A lattice-based representation of power systems dedicated to synchronism analysis

Abstract

An energy-efficient description of electromagnetism based on a reversible interpretation of Faraday’s Law is described. Space aggregation and time reconciliation of all scales involved in power system management are derived from two constants of motion, respectively Gibbs free energy and kinetic energy embedded in the whole system. At the upper space scale, a simplified description of the grid based on the Kuramoto model is proved to be suitable for assessing the stability of the synchronous state of a power system. In addition, the Gibbs free energy embedded in the power grid takes the form of a X − Y lattice-gas Hamiltonian, showing the critical role of reactive power in maintaining synchronism. Finally, the transient stability of the power system is considered from these two constants of motion, before any operation from the dispatch.

Keywords

Thermodynamics power generation transient stability synchronism Kuramoto’s model

1. Introduction

Power system generation has to adapt in real time to consumption, which is highly fluctuating and only partly predictable. Two intricate problems are concerns for grid operators and utilities, involving several time-scales [1]:

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Keeping the power system quantities within the security margins during normal and transient operations, notably frequency and voltage plan [2];
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Handling the evolution of power systems which is driven by political, economic, social and environmental issues addressed through long-term energy planning models.

In order to address the considerable challenge of grid decarbonization and the subsequent massive introduction of intermittent electricity production, a general framework is needed that:

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aggregates the space characteristics of the power grid;
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reconciles the short-term dynamics of power system management and the prospective analysis in the long-term;

As expected based on uniform time [3], this contribution derives two energy-based constants of motion from first principles (Section 2). Then:

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the transient stability of power systems can be discussed (sections 3 and 4); and subsequently
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the conditions of operations can be endogenized in prospective studies.

2. Multi-scale description

Usually, thermodynamic approaches to electromagnetics do not consider any extension towards time-varying regimes [4]. Although some improvements for steady-states regimes are summarized in [5], no general contribution covers transient regimes. As a general rule in this paper, variational parameters and functionals are denoted by italics. A roman font is used to specify their minimum values. The magnetodynamic behavior of any electrical system that externally receives net mechanical power P_mech and embeds kinetic energy E_kin is derived from the functional expressing the second principle of thermodynamics and establishing the difference between the mechanical power received by the field from the actuators $(\text{P}_{\text{mech}}-\frac{\text{d}\text{E}_{\text{kin}}}{\text{d}t})$ and the time variations of Gibbs free-energy G [6]: $\begin{eqnarray}\displaystyle \text{P}_{\text{mech}}-\displaystyle \frac{\text{d}\text{E}_{\text{kin}}}{\text{d}t}-\displaystyle \frac{\text{d}\text{G}}{\text{d}t}=\mathop{\min }_{\boldsymbol{H},\boldsymbol{E}}\left(\int _{C}{\sigma}^{-1}(\mathbf{curl}∼\boldsymbol{H})^{2}∼\text{d}^{3}r+\displaystyle \frac{\text{d}}{\text{d}t}\displaystyle \int (\mathbf{B}\cdot \boldsymbol{H}+\mathbf{D}\cdot \boldsymbol{E})∼\text{d}^{3}r\right) & & \displaystyle\end{eqnarray}$ (1) where the functional in the RHS exhibits:

–

the magnetic field H related to the free current density J and the displacement density ∂_t D according to the Maxwell-Ampere equation: curl H = J + ∂_t D . The quasi-static approximation enforces D ≡ 0 in conductors (i.e. no free charge);

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the Joule losses monitored in conductors. This term is quadratic to respect invariance of losses with inversion of time (σ⁻¹ > 0 is the resistivity);

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the variation with time of the electromagnetic energy coupling the field with the generator supplied by the net current I₀ and the mass at the potential V₀;

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the magnetic B( h ) and electrostatic D( e ) behavior laws derived from the thermostatistic equilibrium of the Gibbs potential: $\begin{eqnarray}\displaystyle G(\text{I}_{0},\text{V}_{0})=\displaystyle \int \left(\int _{0}^{\boldsymbol{H}}(-\boldsymbol{B})\cdot {\delta}\boldsymbol{h}+\int _{0}^{\boldsymbol{E}}(-\boldsymbol{D})\cdot {\delta}\boldsymbol{e}\right)∼\text{d}^{3}r. & & \displaystyle\end{eqnarray}$ (2)

Extending the electric field in the conductor according to Ohm’s Law with motion: E = σ⁻¹ J − V × B, Maxwell-Faraday’s equation: curl E = −∂ _t B may be viewed as acting locally to globally check the tendency towards reversibility expressed by the minimization in the RHS of (1). This striking property provides a thermodynamic-oriented insight into the variational theory of electromagnetism [7].

For a local focus on design or power management, the electrical power: $\begin{eqnarray}\displaystyle \text{P}_{\text{elec}}({\Theta})=-\oint _{\partial {\Theta}}(\mathbf{E}\times \mathbf{H})\cdot \mathbf{n}∼\text{d}^{2}r & & \displaystyle\end{eqnarray}$ (3)expresses the power deviation in the domain Θ to locally enforce the integral form of the Poynting’s conservation equation $\begin{eqnarray}\displaystyle \text{P}_{\text{elec}}({\Theta})+\text{P}_{\text{mech}}({\Theta})=\text{P}_{\text{Joule}}({\Theta})+\displaystyle \frac{\text{d}\text{F}}{\text{d}t}({\Theta})+\displaystyle \frac{\text{d}\text{E}_{\text{kin}}}{\text{d}t}({\Theta}) & & \displaystyle\end{eqnarray}$ (4) and globally satisfy: $\begin{eqnarray}\displaystyle \displaystyle \mathop{\sum }_{{\Theta}}P_{\text{elec}}({\Theta})=0 & & \displaystyle\end{eqnarray}$ (5) where F denotes the field’s Helmholtz’ free energy at thermodynamic equilibrium of the Gibbs potential (2): $\begin{eqnarray}\displaystyle \text{F}({\Theta})=\text{G}({\Theta})+\displaystyle \int _{{\Theta}}(\mathbf{H}\cdot \mathbf{B}+\mathbf{E}\cdot \mathbf{D})∼\text{d}^{3}r. & & \displaystyle\end{eqnarray}$ (6) Hence, the contribution of Θ to (1) reads: $\begin{eqnarray}\displaystyle \text{P}_{\text{mech}}({\Theta})-\displaystyle \frac{\text{d}\text{E}_{\text{kin}}}{\text{d}t}({\Theta})-\displaystyle \frac{\text{d}\text{G}}{\text{d}t}({\Theta})+\text{P}_{\text{elec}}({\Theta}). & & \displaystyle\end{eqnarray}$ (7)

As matter of fact, free current density exhibits well-split scales allowing a mean-field hypothesis where the global minimization in (1) is replaced by embedded minimizations on the various scales involved by the power conversion of the electrical system [8], from deep within the material [9] to the whole electrical system [10]. In addition, the power functional (1) exhibits two energy-based invariants under steady-state evolution, namely the kinetic energy and the Gibbs free energy. They are also called constants or integrals of motion [3]. While the latter enforces synchronism between the machines and provide a way to aggregate kinetic energy over the whole grid (Section 3), the former acts to absorb any transient disturbances (Section 4).

Fig. 1.

Phasor diagram of the generator 𝛩_i with an internal admittance X _{s
_i} connected to the machine 𝛩_j by the admittance X _ij of the power grid. While θ_i(t) is the terminal voltage angle with respect to the synchronous reference frame (denoted by 0), δ_i is the electrical internal voltage phase angle – also called power angle – of the machine i. The mechanical rotor angle is ${\alpha}_{i}(t)=\frac{{\pi}}{2}+{\delta}_{i}+{\theta}_{i}(t)$ with respect to the synchronous reference frame. When the rotor is tilted slightly of Δ𝛼_i (double-mixed lines) and the stator current I _i is maintained unchanged, the phase of the terminal voltage θ_i undergoes the deviation Δθ_i = Δ𝛼_i. All the angles are counted anti-clockwised.

3. Synchronism analysis

It is convenient to perform a partition of the whole power system between the machines and the grid elements. The per phase contribution to the Gibbs free energy reads: $\begin{eqnarray}\displaystyle \text{G}=\mathop{\sum }_{i}\text{G}_{i}({\alpha}_{i},{\theta}_{i})+\mathop{\sum }_{\langle i,j\rangle \in {\varepsilon}_{{\mathcal{G}}}}\text{G}_{ij}({\theta}_{i},{\theta}_{j})=\mathop{\sum }_{i}\left(\text{G}_{i}({\alpha}_{i},{\theta}_{i})+\frac{1}{2}\mathop{\sum }_{j}\text{G}_{ij}({\theta}_{i},{\theta}_{j})\right) & & \displaystyle\end{eqnarray}$ (8) where (Fig. 1):

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the first summation is performed on the generators, G_i denoting the contribution of the machine i; while

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the second summation is performed on the set of edges ${\varepsilon}_{{\mathcal{G}}}$ underlying the graph ${\mathcal{G}}$ of admittance of the power grid, G_ij being the contribution of the line 〈i, j〉 connecting the machines i and j.

Noticing the similar variations for the mechanical rotor angle 𝛼_i and the terminal voltage angle θ_i, the per phase contribution to the averaged electrodynamic torque acting on the rotor is derived within a quasi-static evolution of the machine i from a stationnary condition on G: $\begin{eqnarray}\displaystyle \text{T}_{i}=-\displaystyle \frac{\partial \text{G}_{i}}{\partial {\alpha}_{i}}=\mathop{\sum }_{j}\displaystyle \frac{\partial \text{G}_{ij}}{\partial {\theta}_{i}}. & & \displaystyle\end{eqnarray}$ (9) Assuming a linear time-harmonic regime, the averaged Gibbs free energy stored between the nodes i and j is conveniently derived by (6) from the reactive power $\text{Q}_{ij}=\frac{\text{d}\text{F}_{ij}}{\text{d}t}$ flowing in the lumped admitance X_ij: $\begin{eqnarray}\displaystyle \text{G}_{ij}=-\displaystyle \frac{\text{Q}_{ij}}{2{\omega}_{i}}=-\displaystyle \frac{1}{2{\omega}_{i}}\text{X}_{ij}|\text{I}_{ij}|^{2} & & \displaystyle\end{eqnarray}$ (10) where ω_i is the pulsation observed at the node i. For a lossless power grid, the Law of Cosines expressed from the voltage drop between U_j and U_i (Fig. 1) provides: $\begin{eqnarray}\displaystyle \text{G}_{ij}({\theta}_{i},{\theta}_{j})=-\displaystyle \frac{1}{2{\omega}_{i}\text{X}_{ij}}(|\text{U}_{j}|^{2}+|\text{U}_{i}|^{2}-2|\text{U}_{j}||\text{U}_{i}|\cos ({\theta}_{j}-{\theta}_{i})) & & \displaystyle\end{eqnarray}$ (11) and the per-phase electrodynamic torque acting on the rotor of the machine i reads: $\begin{eqnarray}\displaystyle \text{T}_{i}=-\mathop{\sum }_{j}\displaystyle \frac{|\text{U}_{j}||\text{U}_{i}|}{{\omega}_{i}\text{X}_{ij}}\sin ({\theta}_{i}-{\theta}_{j}). & & \displaystyle\end{eqnarray}$ (12) This relation:

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expresses a trend to align the terminal voltage angle of the machine i with that of machine j;

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conserves the flow between the active power supplied by the lossless machine i and the electric power transferred from node i to the grid;

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illustrates that the Gibbs free energy and the electrodynamic torque are dependant on the voltage plan.

By introducing the moment of inertia J_i, the linear friction 𝜅_i and the apparent power S_i of the machine i, the normalized swing equations for each connecting point i, tracked from the deviation of the terminal voltage phase θ_i, are obtained by multiplying the dynamic equation of the rotor with the rated pulse ω_i [2]: $\begin{eqnarray}\displaystyle \text{m}_{i}\ddot{{\theta}}_{i}+\text{d}_{i}\dot{{\theta}}_{i}=\text{p}_{i}-\mathop{\sum }_{j}\text{s}_{ij}\sin ({\theta}_{i}-{\theta}_{j}) & & \displaystyle\end{eqnarray}$ (13) where, for small and slow deviations of the frequencies of the machines due to their huge inertia:

–

the reduced inertia of the machine obeys: $\begin{eqnarray}\displaystyle \text{m}_{\text{i}}=\displaystyle \frac{2}{{\omega}_{i}}\frac{\frac{1}{2}\text{J}_{i}{\omega}_{i}^{2}}{\text{S}_{i}}=\displaystyle \frac{2}{{\omega}_{i}}\frac{\text{E}_{\text{kin},i}}{\text{S}_{i}} & & \displaystyle\end{eqnarray}$ (14)

–

the reduced damping term reads: $\begin{eqnarray}\displaystyle \text{d}_{i}=\displaystyle \frac{{\kappa}_{i}{\omega}_{i}}{\text{S}_{i}} & & \displaystyle\end{eqnarray}$ (15)

–

the per unit net mechanical power supplied to the grid follows: $\begin{eqnarray}\displaystyle \text{p}_{i}=\frac{\text{P}_{\text{mech},i}}{\text{S}_{i}} & & \displaystyle\end{eqnarray}$ (16)

–

the per unit apparent power in the lossles line 〈i, j〉 is given by: $\begin{eqnarray}\displaystyle \text{s}_{ij}=\displaystyle \frac{3\frac{|\text{U}_{j}||\text{U}_{i}|}{\text{X}_{ij}}}{\text{S}_{i}}. & & \displaystyle\end{eqnarray}$ (17)

The coupled oscillator model (13) is a second order system of X − Y spins [11] that aim to minimize the valuations of the bidimensional lattice-gas Hamiltonian with respect to the terminal voltage phase θ_i: $\begin{eqnarray}\displaystyle {\mathcal{H}}({\Theta})=\mathop{\sum }_{\langle i,j\rangle }\text{J}_{ij}(1-\cos ({\theta}_{i}-{\theta}_{j}))-\mathop{\sum }_{i}h_{i}\cdot {\theta}_{i} & & \displaystyle\end{eqnarray}$ (18) where, for voltage amplitude over the grid maintained at the constant level U:

–

the first term provides a pair-wised trend to the alignement of the spins: $\begin{eqnarray}\displaystyle \text{J}_{ij}=\displaystyle \frac{3\text{U}^{2}}{{\omega}_{i}\text{X}_{ij}}. & & \displaystyle\end{eqnarray}$ (19) This term has been extensively studied in the context of phase transitions and critical phenomena (on a regular and infinite lattice). Briefly summarized, no long-range order exists in two-dimensional systems with short range interaction between Heisenberg magnets [12] whereas Onsager provided an exact resolution of the two-dimensional Ising model with first neighbor interactions [13]. Hence, the X − Y model appears as a marginal case where the long-range ordering may vanish through a weak singularity subject to external perturbation [14]. As a result, the “natural” synchronism between rotors is questionable, especially under the effect of

–

the second term (which) represents the energy of an external field driving the spins away from the “all-aligned” state: $\begin{eqnarray}\displaystyle h_{i}=\displaystyle \frac{\text{P}_{\text{mech},i}-{\kappa}_{i}{\omega}_{i}^{2}}{{\omega}_{i}}. & & \displaystyle \end{eqnarray}$ (20) For sufficiently low mechanical power, ${\mathcal{H}}({\theta})$ features a phase cohesive minimum if every pair of connected machines has a distance smaller than some angle ${\gamma}\in \left[0,\frac{{\pi}}{2}\right]$ , that is $|{\theta}_{i}-{\theta}_{j}|\lt {\gamma}\lt \frac{{\pi}}{2}$ for every $\langle i,j\rangle \in {\varepsilon}_{{\mathcal{G}}}$ .

From a dynamic point of view, it is convenient to study the set of Eq. (13) within a second order Kuramoto’s model [15]. This model was initially proposed to describe the first order evolution between any population of non-linear coupled oscillators that are all inter-connected [16]. Kuramoto’s model has been subsequentlly extended to more general, not necessarily complete, networks [17,18]. It has also been adapted to a power grid context. Sufficient conditions for steady-state solutions of (13) being locally and exponentially stable were established in [19,20]. These conditions are expressed as functions of grid algebraic connectivity ${\lambda}_{2}({\mathcal{G}})$ of the graph ${\mathcal{G}}$ underlying the power flowing in the grid, and must be higher than a norm on the product between the incidence matrix B of the admittances of the grid and the vector of powers connected to the grid. The same author has proven that this condition is fulfilled for each kind of graph with a square norm and in most cases with an infinite norm, and is extremely accurate for power networks : $\begin{eqnarray}\displaystyle {\lambda}_{2}({\mathcal{G}})\geq \Vert \text{B}^{T}∼\text{P}_{\text{mech},i}\Vert _{\infty }=\max _{\langle i,j\rangle \in {\varepsilon}_{{\mathcal{G}}}}|\text{P}_{\text{mech},i}-\text{P}_{\text{mech},j}|. & & \displaystyle\end{eqnarray}$ (21) When the condition (21) is met, the power system is stable in the sense that it will naturally and exponentially come back to a synchronous state after any small disturbance. Then the kinetic inertia of the turbines can be aggregated at the scale of the whole power system.

4. Transient stability

At the power management level, the steady state of any power grid is operated within a time-harmonic regime at the frequency $f=\frac{{\omega}}{2{\pi}}$ , typically rated at 50 Hz or 60 Hz in continental contexts. A further analysis of the sides of the equality (4) over the whole grid: $\begin{eqnarray}\displaystyle \text{P}_{\text{mech}}(t)=\text{P}_{\text{Joule}}(t)+\displaystyle \frac{\text{d}\text{F}}{\text{d}t}(t)+\displaystyle \frac{\text{d}\text{E}_{\text{kin}}}{\text{d}t}(t) & & \displaystyle\end{eqnarray}$ (22) provides a fundamental description of the instantaneous phenomenon of stability, i.e.:

–

The RHS of (22) is focused on the transient stability of the system, providing reactive power and kinetic energy to deal with any sudden disturbances, before any control operation from the dispatch. Besides,

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The LHS of (22) is related to system adequacy thanks to the primary, secondary and tertiary controls, consisting in an adjustment of the mechanical power supplied to the system to successively restore: (i) a balance between supply and demand, (ii) the rated point, and (iii) an economic optimum for the utility.

In order to have the kinetic energy as high as possible whithout lowering it in Joule losses in the face of a sudden disturbance and to delay any operations of power management as long as possible, it appears critical to synchronize the grid to aggregate the available kinetic energy over the whole system.

5. Conclusions

As expected based on uniform time, two energy-based constants of motion were theoretically derived from the second principle of thermodynamics expressed in a power system. These are field-related and provide the minimum set of conditions by which the power system may be operated:

The Gibbs free energy of the electromagnetic field is related to the synchronism between the machines occuring at the rated frequency. Synchronism results from competition between the grid’s connectivity and the expected congestion and guarantees that the power system will naturally go back to a phase cohesive state after a small disturbance. Thus, a suitable level of Gibbs free energy allows a space aggregation of the kinetic energy over the whole grid.

The kinetic energy embodied in the power system is related to the motion of the machines in the Galilean earth frame. Provided that the synchronism is maintained over the grid, the kinetic energy deals with any sudden disturbance wherever its location and before any control operation from the dispatch (typically for a few dozens of seconds).

A reliability constraint on kinetic energy was endogenized in the technical optimal TIMES model [21] to reconcile short-term dynamics of power management with long-term prospective analysis. For the specific case of Reunion Island and on the condition that the grid could be made significantly stronger, it demonstated that [22]:

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electrochemical storage technology potentially contributes to power dynamics and its reliability; and

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a high share of variable renewable plants can be considered without jeopardizing power transmission.

A similar study is under way for France [23].

Footnotes

Acknowledgements

This research was supported by the Modeling for Sustainable Development Chair, led by MINES ParisTech, Ecole des Ponts ParisTech, AgroParisTech, and ParisTech; and supported by ADEME, EDF, GRT Gaz and SCHNEIDER ELECTRIC.

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