Bridging between Load-Flow and Kuramoto-like Power Grid Models: A Flexible Approach to Integrating Electrical Storage Units

TL;DR

This paper introduces a novel approach combining load-flow and Kuramoto-like models to better incorporate electrical storage units into power grid simulations, aiding in smart storage control and system stability analysis.

Contribution

It extends Kuramoto-like power grid models to include arbitrary grid components like storage units, enabling more comprehensive analysis of grid dynamics and storage integration.

Findings

Demonstrates integration of storage units into Kuramoto models.

Shows potential for improving frequency stability with storage control strategies.

Identifies limitations related to storage capacity and response time.

Abstract

In future power systems, electrical storage will be the key technology for balancing feed-in fluctuations. With increasing share of renewables and reduction of system inertia, the focus of research expands towards short-term grid dynamics and collective phenomena. Against this backdrop, Kuramoto-like power grids have been established as a sound mathematical modeling framework bridging between the simplified models from nonlinear dynamics and the more detailed models used in electrical engineering. However, they have a blind spot concerning grid components, which cannot be modeled by oscillator equations, and hence do not allow to investigate storage-related issues from scratch. We remove this shortcoming by bringing together Kuramoto-like and algebraic load-flow equations. This is a substantial extension of the current Kuramoto framework with arbitrary grid components. Based on this concept, we provide a solid starting point for the integration of flexible storage units enabling to address current problems like smart storage control, optimal siting and rough cost estimations. For demonstration purpose, we here consider a wind power application with realistic feed-in conditions. We show how to implement basic control strategies from electrical engineering, give insights into their potential with respect to frequency quality improvement and point out their limitations by maximum capacity and finite-time response.

Equations

$$m_i{\ddot{\delta }}_i=$$

$${\alpha }_i{\dot{E}}_i=$$

$$+{\chi }_i\sum _{j=1}^N{B}_{ij}{E}_j\cos {\delta }_{ij}.$$

$$P_{\mathrm{SCU},i}^{\mathrm{out}}=\sum _{j=1}^N{B}_{ij}{E}_i{E}_j\sin {\delta }_{ij},$$

$$P_{\mathrm{WPP}}(t)=|P_{\mathrm{sys}}^{\mathrm{cons}}|+x(t)$$

$$\dot{y}$$

$$\dot{\tilde{x}}$$

$$q_{\bar{\omega }}=\{\begin{array}{l}{\int }_{\bar{\omega }}^{\infty }p(\omega )\mathrm{d}\omega \mathrm{for}\bar{\omega }>0,\\ \\ {\int }_{-\infty }^{-\bar{\omega }}p(\omega )\mathrm{d}\omega \mathrm{for}\bar{\omega }<0,\end{array}$$

$$P_{\mathrm{droop}}(t)=k_{\mathrm{DC}}({\omega }_{\mathrm{sys}}^{\mathrm{nom}}-{\omega }_{\mathrm{sys}}(t))=-k_{\mathrm{DC}}\omega (t),$$

$$P_{\mathrm{droop}}(t)=\{\begin{array}{l}-k_1^{\mathrm{DC}}{\omega }_{\mathrm{sys}}(t)\forall {\omega }_{\mathrm{sys}}\ge 0,\\ k_2^{\mathrm{DC}}({\omega }_{\mathrm{sys}}(t){)}^n\forall {\omega }_{\mathrm{sys}}<0.\end{array}$$

$$\Delta P_{\mathrm{ramp}}(t)=\frac{P_{\mathrm{in}}(t)-P_{\mathrm{ref}}}{P_{\mathrm{norm}}}$$

$$|\Delta P_{\mathrm{ramp}}|\le r_{\mathrm{tol}}.$$

Full text

††preprint: AIP/123-QED

I Introduction

The transition of the electrical energy system towards sustainability is paralleled by grid decentralization and increasing percentage of renewables. This development requires novel grid operation and design concepts. Electrical storage will be a key component of future energy systems to balance feed-in variations and mitigate power quality problems induced by stochastic renewables Albadi and El-Saadany (2010); Ibrahim et al. (2011); Ren et al. (2017). Therefore, new research issues emerge concerning optimal grid embedding and sizing of storage facilities as well as smart storage control strategies, which are customized to the specific application purpose and feed-in properties.

Wind and solar have characteristic non-Gaussian statistics over a broad range of time scales from seasonal and diurnal imbalances down to sub-second fluctuations Anvari et al. (2016); Ren et al. (2017). Short-term fluctuations on the second and sub-second scale are a particular challenge for power system operation, since standard load balancing such as primary controlent (2009) does not operate yet on these time scales. As a consequence, frequency quality is significantly reduced Albadi and El-Saadany (2010); Ibrahim et al. (2011); Schmietendorf, Peinke, and Kamps (2017). This problem is exacerbated by a side effect: as conventional power plants are progressively substituted by renewables, system inertia is decreased and the grid becomes more sensitive to sudden perturbations in terms of feed-in fluctuations ent (2017); Tielens and Van Hertem (2012).

Against this backdrop, the focus of power grid research shifts towards short-term dynamics and multidisciplinary approaches including self-organization and collective phenomena. This requires a profound mathematical modeling framework mediating between the simple conceptual models from nonlinear dynamics and the detailed models used for case studies in electrical engineering. Over the past decade, the Kuramoto-like modeling framework has been established as a suitable instrument for this purpose. It is derived from the original Kuramoto model, which describes the phase dynamics of coupled oscillators, in particular the phase transition from incoherence to self-organized synchronization Kuramoto (1975); Strogatz (2000). Kuramoto-like models have been used to address various issues of power system dynamics and topology-stability interplay Filatrella, Nielsen, and Pedersen (2008); Menck et al. (2014); Motter et al. (2013); Rohden et al. (2012, 2016, 2017); Witthaut et al. (2016); Wolff, Lind, and Maass . In a previous work Schmietendorf, Peinke, and Kamps (2017), it was shown how the turbulent-like character of wind feed-in, in particular its intermittency, is directly transferred into frequency and voltage fluctuations. This was confirmed by real-world frequency measurements Haehne et al. (2018). Other recent studies on Kuramoto-like grids with stochastic feed-in investigated the propagation of frequency quality deterioration Auer et al. (2017); Zhang (2017) and potential routes to system instability Schäfer et al. (2017). However, the current Kuramoto-like framework does not allow to implement grid components, which are not modelled by oscillator equations. This shortcoming affects the integration of storage units with arbitrary control strategies from scratch and hence prevents from fundamental investigations of storage-related issues.

With our study, we fill this gap. The primary target was to integrate a flexible storage model, which does not imply any restrictive assumptions on storage features or control strategies beforehand. For this purpose, we introduce a novel approach by bringing together Kuramoto-like differential and algebraic load-flow equations, which are a standard tool in power-flow analysis. The general idea of embedding grid components by means of load-flow equations has a broader range of application: it can serve as a starting point to implement arbitrary grid components into Kuramoto-like power grids, e. g. power inverters with various types of control or nodes connecting different grid levels. This broadens the scope of KM-like models significantly. At the same time, the modeling framework is still a reduced approach compared to the detailed models used in electrical engineering and yet simple enough to address power grid dynamics from the viewpoint of self-organization and collective dynamics, i. e. methods beyond the standard engineering practice.

For demonstration purpose, we consider frequency quality improvement by means of a storage facility with limited capacity in a simplified power system subjected to realistic wind feed-in. This application example has been identified as one of the key issues on the road to power systems with high percentage of wind and solar by electrical engineering communities Albadi and El-Saadany (2010); Jabir et al. (2017); Li, Hui, and Lai (2013); Ibrahim et al. (2011); Zhao et al. (2015). In order to provide a guide to the implementation of storage units as a starting point for follow-up research, we demonstrate basic control strategies adopted from electrical engineering and give insights into their potential and limitations with respect to different aspects of frequency quality improvement.

The paper is organized as follows: First, we briefly address electrical storage in wind and solar applications and list the features of real storage units, which should be implementable into the model. Then we outline Kuramoto-like power grid modeling and describe how to integrate storage units by means of load-flow equations. After that, we specify the simplified power system with realistic wind power input, which we use in this study. We close the subsection with an explanation of the frequency quality assessment we use, and how this is related to established electrical engineering practice. Then we turn to the application example: We start with a preliminary performance assessment by considering the ability to ensure stationary operation as function of maximum capacity. Subsequently, we demonstrate how to implement three basic control strategies, namely: state-of-charge feedback reinterpreted as storage resource management, droop control and ramp-rate control. We investigate their potential with respect to frequency quality improvement. It shows that these control concepts have different advantages according to their underlying main target. It is pointed out that the ambition in terms of control strength or tolerance range has to be carefully adjusted to the storage dimension in order to perform optimally. Finally, we demonstrate that these strategies are sensitive against finite-time response and confirm that short-term frequency quality applications require storage and control systems with rapid response. We conclude with a summary of the main results and give an outlook to storage-related problems, which can now be addressed within the context of Kuramoto-like power grid models.

II Model and Methods

II.1 Electrical storage

Electrical energy storage Luo et al. (2015) denotes the process of converting surplus electrical power into a storable form and reserving it, until it is converted back when required. It is commonly categorized by the form of energy stored, but also in terms of their technical features like response time or capacity, or their function. Electrical storage is already or considered as a promising candidate for various wind and solar power applications. The type of storage follows its function, or to be more precise, the underlying time scale of power variability. For long-term storage applications like time-shifting, peak-shaving, seasonal storage and mid-term frequency control, storage types with large energy dimensions are used, which do not necessarily feature fast response, e. g. pumped hydro, hydrogen-based or compressed air storage. Short-term frequency quality improvement and power output smoothing require rapid response (ranging from few seconds to milliseconds), which is usually paralleled by smaller energy capacity. Candidates for this application are flywheels, batteries, superconducting magnetic energy storage and (super) capacitors. Díaz-González et al. (2012); Jabir et al. (2017)

The storage model to be developed has to meet two requirements: On the one hand, simplifications are essential in order to fit the model into the Kuramoto-like framework. On the other hand, all relevant characteristics of real storage operation have to be implementable, namelyDíaz-González et al. (2012); Jabir et al. (2017); Luo et al. (2015):

• •

*Efficiency

*In practice, the energy conversion processes can not be realized without losses. The efficiency factor $\eta$ gives the ratio of input to output energy. It depends on the type of storage.

• •

Maximum energy capacity and *power rating

*These define the main dimensions of the storage facility. The energy capacity is the maximum energy the storage unit is able to deliver and hence serves as an upper limit for the amount of energy stored. The power rating corresponds to the maximum instantaneous supply.

• •

*Control strategies

*The storage control strategy determines the storage output at time $t$ as a function of one or more feedback variables. It can be used to manage the storage resources or to provide system services like frequency control and power output smoothing.

• •

*Response time

*The storage unit has a finite response time effecting a time delay between the feedback signal and its reaction. The response time depends on the type of storage and the underlying control mechanism.

II.2 Kuramoto-like power grid models

Kuramoto-like grid models are based on networks of synchronous machines with producers (generators) and consumers (motors) converting mechanical power into electrical power and vice versa. Real and reactive power is transferred among these nodes via transmission lines. The topology of the underlying network is condensed in the nodal admittance matrix, $\{Y_{ij}\}_{i,j=1,..,N}$, with $N$ being the number of nodes. The common assumption of lossless transmission yields $Y_{ij}\approx\mathrm{i}\mathfrak{Im}(Y_{ij})=\mathrm{i}B_{ij}$ with susceptance $B_{ij}$. Each node $i\,\in\mathcal{M}_{\mathrm{grid}}$ of the grid is associated with a complex nodal voltage $\bm{E}_{i}=E_{i}\mathrm{e}^{\mathrm{i}\delta_{i}}$ with $E_{i}$ being the voltage magnitude and $\delta_{i}$ the phase with respect to a reference frame rotating with nominal frequency. (Hence, $\dot{\delta}_{i}=\omega_{i}=0$ means that node $i$ is at nominal frequency.)

The coupled frequency-voltage dynamics of the synchronous machines are given by Machowski, Bialek, and Bumby (2008); Schmietendorf et al. :

[TABLE]

The parameters $m_{i}$ and $\gamma_{i}$ denote the total inertia and effective damping, $P_{i}$ is the mechanical power feed-in or consumption. $P_{ij}=B_{ij}E_{i}E_{j}\sin\delta_{ij}$ is the real power transfer between nodes $i$ and $j$, and the interaction term in Eq. (1b) is related to reactive power flows. $\alpha_{i}$, $C_{i}$, and $\chi$ can be calculated from machine parameters. The term $\beta_{i}(\bar{E}_{i}-E_{i})$ mimics a proportional voltage controller, which pulls the voltage towards its nominal value Schmietendorf, Peinke, and Kamps .

II.3 Integration of storage units by means of load-flow equations

We now assume a power network consisting of a set of conventional synchronous machines $\mathcal{M}_{\mathrm{syn}}$ (modeled acc. to Eqs.(̇1a) and (1b)) and a set of storage units with control equipment $\mathcal{M}_{\mathrm{SCU}}$. The storage units are also associated with nodal voltages $\bm{E}_{i}=E_{i}\mathrm{e}^{\mathrm{i}\delta_{i}}$. However, their dynamics differ from synchronous machines in that they lack inertia and have no inherent physical relationship between frequency and electrical power output. The most direct approach, which does not include any restrictive assumptions on storage features or control, is to calculate the phase $\delta_{i}$, $i\in\mathcal{M}_{\mathrm{SCU}}$, by solving the algebraic load flow equation Kundur, Balu, and Lauby (1994); Machowski, Bialek, and Bumby (2008)

[TABLE]

with the nodal voltage $E_{i}$ assumed to be constant. $P_{\mathrm{SCU},i}^{\mathrm{out}}$ is the power being injected into the grid by storage-control unit.

Load-flow analysis is a standard tool in electrical engineering for power-flow calculations on power networks. It is can be derived from basic physical relationships given by Kirchhoff’s and Ohm’s laws. This approach is particularly qualified as a starting point for the investigation of smart control since the power fed into the grid $P_{\mathrm{SCU},i}^{\mathrm{out}}$ can be determined by arbitrary control strategies and the model can flexibly be complemented with the other realistic storage characteristics listed above. The combination with load-flow equations provides a general method for the straight-forward implementation of arbitrary grid components into Kuramoto-like networks. This includes, for example, for power inverters and nodes linking different voltage levels or microgrid-macrogrid connections.

II.4 Simplified power system with wind feed-in and storage facility

In this study, we demonstrate the operation of a storage unit with control equipment by means of a simplified system consisting of a generating unit with wind power feed-in and a synchronous machine mimicking the response of the grid in terms of frequency $\omega_{\mathrm{sys}}$ and voltage $E_{\mathrm{sys}}$ in a coarse-grained view (see Fig. 1). We consider its application with respect to frequency quality under fluctuating wind power feed-in.

The storage unit is assumed to have a finite maximum capacity $K_{\mathrm{max}}$. The actual capacity $K(t)$, or in more casual terms: the “filling level” at time $t$, corresponds to the state of charge with respect to batteries. $K_{\mathrm{max}}$ serves as an upper bound: if $K(t)=K_{\mathrm{max}}$, surplus energy cannot be stored and has to be discarded. On the other hand, the storage unit can only provide balancing power, if $K(t)$ is sufficient. For $K(t)=0$, only positive power mismatch can be mitigated, whereas the system is exposed to negative power deficiencies.

For the sake of simplification, we neglect efficiency and limitations due to power rating here. This means we assume lossless conversion and that the power to be delivered according to the specific control strategy is provided completely if permitted by the storage filling $K(t)$. The storage unit can be equipped with different control strategies. Three standard strategies adopted from engineering practice (state-of-charge dependent resource management, droop control, ramp rate control) are specified and investigated below. Our intention is to demonstrate the general impact of different basic storage strategies and their limitations due to maximum capacity and time-delay on system behaviour rather than to model a detailed situation or derive concrete guidelines. The approach can be applied to more concrete situations in the course of follow-up research. Complex optimization problems may evolve depending on multiple factors such as operation conditions and technical system requirements, cost concerns, legal and economic framework etc.

II.5 Wind power feed-in

Due to atmospheric turbulence, wind power has specific turbulent-like characteristicsMilan, Wächter, and Peinke (2013); Anvari et al. (2016): extreme events, correlations, Kolmogorov power spectrum, and intermittent increment statistics. We implement realistic wind feed-in time series taking these basic properties into account

[TABLE]

with the constant part $|P_{\mathrm{sys}}^{\mathrm{cons}}|$ meeting the consumption of the system. The fluctuating time series $x(t)$ is generated as follows Schmietendorf, Peinke, and Kamps (2017): first, a time series $\tilde{x}(t)$ is generated by means of the Langevin-type system of equations

[TABLE]

with $\gamma=1.0$, $g=0.5$, $x_{0}=2.0$, $D=2.0$ and $\delta$-correlated Gaussian white noise $\Gamma$. Then the corresponding Fourier spectrum is modified so that the final power spectrum $S(f)=|F(f)|^{2}$ roughly reproduces real data sets, in particular the Kolmogorov $\frac{5}{3}$-decay. Transforming back to real space yields $x(t)$ (see Fig. 1(a) for an exemplary part of the feed-in time series $P_{\mathrm{WPP}}(t)$). Since $\langle x\rangle=0$, power balance is given over time: $\langle P_{\mathrm{WPP}}\rangle=|P_{\mathrm{sys}}^{\mathrm{cons}}|$. Due to the generating process, $P_{\mathrm{WWP}}$ features a smallest frequency mode. Lower frequencies corresponding to power variations on longer time scales are assumed to be handled by other mechanisms like standard load balancing. In practice, different time scales can actually be divided up and assigned to different control mechanisms by low-pass filtering Zhao et al. (2015).

II.6 Power quality assessment

Power quality is a wide ranging notion, which includes different aspects of voltage and frequency stability and supply reliability. In this study, we focus on short-term frequency quality. We use different criteria for performance assessment, which in combination give a more detailed picture of frequency quality111Electrical engineering literature indicates power output smoothing as another major issue with respect to stochastic feed-inJabir et al. (2017); Li, Hui, and Lai (2013). This is related to frequency quality are related, as system frequency and power balance are coupled..

Frequency quality ent (2009, 2013, 2017) refers to the systems ability to maintain nominal frequency $\omega_{\mathrm{sys}}^{\mathrm{nom}}$, or keep the frequency within a pre-defined range (the standard frequency range) for a large percentage of operation time. It can be evaluated on different time scales. Short-term frequency quality is referred to instantaneous frequency deviations in electrical engineering. It is commonly evaluated by means of the percentage of time the system frequency is outside the standard frequency range ent (2013).

Against this practical backdrop, we define

[TABLE]

with $\bar{\omega}$ denoting the bound given by the standard frequency range $\omega_{\mathrm{sys}}^{\mathrm{nom}}\pm\bar{\omega}$ and $p(\omega)$ being the probability distribution of frequency deviations $\omega(t)=\omega_{\mathrm{sys}}(t)-\omega_{\mathrm{sys}}^{\mathrm{nom}}$. In real power grids, the nominal frequency is 50 Hz or 60 Hz. Kuramoto-like power grid models are usually transformed into a reference frame rotating with nominal frequency so that here $\omega_{\mathrm{sys}}^{\mathrm{nom}}=0$. For sufficiently long simulation, $q_{\bar{\omega}}$ corresponds to the percentage of time the system is expected to operate outside of the frequency range defined by $\bar{\omega}$ on average. With a specified standard frequency range $\bar{\omega}$, $q_{|{\bar{\omega}}|}=q_{-\bar{\omega}}+q_{\bar{\omega}}$ was introduced as the exceedance measureAuer et al. (2017). In this study, we evaluate $q_{\bar{\omega}}$ as a function of $\bar{\omega}$ rather than for one specified $\bar{\omega}$ for two reasons: firstly, this gives a more informative picture of system dynamics; secondly, the value of standard frequency range is not unequivocally defined222In ent (2013), this fact is mentioned with view to the different characteristics of transmission grid areas. This applies even more for the heterogeneous operation conditions on the distribution grid level and in islanded microgrids..

Frequency quality not only involves deviations from nominal frequency, but also the time derivative $\mathrm{d}\omega/\mathrm{d}t$, commonly referred as the rate of change of frequency ent (2013, 2017). The rate of change of frequency reflects the sensitivity against sudden perturbations and is inversely proportional to system inertia Tielens and Van Hertem (2012). In former times, it was of interest mainly during transient periods after significant imbalances ent (2017). Nowadays, due to the loss of system inertia and the increasing percentage of stochastic renewables inducing continuous perturbations, the rate of frequency change becomes relevant also during “normal operation”.

The frequency changes can be related to the increments $\Delta\omega_{\mathrm{sys}}=\omega_{\mathrm{sys}}(t+\Delta t)-\omega_{\mathrm{sys}}(t)$. It was shown that intermittency of wind power in terms of heavy-tailed probability density functions is directly transferred into frequency fluctuations and significantly contribute to frequency quality decrease Schmietendorf, Peinke, and Kamps (2017). Therefore, we here capture increments statistics not only by their mean value $\mu_{|\Delta\omega|}$ and standard deviation $\sigma_{\Delta\omega}$ (as it is standard in electrical engineering) but also their kurtosis $\kappa_{\Delta\omega}=\mu_{4,\Delta\omega}/\sigma_{\Delta\omega}^{4}$ (with $\mu_{4}$ denoting the fourth moment of the distribution)333The statistical measures $\mu_{|\Delta\omega|}$, $\sigma_{\Delta\omega}$ and $\kappa_{\Delta\omega}$ refer to a given time lag $\Delta t$, i. e. strictly speaking we have $\mu_{|\Delta\omega|}^{\Delta t}$ etc. We drop the upper index, but specify $\Delta t$ in the following analysis. The kurtosis serves as a measure for the tailed-ness of the distribution. With $\kappa=3$ being the value of the Gaussian distribution, $\kappa>3$ means that there are more extreme events or outliers than in the Gaussian case, and vice versa for $\kappa<3$. Note that $\kappa$ entails information about the shape of the distribution, not about the magnitude of the outliers.

As we will see, the extreme events observed in the increment statistics in this study have two reasons: on the one hand, the system is exposed to the feed-in fluctuations $P_{\mathrm{WPP}}(t)$ during time intervals, in which the storage facility is not able or supposed to fully compensate power imbalances. As stated above, these fluctuations are known to transfer intermittency into the frequency statistics. On the other hand, new extreme events can be induced when the storage steps in. For example, if the system runs out of storage in the course of a longer time period with power deficiency or discontinues balancing quite suddenly due to its control specifications, the frequency may face an instantaneous drop. The following analysis will show that the kurtosis serves as a good indicator for an inaccurate adjustment of control strength.

III Results

III.1 Storage control limited by maximum capacity

We start with a simple storage strategy, which provides a first insight into the performance of the storage facility as a function of its maximum capacity. We assume the storage facility to have a maximum energy capacity $K_{\mathrm{max}}$. The actual power mismatch is $\Delta_{P}(t)=P_{\mathrm{WPP}}(t)-P_{\mathrm{SCU}}(t)$ (see Fig. 1). In this simplified operation mode, the storage unit is intended to ensure power balance between feed-in and consumption whenever possible:

• •

For positive power mismatch $\Delta_{P}(t)\geq 0$, $P_{\mathrm{SCU}}^{\mathrm{out}}(t)=|P_{\mathrm{sys}}^{\mathrm{cons}}|$ is fed into the system. The corresponding energy surplus $\Delta_{K}$ is stored with the maximum capacity $K_{\mathrm{max}}$ being an upper bound.

• •

For $\Delta_{P}(t)<0$, the power mismatch is

• •

either fully compensated, i. e. $P_{\mathrm{SCU}}^{\mathrm{out}}(t)=|P_{\mathrm{sys}}^{\mathrm{cons}}|$, if the storage is sufficiently filled,

• •

or the rest capacity is used to provide $P_{\mathrm{SCU}}^{\mathrm{out}}(t)$ with $P_{\mathrm{WPP}}(t)<P_{\mathrm{SCU}}^{\mathrm{out}}(t)<|P_{\mathrm{sys}}^{\mathrm{cons}}|$.

• •

While $K(t)=0$, no balancing power can be provided.

In this mode of operation, the system dynamics alternate between stationary operation and time intervals with $\omega_{\mathrm{sys}}<0$, in which the system either fluctuates in reaction to the stochastic feed-in, or is on its way to return to stationary operation (see Figure 2 (a)). Figure 2 (b) shows how the percentage of non-stationary operation time $q_{\mathrm{non-stat}}=q_{|\mathrm{10^{-3}}|}$444We define $|\omega_{\mathrm{sys}}(t)|<|10^{-3}|$ as stationary operation. This is in accordance with real power grids in the sense that, strictly speaking, these are constantly subjected to disturbances and hence, never in a stationary state with $\omega_{\mathrm{sys}}=\omega_{\mathrm{sys}}^{\mathrm{nom}}$ and $\frac{\mathrm{d}}{\mathrm{d}t}\omega_{\mathrm{sys}}=0$. decreases to zero with maximum storage capacity $K_{\mathrm{max}}$. The behaviour for the $K\rightarrow\infty$ limit is trivial in qualitative respects, as it implies that a sufficiently large storage capacity is able to continually balance power differences and guarantee stationary operation. It was to be expected as we restricted our analysis to a limited time scale, i. e. we assumed the feed-in fluctuations to be balanced by other load control mechanisms on longer time scales.

However, our analysis so far was only to serve as a first storage capacity assessment. In reality, the equipment of wind and PV plants with large storage capacity is cost expensive. In the following, we therefore consider the more realistic and less trivial situation of a storage facility with “insufficient capacity”.

III.2 Storage control strategies

We now investigate different basic storage control strategies with regard to their potential to improve frequency quality. We set the maximum storage capacity $K_{\mathrm{max}}=2.0$. This would allow for stationary operation in about 90 % of time assuming the simplified strategy considered above. The control strategies refer to accepted methods in engineering practice, and rely on different control feedback signals:

• •

the actual storage level $K(t)$, or the state-of-charge (here used for the purpose of storage resource management),

• •

system frequency as an indicator for power imbalances (droop control), and

• •

power differences between certain time steps (ramp rate control).

These methods cover the elementary strategies for power quality improvement and therefore provide a basic structure for the development of smart control techniques by refining the conventional strategies or composing hybrid systems.

III.2.1 Storage resource management

In the current technical application, the state-of-charge is applied as a feed-back signal in battery storage systems mainly to guarantee operation within proper state-of-charge range and to prevent shut-down due to over-charge Li, Hui, and Lai (2013). Here, we shift the scope of application to the grid side and reinterpret the basic idea as a form of intelligent storage management.

We complement the simple storage strategy presented above and make the balance power at time $t$ dependent on the current capacity $K(t)$. To be specific, the power to be delivered by the storage is the power deficiency $\Delta_{P}$ multiplied by a factor $f=f(K(t))\in[0,1]$ with $f(0)=0$ and $f(K_{\mathrm{max}})=1$. Here, we consider the different realizations for $f(K)$ depicted in Fig. 3 (a): $f(K)=-(1-K/K_{\mathrm{max}})^{n}+1$ for $n=12,\,4,\,2$ (denoted as scenarios I, II and III) and $f(K)=(K/K_{\mathrm{max}})^{n}$ for $n=1,\,2,\,4,\,12$ (scenarios IV-VII).

Fig. 3 (b) shows $q_{\bar{\omega}}(\bar{\omega})$ for the different realization of storage resource management in comparison to the simplified storage strategy described in the previous subsection. With proper a choice of $f(K)$, the proposed storage resource management can in fact serve to prevent large frequency deviations. Of course, a higher percentage of small deviations has to be tolerated in exchange. The best option can only be chosen in knowledge of the operational circumstances and the specific guidelines for $\bar{\omega}$. Furthermore, storage resource management can be applied as part of a multi-pronged control strategy in combination with other storage control mechanisms.

III.2.2 Droop control

Droop control is based on the relationship between power imbalances and system frequency: a positive power mismatch is paralleled by frequency increase, whereas negative mismatch leads to frequency decrease. This fact can also be observed in Kuramoto-like grids.

Droop control has a broad range of application, which includes frequency control services provided by wind power plants Zhao et al. (2015). The standard practice is to use a linear droop control mechanism, whose slope is given by the control strength $k_{\mathrm{DC}}$. The balancing droop power is

[TABLE]

as $\omega_{\mathrm{sys}}^{\mathrm{nom}}=0$ here. If $\omega_{\mathrm{sys}}(t)<0$, this is interpreted as an indicator of negative power balance, and consequently more power is injected into the system in order to keep system frequency close to its nominal value. For $\omega_{\mathrm{sys}}(t)>0$, power feed-in is reduced accordingly.

This adjustment of power input to the actual system frequency obviously requires storage capacity in the background. A specific type of inverter with linear droop control was shown to behave analogue to a synchronous machine Schiffer et al. (2013), and was already implemented into Kuramoto-like grids with stochastic feed-in Auer et al. (2017). Our approach here is different in the sense that we explicitely take into account the limits of the installed background storage capacity $K_{\mathrm{max}}$ but do not assume any further specifications on the grid feed-in process. As explained above, in case of $\omega_{\mathrm{sys}}(t)<0$, the balancing power $P_{\mathrm{droop}}>0$ can only be provided to the extent that the storage level $K(t)$ is sufficient, and for $\omega_{\mathrm{sys}}(t)>0$, surplus power can only be stored with $K_{\mathrm{max}}$ as an upper bound.

We first investigate system performance under standard droop control according to Eq. (7) for fixed maximum storage capacity $K_{\mathrm{max}}=2.0$ and varying control strength $k_{\mathrm{DC}}$. Fig. 4(a) shows system frequency in response to the same power feed-in $P_{\mathrm{WPP}}(t)$ for different $k_{\mathrm{DC}}$. With increasing control strength, the positive frequency deviations are more and more eliminated, as it is always possible to feed in less power than available. In contrast, balancing negative frequency deviations requires sufficient storage level. This asymmetry can also be seen in Fig. 4(b1), which reveals the dilemma of standard droop control under limited storage capacity: On the one hand, for sufficiently large control strength $k_{\mathrm{DC}}$, the positive frequency deviations can be more or less eliminated; but in this case, the storage unit runs out of capacity quickly at the beginning of longer timer periods with negative power mismatch. Fig. 4(a) highlights an concrete example of a frequency dip not being prevented due to overambitious control strength. On the other hand, for small control strength, the storage facility performs better in the sense that it reduces the probability of large negative frequency deviations. But at the same time, the droop control mechanism remains sub-optimal with respect to positive frequency deviations.

To overcome this problem, we propose a non-symmetric droop control strategy, which treats positive an negative frequency deviations differently:

[TABLE]

We choose large control strength $k^{\mathrm{DC}}_{1}$ in order to counteract the positive frequency deviations and tested two control schemes for the negative frequency range: (i) quartic droop control555Nonlinear droop control is technically feasible Prabhakaran, Goyal, and Agarwal (2018). ; and (ii) linear droop control with small control strength $k^{\mathrm{DC}}_{2}$. Fig. 4(b2) shows that these alternative strategies combine the best of both small and large control strength in standard droop control: they effectively mitigate positive deviations and prevent large frequency dips. A nonlinear control-term, inter alia, gives the opportunity to focus the onset of control to a specific frequency bound. For example, the drop of $q_{\bar{\omega}}(\bar{\omega})$ indicates that the quartic control term actually starts acting around $\omega_{\mathrm{sys}}\approx 0.05$.

With view to the frequency increments statistics (see Fig. 4(c)), the mean value $\mu_{|\Delta\omega|}$ and standard deviation $\sigma_{\Delta\omega}$ decrease with increasing control strength, finally converging to a minimum value. However, the non-Gaussianity in terms of kurtosis $\kappa_{\Delta\omega}$ grows, even when $\mu_{|\Delta\omega|}$ and $\sigma_{\Delta\omega}$ have nearly approached their minima and barely change666The corresponding values of the statistical measures for the alternative strategies (i) and (ii) are: (i) $\mu_{|\Delta\omega|}=2.25\cdot 10^{-5}$, $\sigma_{\Delta\omega}=3.21\cdot 10^{-5}$, $\kappa_{\Delta\omega}=9.00$, and (ii) $\mu_{|\Delta\omega|}=2.26\cdot 10^{-5}$, $\sigma_{\Delta\omega}=3.16\cdot 10^{-5}$, $\kappa_{\Delta\omega}=8.50$.. This is an indicator that the control strength $k_{\mathrm{DC}}$ is getting too ambitious and the storage facility runs out of capacity more frequently. It therefore becomes evident that increment statistics are an essential part of a comprehensive picture of frequency quality.

III.2.3 Ramp rate control

A power ramp is defined as a normalized power change or power increment:

[TABLE]

with input power $P_{\mathrm{in}}(t)$ and reference power $P_{\mathrm{ref}}$. Ramp rate control Marcos et al. (2014); Schnabel and Valkealahti (2016) aims at keeping power ramps within specified tolerance bounds:

[TABLE]

It is utilized in wind and solar power applications. In the latter case, power ramps play a even major role due to passing clouds.

The basic idea opens up numerous opportunities for concrete realization depending of the choice of $P_{\mathrm{ref}}$. For example, it can be given by prior values $P(t-\Delta t)$ defined by a sampling time $\Delta t$ or be calculated as a function of the actual demand. We here demonstrate a version of ramp rate control, which mainly targets on short-term ramps: First, we set $P_{\mathrm{ref}}=P_{\mathrm{SCU}}^{\mathrm{out}}(t-\Delta t)$ with $\Delta t=0.005$. As long as the ramp condition Eq. (10) is satisfied, no balancing is necessary and $P_{\mathrm{SCU}}^{\mathrm{out}}(t)=P_{\mathrm{WPP}}(t)$. If the condition is violated, the storage facility steps in: For $\Delta P_{\mathrm{ramp}}>0$ (upward ramps), $P_{\mathrm{SCU}}^{\mathrm{out}}$ is decreased so that $|\Delta P_{\mathrm{ramp}}|=r_{\mathrm{tol}}$ and surplus power is stored. For $\Delta P_{\mathrm{ramp}}<0$ (downward ramps), the storage is supposed provide balance power in order to fulfill $|\Delta P_{\mathrm{ramp}}|=r_{\mathrm{tol}}$. Again, the storage of surplus power is limited by the maximum capacity $K_{\mathrm{max}}$ and balancing power can only be delivered if the actual storage level $K(t)$ is sufficient.

The performance of ramp rate control is usually assessed with respect to the power feed-in statistics. Here, we consider frequency statistics instead, for two reasons: first, this is the scope of our study and consistent with the previous analysis. Secondly, we investigate a consequential phenomenon, as power fluctuations are directly transferred into frequency variations. Fig. 5 (a) and (b) show how the likelihood of tolerance bound violations in terms of $q_{\bar{\omega}}(\bar{\omega})$ and the frequency increment statistics evolve as functions of the tolerance ramp rate $r_{\mathrm{tol}}$. It shows that the ramp rate control strategy fulfills its main purpose with view to the increment statistics: by suppressing power ramps, frequency increments can be mitigated significantly. In parallel, the percentage of operation time beyond certain tolerance bounds can be decreased.

Again, the ambition of control, here in terms of $r_{\mathrm{tol}}$, has to be chosen carefully. On the one hand, if $r_{\mathrm{tol}}$ is too large, the control does not achieve its potential. It has no influence on $q_{\bar{\omega}}(\bar{\omega})$ and barely improves the increment statistics. On the other hand, if $r_{tol}$ is too small, the storage tends to run out of capacity. This is indicated by a steep rise of $\kappa_{\Delta\omega}$ and increasing likelihood of large negative frequency deviations. Compounding the problem in this specific version of ramp-rate control is the fact that if the power input $P_{\mathrm{SCU}}^{\mathrm{out}}$ drops to a low $P_{\mathrm{WPP}}$ during a feed-in deficit period with empty storage, this value serves as the new reference $P_{\mathrm{ref}}$. In the following, the input power and system frequency can return to their nominal values only slowly due to the tight tolerance range, even if storage capacity is available. As explained before, running out of storage is paralleled by sudden frequency drops, which is indicated by the drastic increase of the non-Gaussianity of the increment distribution.

Note that the dissymmetry between positive and negative frequency deviations (which can be seen in Fig. 5 (a)) is not completely analogue to the droop control case. First, ramp rate control responds to power input fluctuations (which cause of frequency fluctuations) and not to the deviation from nominal frequency directly. Secondly, upward ramps can always be balanced, irrespective whether the actual system frequency is below or above its nominal value. In contrast, balancing downward ramps requires sufficient storage. This particularly affects power deficit periods accompanied by $\omega_{\mathrm{sys}}<0$, during which the storage is depleted.

Comparing droop control and the applied version of ramp rate control, the latter has the advantage to be able to mitigate frequency increments to a certain extent without being paralleled by increasing non-Gaussianity. For example, droop control with $k_{\mathrm{DC}}=6.0$ and ramp control with $r_{tol}=0.03\sigma\%$ both reduce the mean value to $\mu_{\Delta\omega}\approx 1.6\cdot 10^{-5}$. At the same time, the statistics for the droop control case contain considerably more extreme events ($\kappa_{\Delta\omega}=13.7$) than the system with ramp rate mechanism ($\kappa_{\Delta\omega}=1.8$). On the other hand, the ramp rate control does not take into account the absolute deviation from nominal frequency, and hence is not designed to prevent large frequency excursion as efficiently as droop control.

III.3 Finite response time

Real control equipment does not react instantaneously but in response to the feedback signal at time $t-\tau$. In the following, we investigate and compare the sensitivity of the three control strategies introduced in the previous section. We implemented finite time response as follows777Alternatively, the time delay could be modelled by an ordinary differential equation with an appropriate time constant.:

• •

In case of storage resource management, the balance power to be delivered by the storage facility is $\Delta_{P}\cdot f(K(t-\tau))$888We here assumed finite-time response solely for the filling-level feedback, while the calculation of $\Delta_{P}$ and hence the decision whether the storage facility has to step in, happens instantaneously in response to the actual mismatch $\Delta_{P}(t)$. If this was not the case (this process would perhaps be associated with another time delay $\tau^{\prime}$), the situation would of course be exacerbated and also positive frequency deviations could occur.. We picked the linear storage resource management scenario VI as example.

• •

For droop control, we instance the asymmetric control strategy (ii) with $k^{\mathrm{DC}}_{1}=10.0$ and $k^{\mathrm{DC}}_{2}=0.5$. The balancing power $P_{\mathrm{droop}}(t)$ is calculated on the basis of $\omega_{\mathrm{sys}}(t-\tau)$.

• •

The ramp-rate control realization we presented above is very sensitive due to the short sampling rate. In view of finite time response, we consider another variant of ramp rate control and define the reference power $P_{\mathrm{ref}}=|P_{\mathrm{sys}}^{\mathrm{cons}}|$ $\forall t$ and the tolerance range $r_{\mathrm{tol}}=0.5\sigma$ here. The balancing power at time $t$ is calculated as the response to $\Delta P_{\mathrm{ramp}}(t-\tau)$.

Fig. 6 shows how time delay limits the possibilities for frequency quality improvement with focus on the main target of each control strategy.

Storage resource management was introduced in order to prevent overspending and save capacity to mitigate large frequency deviations. From Figure 6 (a), one can see that finite response time has negligible impact up to $\tau=10.0$. Then the deviations from the instantaneous-response case become more and more apparent. In particular, the control strategy increasingly misses its main objective as the probability of large deviations from nominal frequency grows.

Droop control is intended to mitigate deviations from nominal frequency. Fig. 6 (b) shows that in this respect the system is able to handle a finite response time up to $\tau=0.1$ quite well. Then $q_{\bar{\omega}}$ starts to increase for small $\bar{\omega}$ as the feedback delay causes trouble when the system fluctuates close around nominal frequency. As the control switches between positive and negative balancing too late, oscillations around nominal frequency are induced. A nonlinear droop scheme could mitigate these oscillations as it interfers less for small deviations.

Ramp rate control was shown to be a promising candidate for frequency quality improvement with respect to increment statistics. The version of ramp-rate control considered here is very sensitive towards time delay. Fig. 6 (c) shows that the introduction of finite response time leads to a reduction of frequency quality as $\mu_{|\Delta\omega|}$, $\sigma_{\Delta\omega}$ and $\kappa_{\Delta\omega}$ immediately increase, even beyond the no-control case.

These results indicate that short-term frequency quality applications require rapid response of the underlying control mechanism (on sub-second scale999As we use a greatly simplified power system in dimensionless units in order to point out general relationships, we are careful with specifying concrete values. However, at this point, we want to give a rough idea about the time scale. For $\mathcal{O}(m)\sim 10^{4}\,\mathrm{kg\,m}^{2}$, $\omega^{\mathrm{nom}}_{\mathrm{sys}}=2\pi\cdot 50\,$Hz, $\mathcal{O}(BE_{\mathrm{sys}}E_{\mathrm{SCU}})\sim 1\,\mathrm{GW}$ and $\mathcal{O}(1/\gamma)\sim 0.1\,\mathrm{s}-1\,\mathrm{s}$ (cf. Menck et al. (2014)), $t=\mathcal{O}(t)\sim 0.1\,\mathrm{s}-1\,\mathrm{s}$ and hence the response time of the storage facility has to be in the sub-second range. This coarse evaluation is in accordance with the technical features of electrical storage with the fastest response times being in the range of milliseconds Zhao et al. (2015).). As this is usually cost-expensive, it may be advisable to use hybrid systems and treat the high-frequency and lower frequency fluctuations separately with different storage and control systems.

IV Discussion and Outlook

We extended the current Kuramoto-like modeling framework with flexible storage units. With that, the scope of Kuramoto-like models opens up to one of the most important research topics in power grid engineering. On the way to this goal, we brought together Kuramoto-like equations and load-flow analysis. This is a substantial extension, which can serve as a starting point for the straight-forward implementation of arbitrary grid components.

For demonstration purposes, we considered short-term frequency quality improvement by means of storage facility with maximum capacity in a power system subjected to realistic wind feed-in. Motivated by recent findings, we assessed system performance not only with respect to frequency range violations, but also took into account frequency increment statistics.

We demonstrated how to implement three basic control methods, which cover the elementary strategies for power quality improvement in engineering practice. First, we adopted state-of-charge feedback control and reinterpreted it as a form of storage resource management. It has been proven that this concept can actually serve to save capacity in order to prevent large frequency deviations. Secondly, it was shown that droop control can improve frequency quality not only with view to deviations from nominal frequency but also with respect to frequency increment statistics. We pointed out that, particularly in case of limited capacity, it is favorable to handle positive and negative frequency deviations with different droop schemes and consider non-linear mechanisms. Thirdly, we implemented a version of ramp rate control. Originally designed for power-output-smoothing applications, we demonstrated that this strategy entails frequency quality improvement.

For both droop and ramp-rate control, it became apparent that the corresponding control strength or ramp tolerance range may not be too ambitious and have to be carefully proportioned to the dimensions of the storage facility. Furthermore, it was shown that the finite response time of the control mechanism limits the potential of the storage facility. Short-term frequency quality applications in particular require a rapid response.

With this study, we created a sound starting point for follow-up research on various aspects of storage implementation from the viewpoint of self-organized synchronization and collective phenomena. This includes stability-topology issues like optimal siting of storage units as well as comparative studies on global vs. local storage location or optimal sizing and rough cost-benefit assessment. Another current topic is the development and refinement of smart control strategies, which are customized to the realistic features of wind and solar power and, at the same time, take into account the impact of collective network dynamics. This study has already shown that the presented basic control strategies have different advantages and disadvantages. Against this backdrop, and with view to the impact of finite response times, systems with combined control techniques are conceivable solutions and novel smart control strategies should be developed.

Acknowledgements

Financial support from the Deutsche Forschungsgemeinschaft (PE 478/16-1 and MA 1636/9-1) is gratefully acknowledged.