these two layers is a standard error of popular accounts, and in this article they are separated explicitly.

8. Two houses on one low-voltage busbar

                        MEDIUM VOLTAGE
                               |
                          TRANSFORMER
                          20 kV / 400 V
                               |
                   +-----------+-----------+
                   |                       |
                HOUSE A                 HOUSE B
              load 2 kW              load 5 kW
              PV   6 kW              PV   0
                   |                       |
                   +------ LOW VOLTAGE ----+

House A exports \(6 – 2 = 4\) kW. House B imports 5 kW. Through the transformer above passes

\[P_{\mathrm{MV}} = 5 – 4 = 1\ \text{kW} \;+\; \text{losses}.\]

The solar generation of house A physically serves house B. But here the same caution is required as in §1.1, and it is of principle.

What cannot be asserted. House A, house B, the feeder cable and the secondary winding form one galvanically connected conducting network with branched current paths. The loops “L → load A → N” and “L → load B → N” are topological current paths, not physically isolated populations of carriers. The formulation “separate charge ensembles” applies to the primary and secondary sides of the transformer and is inapplicable to two branches of one low-voltage network. The estimate of §1.1 does not license such a conclusion either: the micrometre is the amplitude of the mean drift component of the ensemble, not a limit on the trajectory of an individual electron.

What can be asserted, and it is stronger. Covering the load of house B by the generation of house A does not require end-to-end directed transport of one and the same population of carriers from the panels of A to the load of B. At 50 Hz the mean drift component is alternating and local; the result is determined by the electromagnetic state of the shared conducting network and by the signed power flows through its ports. Inverter A changed that state so that through port A the mean flow \(\langle u i\rangle\) became negative, and through port B positive.

9. The bridge from the local node to the system: voltage rise

Here lies the link whose absence would make the transition to system-level failures unfounded.

Let the point of connection (PoC) be joined to the transformer busbar by a line with resistance \(R\) and reactance \(X\). No second sign system is introduced here: the same \(P_{\Gamma}\) and \(Q_{\Gamma}\) are used as throughout the article, positive on import (§0.1). For small deviations:

\[\Delta U = U_{\mathrm{PoC}} – U_{\mathrm{bus}} \approx -\,\frac{R\,P_{\Gamma} + X\,Q_{\Gamma}}{U}.\]

The minus sign here is not cosmetic: it is the content of the relation. On import \(P_{\Gamma} > 0\) and the voltage at the point of connection is lower than at the busbar; on active export \(P_{\Gamma} < 0\), and then \(\Delta U > 0\) — the voltage at the point of connection rises. It is this case that concerns us.

Two practical consequences follow from this relation.

First. In low-voltage distribution networks — both cable and overhead — the ratio \(R/X\) is noticeably higher than in transmission networks, and the resistive component makes a substantial contribution to \(\Delta U\). Therefore in low-voltage networks it is precisely active export that is able to raise the voltage appreciably at the point of connection. On reaching the upper permissible limit the inverter is obliged to curtail its output or to disconnect. Specific values of \(R/X\) depend on the cross-section, the material, the length and the construction of the line; a universal range is not fixed in the present text.

Second. In typical transmission networks the reactance substantially predominates over the resistance, and the voltage is governed predominantly by reactive power. Hence the central role of reactive control in system stability.

This is precisely the mechanism connecting an individual solar house to the coordinates of the whole system. A local port cannot change the law of charge conservation — but it can change the voltage at a node, and voltage is a shared system coordinate.

10. Five topologies

The paradox of §0 is dissolved by introducing separate levels of description.

The five-level map below is the analytical framework of this article, constructed on top of the continuity equation, Poynting’s theorem and power-system stability theory. It is not a generally accepted industry classification and is given as a synthetic model of the authors. Each level answers its own question, and confusion between levels generates all the known errors of popular explanations.

The category of the last column is stated separately. The conserved quantities are charge and energy, not topology. Topology — that is, the set of available current paths and connections — is not obliged to be conserved and in a fault is precisely what changes: a breaker opens a branch, a line disconnects, a generator is detached. Confusing these categories would make the article internally contradictory, since the cascade of §11 is explained by exactly this change of connectivity.

A house has a local load-current path

\[\text{L} \to \text{load} \to \text{N} \to \text{secondary winding},\]

but is coupled in energy to the medium-voltage network through the field of the transformer, that one to high voltage, and that one to generation, converters and neighbouring systems:

                        generator
                            |
   PV -> LV = FIELD = MV = FIELD = HV = HV = France
                            |
                        generator

The carriers are local. Energy crosses boundaries. The regime is coupled system-wide. Stability is a property of the whole.

11. 28 April 2025: a loss of stability, not a violation of conservation

The final report of the ENTSO-E expert panel was published on 20 March 2026. It was prepared by a panel of 49 participants — representatives of transmission system operators, regional coordination centres, ACER and national regulators — and contains a set of recommendations for strengthening the resilience of the European system^[1].

The conclusion of the report. The outage arose not from a single cause but from a combination of interacting factors: oscillations, gaps in voltage and reactive-power control, differences in voltage-regulation practices, rapid reductions of output and disconnections of generators in Spain, and uneven stabilisation capabilities. These factors led to a rapid rise of voltage and to cascading disconnections of generation^[1].

Recorded quantitative features. Oscillatory instability at frequencies of 0.63 Hz (a local mode) and 0.2 Hz (an inter-area mode). In the interval 12:32:00–12:32:48 the output of large renewable generation installations above 5 MW fell by approximately 500 MW. By 12:33:16 disconnections in the Badajoz area had removed 727 MW of photovoltaic and concentrated solar generation; over the following two seconds a further 928 MW disconnected across five provinces. In total more than 2.5 GW was lost at voltages exceeding 435 kV. At 12:33:19 the systems of Spain and Portugal lost synchronism with the continental European network^[1][2].

Continental Spain and Portugal were affected. The island systems and Ceuta and Melilla were not part of the synchronous area of the event.

11.1 Why this is not a violation of a conservation law

For any chosen control volume, throughout the whole event,

\[\oint_{\Gamma}\mathbf{J}\cdot d\mathbf{A} + \frac{dQ_{V}}{dt} = 0\]

held. Not one coulomb disappeared or appeared. The laws of conservation of charge and of energy were not violated at any moment of the event.

What became unstable was the operating regime of the system: voltages, angular dynamics, reactive reserves, the composition of generation and the switching topology ceased to support a stable synchronous operating regime. The term “operating point” is inapplicable here: it describes a steady state, whereas a cascade is a dynamic trajectory. The consistency of the shared system coordinates

\[U,\quad f,\quad \varphi,\quad P,\quad Q_{\mathrm{r}}\]

was not maintained. In the terms of §10, the fourth and fifth levels failed — stability and protection. This is a statement about the regime, not about the “correctness” of the first two levels: energy topology is not a binary quantity, and flows and stores were changing throughout the event.

11.2 The dynamic coordinate: inertia and the rate of change of frequency

In a classical synchronous system, frequency dynamics is directly related to the balance of mechanical and electrical power of synchronous machines and to their stored kinetic energy. For an aggregated simplified model over the initial interval:

\[\frac{2H}{f_{0}}\,\frac{df}{dt} \simeq \frac{P_{m} – P_{e}}{S_{\mathrm{base}}},\]

where \(P_{m}\) is the total mechanical power, \(P_{e}\) the total electrical power, \(H\) the inertia constant (s), and \(S_{\mathrm{base}}\) the base apparent power of the synchronously connected machines. The sign is determined precisely by the difference \(P_{m} – P_{e}\) and is nowhere redefined; writing this in terms of a “power deficit” without an explicit sign definition is inadmissible.

The meaning of the relation is direct: as the equivalent store of kinetic energy of the synchronously rotating masses decreases — represented in this simplified model by the product \(H S_{\mathrm{base}}\), and not by an arithmetic sum of the inertia constants of individual machines — the same imbalance produces a greater rate of change of frequency. This does not change the balance — it shortens the time available to protections and regulators.

In a converter-rich system the picture is supplemented by fast power-electronic control, frequency-dependent loads, storage and synthetic inertia. The correct formulation is therefore this: replacing synchronous machines with ordinary grid-following converters without an equivalent frequency-support function reduces the natural electromechanical inertia of the system and changes its frequency dynamics. Converter resources are capable of providing fast frequency response, synthetic inertia and droop — but this is a property of the control architecture and of the available energy reservoir, not an automatic consequence of connecting a converter.

11.3 The mechanism: fixed power factor and overvoltage

Peer-reviewed analysis of the event describes what occurred as a cascading disconnection of renewable generation operating under fixed-power-factor control, triggered by overvoltage protections. The mechanism is named an overvoltage-driven blackout and differs from the widely studied collapse by undervoltage^[4]. Among the critical leading factors are named: renewable generation with a fixed power factor that does not provide dynamic voltage control; insufficient capacity for reactive-power absorption; and a lightly loaded transmission grid^[5].

The physical meaning of a fixed power factor is transparent. If

\[\cos\varphi = \mathrm{const}, \qquad \text{then} \qquad Q_{\mathrm{r}} = P\tan\varphi,\]

that is, reactive power is rigidly tied to active power and ceases to be an independent control coordinate. In such a control mode \(Q_{\mathrm{r}}\) is not used as an independently regulated coordinate of dynamic voltage control — precisely the degree of freedom that stands in the relation of §9. What is at issue is the control mode, not a fundamental incapacity of the converter: the hardware ability to supply or absorb reactive power may well be present.

The subsequent development has the character of positive feedback:

  available Q absorption capacity insufficient  AND  generation PF ~ const
                    |
                    v
                  U up
                    |
                    v
     overvoltage protections disconnect generation
                    |
                    v
          change of P, Q and switching topology
                    |
                    v
              U up at other nodes
                    |
                    v
                further disconnections

Hence the exact formulation of the link between §9 and §11. It must not be stronger than this:

It is precisely the cascading character of the feedback that distinguishes §11 from a simple voltage rise on a feeder. The local sensitivity of voltage to power flows is statics; the national outage is the dynamics of that same relation in a system with control and protections.

11.4 Statement of the result

The outage of 28 April 2025 demonstrates exactly the thesis for which this article was written: one can build a system in which the law of charge conservation holds in every local control volume while the system as a whole remains dynamically unstable. Local fulfilment of conservation laws does not sum into system stability.

The word “flawless” is deliberately not applicable here. Satisfaction of the continuity equation does not mean that every local current balance was flawless in the engineering sense: during switching there are accumulation, displacement currents, leakage, transients and the very change of the control volumes. What is asserted is exactly what is asserted — conservation holds, stability is lost.

12. Who sets the boundary electromagnetic state

Everything set out above reduces to one question, which the charge model formulates precisely.

There is no strict, generally accepted boundary between converter control modes: the literature distinguishes grid-following, grid-supporting, grid-forming, the virtual synchronous machine, power synchronisation, the virtual oscillator and hybrid schemes, including ones with simultaneous behaviour of both types. For the distinction considered here it is enough to single out two limiting modes.

Limiting mode 1 — following the regime. The device measures the already existing \(U\), \(f\), \(\varphi\) and injects a current synchronised to the measured phase. Its equation is \(i(t) = f\bigl[\hat\theta(u_{\mathrm{grid}})\bigr]\). Such a controller presumes an external voltage and frequency reference and is not intended to form an autonomous bus; on disappearance of the reference voltage it is obliged to disconnect. The typical grid-connected PV inverter belongs here.

Limiting mode 2 — setting the regime. The device itself forms the boundary condition for a local charge ensemble: it sets \(U\), \(f\), \(\varphi\) and the waveform, and holds them under changing load. Its equation is \(u(t) = f[\text{internal state}]\), and the current is a consequence of the connected load. It is precisely such behaviour that makes possible the existence of a local alternating-current bus as an independently formed regime.

The distinction is not quantitative. It is the distinction between a device that participates in an externally established regime and a device that establishes the regime itself.

12.1 The position of VENDOR.Max

Terminological precision is required here, and it matters more than convenience of formulation.

In the classical alternating-current context used in §4, §11 and the opening of §12, grid-forming means forming the voltage together with its temporal reference — frequency and phase. It cannot, however, be asserted that the term belongs exclusively to the alternating-current domain: in the literature on direct-current microgrids, grid-forming is applied to the formation and stabilisation of the DC bus voltage as well, and the authors note that the GFL/GFM classification was developed for alternating-current systems and that its transfer to direct current requires a separate definition^[9].

Hence the position of this article, and it is deliberately narrow. The target output of VENDOR.Max is a regulated −48 V direct-current interface, for which \(f\) and \(\varphi\) are not coordinates of the interface at all. The article does not classify VENDOR.Max as a grid-forming device — neither in the alternating-current nor in the direct-current sense. A narrower and verifiable formulation is used instead: the formation of its own regulated direct-current boundary condition for the connected load.

The stability of that boundary condition is provided by the canonical hierarchy of four time levels, each with its own carrier:

The functions of the levels are canonically distinguished and are not mixed: the transient gap is bridged by the accumulator, while supervisory control moves the system to the new operating point.

From the standpoint of this article this means the following. A site powered from VENDOR.Max forms a local node whose boundary condition is set from inside the site rather than read from outside. The existence of the regime of that node does not require an external grid phase and frequency. The stability topology of such a direct-current node becomes a local task of controlling its own bus, rather than a task of tracking external \(f\) and \(\varphi\).

12.2 Mandatory qualification on the boundaries of the claim

Four limitations are fixed explicitly, and none of them is softened:

1. An autonomous output and parallel operation with the grid are different classes of product. The ability to form a voltage for one’s own load does not mean readiness for synchronous parallel operation with the national grid. The latter requires synchronisation, phase holding, control of flow direction, prevention of unintended energisation of a de-energised line, and compliance with grid codes — that is a separate class of testing and certification.
2. This article contains no claim about grid services. Everything stated relates to the formation of the regime of a local node, not to participation in the voltage or frequency regulation of the power system.
3. Status of the result. A result established in engineering by the developer and a result independently confirmed metrologically are two different statuses. The transition between them is provided by independent measurement at the boundary of the product according to the published protocol.
4. Delimitation of terminology. This article does not assign to the present direct-current-interface product a grid-following or grid-forming classification. A single property is claimed for it — the formation of a regulated direct-current boundary condition. Any broader control classification, as well as the concepts of synchronisation, islanded operation, anti-islanding protection, fault ride-through and participation in voltage regulation, requires a separate definition, separate testing and separate validation.

The question raised in the title of this section remains open for the industry as a whole. The Iberian event showed its price.

13. Summary

1. In a steady periodic alternating-current regime without a direct component, the algebraic charge transfer across a cross-section over a full period is zero, whereas the energy transfer may be non-zero. Knowing the quantity of charge that has passed is insufficient to determine the energy transferred: the potential at which the transfer occurs is required.
2. Energy is carried by the electromagnetic field; the conductor sets the geometry and the boundary conditions.
3. A photovoltaic inverter has its own direct-current circuit, but on the alternating-current side it does not form a separate line to the power station: it becomes a second controlled port of an existing node. In a transformerless design, galvanic isolation between the PV side and the grid is not guaranteed.
4. In one service measurement cross-section, the meter determines active energy from \(u(t)\,i(t)\), while the RCD determines the residual current \(\sum_k i_k\). These are two different measurement principles, associated respectively with the energy balance and the charge balance. The service measurement plane is not, however, a complete energy boundary.
5. Under the convention adopted throughout this article, that import is positive, the relation of voltage to power flows takes the form \(\Delta U \approx -\,(R P_{\Gamma} + X Q_{\Gamma})/U\). Therefore on active export \(P_{\Gamma} < 0\), and a local port is able to raise the voltage at the point of connection — this is how it influences a shared system coordinate.
6. The outage of 28 April 2025 violated no conservation law: what was lost was the stability of the coupled regime. The official investigation established a combination of interacting factors; the peer-reviewed analysis examined in §11.3 shows one of the key mechanisms of the cascade — the link between insufficient reactive absorption, generation at a fixed power factor, overvoltage and the action of protections.
7. The decisive distinction between devices is not power, but whether the device follows an externally established boundary electromagnetic state or sets its own.
8. For a direct-current interface this distinction is formulated without frequency and phase: the device either accepts the boundary condition from outside or forms its own regulated condition on its bus.