Power System Synchronism in Planning Exercises - Science Direct

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ScienceDirect Energy Procedia 105 (2017) 2712 – 2717

The 8th International Conference on Applied Energy – ICAE2016

Power system synchronism in planning exercises: From Kuramoto lattice model to kinetic energy aggregation Vincent Krakowskia, Xiang Lia,b, Vincent Mazauricb,a*, Nadia Maïzia a

MINES ParisTech – PSL Research University, Centre for Applied Mathematics, BP 207, 06902 – Sophia Antipolis, France b Schneider Electric, Corporate Research, 38TEC, 38050 – Grenoble cedex 9, France

Abstract In order to assess the stability of the synchronous state of a power system, we introduced a dedicated description of the grid based on the Kuramoto model. We derived a coherent approach suitable for long-term power systems analysis and implemented in the case of the French Reunion Island. Keywords: Energy planning; Power generation; Intermittency; Synchronism; Kuramoto model; Transient reliability; Reunion Island.

1. Introduction Power system generation has to adapt to consumption in real time, involving strong fluctuations that are only partly predictable. Two intricate time-based issues are of concern to grid operators and utilities to [1]: x Maintain the power system quantities within the security margins under normal and transient operations, notably frequency and voltage plan [2]; x Handle the evolution of power systems, which is driven by political, economic, social and environmental issues addressed through long-term energy planning models. Tackling the considerable challenge of grid decarbonization and the subsequent massive introduction of intermittent electricity production requires a general framework that: x Aggregates the space characteristics of the power grid; x Reconciles the short-term dynamics of power system management with long-term prospective analysis. This paper is an attempt to derive a set of conditions on which reliability and robustness of the power grid could be discussed and endogenized in prospective studies. In the next section, we derive the global condition for transient reliability, and then in the third section discuss the adequate conditions for a power system to naturally maintain synchronism. This theoretical

* Corresponding author. Tel.: +33 4 76 57 74 73; fax: +33 4 76 57 32 32. E-mail address: [email protected].

1876-6102 © 2017 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the scientific committee of the 8th International Conference on Applied Energy. doi:10.1016/j.egypro.2017.03.921

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framework is finally applied to a prospective analysis for the Reunionese power system in the fourth section. 2. Power system transient reliability For local design or power management purposes, the electrical power Pelec is expressed from the power deviation in the domain 4, typically surrounding an electrical machine, to [1]: x Locally enforce the integral form of Poynting’s conservation equation: dF dE Pmech ሺȣሻ ൅ Pelec ሺȣሻ ൌ PJoule ሺȣሻ ൅ ሺȣሻ ൅ kin ሺȣሻ (1) x Globally satisfy:

d௧

d௧

σ஀ ‡Ž‡… ሺȣሻ ൌ Ͳ

(2) where: x Pmech denotes the net mechanical power received externally by the actuators; and x PJoule the Joule losses; x F is the electromagnetic (Helmoltz free-) energy; and x Ekin the kinetic energy. Basically, the steady state of any power grid is operated within a time-harmonic regime at the frequency f = Z/2S typically operated at 50 or 60 Hz. Each machine is then mechanically balanced between a running electrodynamic torque – obtained with a suitable voltage plan – and a motor torque. Under admissible load fluctuation, the huge inertia of the machines prevents an abrupt frequency deviation. Therefore, such deviations should only occur over several periods. However, some frequency discrepancies can be observed between the domains 4 according to the location of the fluctuation. Denoting harmonic peak-to-peak values with ^ and long-time averaged values with ~, the power dynamics experienced by the whole power grid follow after a low-pass filtering on (1): ෩ ෩ ሺ‫ݐ‬ሻ ൅ dEkin ሺ‫ݐ‬ሻ ෩ (3) mech ሺ‫ݐ‬ሻ ൌ ෩ Joule ሺ‫ݐ‬ሻ ൅  d௧

෩ ൌ † ൫ ෠Τʹ൯ is the so-called reactive power, i.e. exchanged with the electromagnetic field. In where  †೟ addition, the kinetic energy acting as a dynamic reserve for the whole power grid obeys the set of inequalities: ෩kin ሺ‫ݐ‬ሻ ൑ σ஀ Ekin ሺȣሻ min஀ Ekin ሺȣሻ ൑ E (4) where the upper bound is reached only if the synchronism is achieved over the whole power grid during the transient regime. In other words, some locally available kinetic energy has been lowered in Joule losses by the inrush currents during the transient and discarded by the low-pass filtering performed on (1). In order to keep the kinetic energy as high? as possible in the face of a sudden disturbance, it appears critical to enforce the synchronism in order to discuss power system reliability. In the following, an X-Y lattice model is adopted to describe the interaction between the magnetic momentum carried by the rotor of a given machine, also denoted as 4, and the magnetic field resulting from all of the others [4]. 3. Grid synchronism issue The “natural” synchronism between rotors, given the connection with ordering in two-dimensional systems, is questionable, especially in the context of phase transitions and critical phenomena: Although no long-range order exists in two-dimensional lattices with short-range interaction between Heisenberg 3D magnets [5], Onsager provided an exact resolution of the 2D Ising model with first neighbor interactions [6]. Hence, the X-Y model appears as a marginal case in which long-range ordering may vanish through a weak singularity under an external perturbation [7]. In the framework of a power system, it is convenient to tackle this problem using the Kuramoto model. This model was devised to

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describe the interaction between any population of non-linear coupled oscillators 4 given in the phasor space by (T,T’) and all connected to each other [8]: ୏ (5) ߠ௜′ ൌ ߱௜ െ σ௜ •‹൫ߠ௜ െ ߠ௝ ൯ ୒

where Zi is the natural frequency of the ith oscillator, K is a magnetic-based coupling constant, and N is the number of oscillators. Kuramoto’s model has been extended to cover more general, heterogeneous and not necessarily complete networks [9]. It has also been adapted to power grids with a formulation that describes the transient swings in the power angle of generators connected to a grid [2]. A condition for keeping a stable solution – i.e. backing locally and exponentially to a synchronous steady state after any slight disturbance – was established in [10]. This condition expressed that the grid’s algebraic connectivity O2(G) of the graph G underlying the power grid (LHS) is higher than the infinite norm on the product between the incidence matrix B of the grid and the vector of mechanical powers connected to the grid (RHS): ߣଶ ሺ‫ܩ‬ሻ ൒ ԡ ் ‡…Š ԡஶ ൌ ƒšሺ௜ǡ௝ሻ‫א‬ఌಸ ห‡…Šǡ௜ െ ‡…Šǡ௝ ห (6) where HG denotes the set of edges of the graph G and Pmech,i the mechanical power supplied at node i. Hence, the intrinsic stability of a power system, given the grid topology and the list of connected generators, consists in the ratio between the LHS and the RHS of (6). If this indicator is above the unit, the power system will be able to maintain the synchronism, and the kinetic inertia of the turbines can be aggregated at the scale of the whole power system, enabling the calculation of a kinetic indicator [11]. The latter may be interpreted as the time available for an operator to activate regulations when generation and consumption are unbalanced. 4. Results Reunion Island (Fig.1) has set a 100% renewable energy penetration target for electricity production by 2030 with an aim of energy independence. Consequently, this island will face rapid changes in the way it produces electricity, which could jeopardize supply security.

Fig. 1: Strengthened Reunion power grid: current (red) and new lines (orange).

In order to better understand this issue, we built a prospective model from the MARKAL-TIMES family describing the Reunion’s power system and endogenized the kinetic indicator within the model [12],[13]. Despite intermittent sources (wind, solar and ocean energy) that are assumed to provide no kinetic reserves, it was shown that 100% renewable energy scenarios that do not jeopardize kinetic

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reserves could be proposed (Fig. 2).

Fig. 2: Kinetic indicator of Reunion Island’s power system in 2030 in a 100% renewable energy scenario.

This model is completed here with an assessment of the ability of the same power system to maintain synchronism with the indicator derived from (6). This indicator is assessed in a post-processed calculation that makes assumptions on the location of new power plants and electric lines. Here the synchronism indicator is calculated for the two grids displayed in Fig. 1: the current Reunion power grid (dashed lines, Fig. 3) and a strengthened one (solid lines, Fig. 3).

Fig. 3: Synchronism indicator of Reunion Island’s power system in 2030 in a 100% renewable energy scenario with a strengthened grid (solid lines) compared to the same indicator calculated with the current grid (dashed lines). Red indicates periods of the day during which the indicator is lower than the critical value and, green, during which it is higher than this critical value.

Figure 3 shows that Reunion’s power grid would need to be reinforced to deal with the new production of a 100% renewable energy scenario while keeping its ability to maintain synchronism. Even with the new transmission lines considered in this scenario (orange in Fig. 1), there are still some time periods

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during which power system stability is not guaranteed (solid red lines in Fig. 3). These periods occur during the sugar season, when electricity is mainly produced by bagasse-wood power plants (Fig. 4). Indeed, in the studied scenario, these power plants are only installed near two substations, Le Gol and Bois-Rouge. As a result, power production is highly heterogeneous, making the right-hand side of (13) higher than the algebraic connectivity of the grid. In terms of this synchronism indicator, generation distributed close to consumption centers is preferable since it prevents huge electricity transfers on the grid.

Fig. 4: Power production in 2030 on Reunion Island for each sub period of the model in a 100% renewable energy scenario. Note the high share of intermittent renewable production in the 9:00-17:00 time-slice in the summer season, well beyond the 30% legal limit.

5. Conclusion Using a variational formulation of a thermodynamic approach to electromagnetism, combined with a second-order Kuramoto model adapted to power system description, we propose two indicators to analyze a power system’s reliability and robustness: x The kinetic indicator is related to the kinetic energy embodied in the power system. The higher this indicator is, the more time power system operators will have to activate regulations in case of a supply-demand mismatch. However, the synchronous state needs to be achieved to aggregate the kinetic energy on the entire power system. x The synchronism indicator is based on a sufficient condition that guarantees that the power system will naturally and exponentially return to synchronism after a slight disturbance. Only when the latter is higher than the unit is relevant to calculate the kinetic indicator. We assessed both indicators on a modeled 2030 power system of Reunion Island under a 100% renewable energy penetration scenario. The results show that this level of renewable energy could be reached without jeopardizing power system management provided that the grid is significantly strengthened. To date, only the kinetic indicator has been endogenized into a prospective model, with the synchronism indicator being assessed in a post-treatment calculation. By endogenizing both indicators, a reliable evolution of power systems could be proposed.

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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 and the French Ministry of Environment, Energy and Sea. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]

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Biography Vincent Mazauric has been with Schneider Electric (Grenoble, France) since 1995, currently in charge of Scientific Affairs and Patent Policy. He has a PhD in Solid State Physics (Orsay, France) and diplomas in Electrical Engineering (Grenoble, France), Pure Mathematics and Theoretical Physics (Paris, France). He has worked at the French Aerospace Lab (ONERA: 1988-91) and the Center for Extreme Materials (Osaka, Japan: 1992-94). Dr Mazauric is a fellow of the Japanese Society for the Promotion of Sciences, and an expert-evaluator for the R&D Framework Programs of the European Commission. As a member of the Commission on Environment and Energy of the International Chamber of Commerce and of the ParisTech Chair "Modeling for sustainable development", he is closely involved in sustainability and climate-energy initiatives in his role as United Nations Observer for the Framework Convention on Climate Change. He received the 2013 Award from the Japan Society of Applied Electromagnetics and Mechanics and is currently Director (elected) at the Administrative Committee of the IEEE Magnetic Society.

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