Feasibility Study of Flywheel Mitigation Controls Using Hamiltonian-Based Design for E₃ High-Altitude Electromagnetic Pulse Events

Abstract

This paper explores the feasibility of implementing a flywheel energy storage system designed to generate voltage for the purpose of mitigating current flow through the transformer neutral path to ground, which is induced by a high-altitude electromagnetic pulse (HEMP) event. The active flywheel system presents the advantage of employing custom optimal control laws, in contrast to the conventional approach of utilizing passive blocking capacitors. A Hamiltonian-based optimal control law for energy storage is derived and integrated into models of both the transformer and the flywheel energy storage system. This Hamiltonian-based feedback control law is subsequently compared against an energy-optimal feedforward control law to validate its optimality. The analysis reveals that the required energy storage capacity is $13\mathrm{Wh}$, the necessary power output is less than $5\mathrm{kW}$ at any given time during the insult, and the required bandwidth for the controller is around $5\mathrm{Hz}$. These specifications can be met by commercially available flywheel devices. This methodology can be extended to other energy storage devices to ensure that their specifications adequately address the requirements for HEMP mitigation.

1. Introduction

In recent years, there has been increasing interest in mitigating the effects of high-altitude electromagnetic pulses (HEMPs) on the power grid [1,2]. A HEMP results from the detonation of a high-yield nuclear device in the Earth’s atmosphere [3]. HEMP fields can couple with conductive materials and devices, leading to significant operational challenges due to geomagnetically induced currents (GICs). The HEMP insult is conventionally categorized into three distinct time domains. The first time domain, referred to as the $E_1$ component, represents the most impulsive and powerful of the three components. According to IEC Standard 61000-2-9, the $E_1$ portion occurs within the first $0.1\mathrm{\mu }\mathrm{s}$ of the total insult waveform, exhibiting a field strength exceeding $10\mathrm{kV}/\mathrm{m}$ [4]. Mitigation strategies for this portion of the insult typically involve the use of trip coils [5,6].

The second time component of the HEMP insult is designated as the $E_2$ portion, which is agreed upon to occur from the conclusion of the $E_1$ insult up to $1\mathrm{s}$ following the detonation of the nuclear device. The potential field strength of the $E_2$ component is considerably weaker than that of the $E_1$ portion, ranging from $1\mathrm{V}/\mathrm{m}$ to $100\mathrm{V}/\mathrm{m}$. The final component, known as the $E_3$ portion, encompasses the total insult occurring from $1\mathrm{s}$ to $250\mathrm{s}$ post-detonation. This component is further divided into blast and heave segments, which will be elaborated upon in the System Models Section of this paper. The focus of this study is on mitigating the late-time, low-frequency portion of the HEMP insult to protect power grid transformers. The threats associated with this segment of the HEMP insult include, but are not limited to, core over-saturation, dielectric material failure, and electrical creep through insulation materials [7].

A common solution for preventing transformer magnetic core saturation during the late-time portion of a HEMP insult is the utilization of neutral path blocking capacitors [8,9]. In the presence of a DC offset, the neutral blocking capacitor behaves as an open circuit, thereby preventing a DC insult from passing through the core and out through the neutral ground. However, it is important to note that the late-time portion of the HEMP insult does not constitute a true DC signal [10]. The recent literature has demonstrated that this late-time component is sufficiently impulsive to generate system oscillations when interacting with blocking capacitors and system inductances [11]. Moreover, since capacitors function as passive integral controllers, they can lead to instability if poorly tuned [12,13].

This paper proposes an alternative approach to the design of neutral blocking devices for HEMP insult mitigation. Instead of employing a blocking capacitor, this paper explores the use of a flywheel energy storage system to generate the necessary voltage for HEMP mitigation along the neutral path of a standalone transformer. Flywheels have demonstrated potential for stabilizing the grid in the context of an increasing number of stochastic energy generation devices [14,15]. The flywheel energy storage system offers several advantages over other potential HEMP mitigation devices. As an actively controlled system, it allows for the development of custom and optimal control laws that enhance overall performance. Additionally, the maintenance costs associated with flywheel energy storage devices are relatively low over their extended lifespan [16,17,18]. Furthermore, many flywheel devices are already integrated into the power grid, facilitating the reuse of existing energy storage systems for grid stabilization, which can further reduce the costs associated with HEMP mitigation.

The flywheel operates by converting electrical power into kinetic energy for later use. Kinetic energy is stored in the form of a rotating mass and typically housed within a vacuum to minimize friction losses. During the charging phase, the flywheel functions as a motor, drawing power to apply torque to the rotor. When energy is required, the flywheel acts as a turbine, and the rotational velocity decreases as power is extracted from the energy storage system. Notably, the flywheel energy storage system offers advantages over chemical batteries, including increased power density, improved lifecycle, and the ability to release energy more rapidly than conventional battery energy storage systems. This paper outlines specifications for the application of flywheel energy storage systems in mitigating HEMP $E_3$ insults. This is achieved through the development of an optimal control law for energy storage utilizing Hamiltonian feedback control methods [19,20]. It is demonstrated that an energy-optimal control law can effectively prevent any current flow through the neutral path to ground by generating a commanded voltage on a bus connected to the flywheel energy storage system. Early pioneering work on generating flywheel-specific energy storage requirements to support advanced pulsed power weaponry was initially applied to next-generation Navy warships [21].

The use of flywheels provides some additional benefits to grid operators. Additional actively controlled energy storage sources may be used to improve grid stability through voltage and frequency regulation [18,22]. Additionally, flywheel energy storage systems have been shown to facilitate the penetration of renewable energy sources into the grid and provide power in down-times for stochastic energy sources on the grid [23,24]. This is in contrast to supercapacitors, such as the ABB SolidGround system, which are incapable of providing the same responses and controls that active systems, such as flywheels, are capable of providing [25,26]. The flywheel neutral blocking device would be placed along the neutral path of a transformer in lieu of a neutral blocking capacitor, such as SolidGround.

The structure of this paper is organized into six sections. Section 1 serves as the introduction, providing a high-level overview of the paper’s intent and a literature review on the topic. Section 2 introduces the system models employed in the simulations. Section 3 derives control laws and analyzes system stability using Hamiltonian methods. Section 4 presents the results obtained from the simulations. Section 5 offers a discussion and analysis of these results, including the development of an optimal feedforward controller to verify the optimality of the Hamiltonian feedback controller. Finally, Section 6 concludes the paper by summarizing the key findings.

2. System Models

This section presents three models utilized in the simulation. The first subsection introduces the $E_3$ HEMP insult model. The second subsection describes the per-phase transformer model employed in the analysis. The final subsection presents the flywheel/converter system, which provides the control effort $u_n$ to the per-phase transformer. It is assumed that the transformer and flywheel system couple solely with the transformer neutral voltage source $u_n$ and the flywheel output voltage bus $v_B$.

2.1. HEMP Insult Model

A HEMP insult is characterized by three distinct time domain components. The first component, referred to as $E_1$, exhibits the highest frequency and field strength. The second component, $E_2$, has a lower frequency, occurs over a longer duration than the $E_1$ portion, and possesses a slightly weaker peak field strength. The final time component, $E_3$, is further divided into two segments. The first segment is known as the blast, or $E_{3A}$, which is described by Equation [10] as follows:

$$E_{3A}\left(t\right)=\alpha e^{-t/\beta }(\gamma t-\delta t^2)$$

(1)

where $\alpha =9.5$, $\beta =1.4$, $\gamma =26$, and $\delta =8.9$. The second segment is referred to as the heave, or $E_{3B}$, with its field strength defined as

$$E_{3B}\left(t\right)=-\alpha t^2{e}^{-t/\gamma }{\displaystyle \frac{{\beta }^{3}t-3{\beta }^3\gamma +t^4}{\gamma {({\beta }^3+t^3)}^2}}$$

(2)

where $\alpha =1.3·{10}^6$, $\beta =200$, and $\gamma =20$. The total field strength is given by

$$E_3\left(t\right)=E_{3A}\left(t\right)+E_{3B}\left(t\right),$$

(3)

where $E_{3A}$ is defined in (1) and $E_{3B}$ is defined in (2).

These parameters are derived from curve fitting conducted during simulation work at Oak Ridge National Labs [10]. The waveform of the insult is depicted in Figure 1. The insult waveform defined in [10] is intentionally designed to be approximately twice the magnitude of what is conventionally defined for HEMP insult waveforms at the request of the US Department of Energy to provide a safety factor in HEMP mitigation design.

The $E_3$ portion of the HEMP insult is conventionally regarded as a pseudo-DC disturbance; however, this characterization is misleading. As demonstrated by Donnelly et al., the early time component of the $E_{3A}$ insult is sufficiently impulsive to induce ringing throughout a system when utilizing blocking capacitors for mitigation [11]. The heave, or $E_{3B}$, portion occurs concurrently with the $E_{3A}$ segment and is characterized by a sufficiently low frequency that allows it to be treated as a pseudo-DC source to the grid. The field strength equations are expressed in units of V/km. The effective voltage along a transmission line is given by

$$V\left(t\right)=E_3\left(t\right)·l·cos\left(\theta \right),$$

(4)

where $E_3\left(t\right)$ is defined in (3), l is the length of the transmission line, and $\theta$ is the angle between the HEMP field and the transmission line. Longer transmission lines that are more parallel to the HEMP field will experience greater power reception from the HEMP compared to shorter lines that are more perpendicular to the insult.

2.2. Per-Phase Transformer Model

The per-phase transformer model incorporates winding losses and energy storage due to the windings, as well as core losses and a saturable magnetic core. A schematic representation of the transformer is shown in Figure 2.

The corresponding dynamic system of equations is expressed as follows in (5):

$$\left[\begin{array}{c}{L}_1{\dot{i}}_1\\ L_2{\dot{i}}_2\\ {\dot{\lambda }}_m\end{array}\right]=\left[\begin{array}{ccc}-(R_1+R_c)& -R_c& 0\\ -R_c& -(R_2+R_c)& 0\\ R_c& R_c& 0\end{array}\right]\left[\begin{array}{c}{i}_1\\ i_2\\ {\lambda }_m\end{array}\right]+\left[\begin{array}{c}{v}_1-u_n-R_c{i}_m\\ v_2-u_n-R_c{i}_m\\ -R_c{i}_m\end{array}\right]$$

(5)

In (5), $L_1$ and $L_2$ represent the primary- and secondary-side energy storage due to winding inductance, $R_1$ and $R_2$ denote the primary- and secondary-side winding losses, $R_c$ indicates the core losses, $u_n$ is the controlled voltage source on the neutral, $v_1$ is the primary-side voltage, and $v_2$ is the secondary-side voltage, given by $v_2=-i_2{R}_L$, where $R_L$ is the resistive load connected to the transformer. The term $i_m$ represents a nonlinear current through the core and is approximated as

$$i_m=k_{1}tan\left(k_2{\lambda }_m\right)$$

(6)

where $k_1=0.0016$ and $k_2=1.2879$ [11]. The corresponding plot of the current is illustrated in Figure 3.

Figure 3 demonstrates that as long as the magnetic core flux of the transformer is prevented from saturating, it operates linearly. Within the linearized (unsaturated) region, the ratio between $i_m$ and ${\lambda }_m$ is approximately $1/500$, indicating that $L_m\approx 500H$ [11], where $L_m$ represents the linearized inductance of the transformer core.

2.3. Flywheel Model

The flywheel free-body diagram, permanent magnetic (PM) machine, and converter circuit are depicted in Figure 4.

In Figure 4, ${\omega }_f$ represents the flywheel angular velocity, ${\tau }_f$ denotes the torque due to bearing friction in the flywheel, ${\tau }_{pm}$ is the torque produced by the PM machine, $v_u$ is the voltage output from the PM machine, $i_{pm}$ is the current output by the PM machine, $i_u$ is the current output from the converter, $\zeta$ is the variable converter ratio, and $v_B$ is the bus voltage that is fed into Figure 2 through the $u_n$ term. The remaining parameters for the flywheel are summarized in

Table 1. Parameters with simulation values and associated descriptions.

Parameter	Description	Value
$J_f$	Moment of Inertia of Flywheel	25 kg m2
$k_t$	Torque Constant	20 Nm/A
$R_{pm}$	Armature Resistance	0.1 $\mathrm{\Omega }$
$L_{pm}$	Armature Inductance	0.1 mH
$C_u$	Converter Capacitance	1 mF
$R_{cu}$	Converter Resistance	10 M$\mathrm{\Omega }$
$L_u$	Line Inductance	10 mH
$R_u$	Line Resistance	0.1 $\mathrm{\Omega }$
B	Shaft Windage	${10}^{-6}\mathrm{Nm}/{\displaystyle \frac{\mathrm{rad}}{\mathrm{s}}}$

.

The flywheel system is governed by five state equations: one for the flywheel dynamics, one for the PM machine dynamics, two for the converter dynamics, and one for the bus dynamics. The flywheel dynamics are described by

$$J_f{\dot{\omega }}_f=-B{\omega }_f-k_t{i}_{pm}.$$

(7)

The PM machine dynamics are given by

$$L_{pm}{\dot{i}}_{pm}=k_t{\omega }_f-R_{pm}{i}_{pm}-v_u.$$

(8)

The dynamics of the converter are expressed as

$$C_u{\dot{v}}_u=-{\displaystyle \frac{v_u}{R_{cu}}}+i_{pm}-\zeta i_u$$

(9)

and

$$L_u{\dot{i}}_u=-v_B-R_u{i}_u+\zeta v_u.$$

(10)

Lastly, the bus dynamics are represented as

$$C_B{\dot{v}}_B=i_u+i_1+i_2-{\displaystyle \frac{v_B}{R_B}},$$

(11)

with $i_1$ and $i_2$ are the currents through the primary and secondary windings of the per-phase transformer. The converter ratio, $\zeta$, is constrained to $-1\le \zeta \le 1$ and serves as the control variable for the flywheel system. The nature of the control law will be further discussed in the next section.

3. Control Law Development

It has been demonstrated in [19,27,28] that the Hamiltonian framework can be employed to develop energy-minimized control laws and ensure system stability. For system stability, two criteria must be met [19]:

• The Hamiltonian of the system must be positive definite about the equilibrium point.
• The time derivative of the Hamiltonian must be negative definite about the equilibrium point.

Hamiltonian controls are developed with these criteria in mind. The Hamiltonian control law is developed here to minimize transformer magnetic core flux and minimize the energy storage requirements for this controller. This is used as a baseline for future development of other control laws that may have different control objectives. The dynamics of the per-phase transformer are described by the equations in (5). Given that the objective of the optimal control law is to maintain the transformer’s magnetic core flux within the unsaturated region, it is reasonable to assume that ${\lambda }_m$ is linear, allowing for the approximation of $i_m$ as $i_m\approx \frac{{\lambda }_m}{500}$. Consequently, (5) can be reformulated into a matrix form more conducive to the Hamiltonian construction:

$$M\dot{x}=Rx+Bu+Dv,$$

(12)

where

$$\begin{array}{cc}\hfill & M=\left[\begin{array}{ccc}{L}_1& 0& 0\\ 0& L_2& 0\\ 0& 0& L_m\end{array}\right],\hfill \\ \hfill & R=\left[\begin{array}{ccc}-(R_1+R_c)& -R_c& -R_c\\ -R_c& -(R_2+R_c+R_L)& -R_c\\ R_c& R_c& -R_c\end{array}\right],\hfill \\ \hfill & B=\left[\begin{array}{c}-1\\ -1\\ 0\end{array}\right],D=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],x=\left[\begin{array}{c}{i}_1\\ i_2\\ i_m\end{array}\right],\hfill \\ \hfill & u=u_n\left(t\right),i_m\approx {\displaystyle \frac{{\lambda }_m}{L_m}},v=\left[\begin{array}{c}{v}_1\\ 0\\ 0\end{array}\right]\hfill \end{array}$$

(13)

The controls are developed following the formulation in [29]. The total energy of the system is expressed as

$$\mathcal{H}={\displaystyle \frac{1}{2}}{x}^{T}Mx.$$

(14)

Upon examination of the terms in M, it is evident that the Hamiltonian of the system satisfies $\mathcal{H}>0\forall x\ne 0$. Thus, the first criterion for asymptotic stability is fulfilled. However, this condition alone is insufficient for establishing stability. The time derivative of the Hamiltonian is given by

$$\dot{H}=x^{T}M\dot{x}=x^{T}Rx<0\forall x\ne 0.$$

(15)

It can be shown that R is negative definite for all positive and realizable entries. Therefore, to ensure that the second stability criterion is satisfied, the remaining terms must also be negative semi-definite. The remaining terms are set such that

$$x^T(B{u}_n+Dv)=0.$$

(16)

The equation in (16) is expanded so that $u_n$ may be solved for.

$$u_n(i_1+i_2)=v_1{i}_1$$

(17)

By solving for $u_n$ in (17), the control law commanded by the neutral blocking device is derived as

$$u_n={\displaystyle \frac{v_1{i}_1}{i_1+i_2+ϵ}},$$

(18)

where $ϵ$ is a small constant introduced to avoid singularities during implementation. In the case of this paper,

$$ϵ=\left\{\begin{array}{cc}{10}^{-3},\hfill & i_1+i_2\ge 0\hfill \\ -{10}^{-3},\hfill & i_1+i_2<0\hfill \end{array}\right.$$

(19)

This control law satisfies the criteria for asymptotic stability while minimizing the energy required to counteract the HEMP insult. The subsequent subsection will address the stability and control law for the flywheel/converter system. This $u_n$ is used to command a control output from the flywheel subsystem.

The converter ratio, $\zeta$, shown in Figure 4 is the control variable for the flywheel subsystem. The converter in this paper is a bi-directional dual active bridge converter. This allows power to flow in either direction through the converter (and thus permits a positive or negative converter ratio) [30,31]. The converter ratio $\zeta$ is constrained such that

$$-1\le \zeta \le 1.$$

(20)

Following the work of [21], let the converter ratio control law be defined as

$$\zeta \left(t\right)=K_p\tilde{\mu }+K_i{\tilde{\mu }}_i,$$

(21)

where $\tilde{\mu }=u_n-v_b$ and ${\dot{\tilde{\mu }}}_i=\tilde{\mu }$. Consequentially, the system gains a state for ${\tilde{\mu }}_i$. Therefore, (9) and (10) become

$$C_u{\dot{v}}_u=-{\displaystyle \frac{v_u}{R_{cu}}}+i_{pm}-i_u(K_i{\tilde{\mu }}_i+K_p(u_n-v_b))$$

(22)

and

$$L_u{\dot{i}}_u=-v_B-R_u{i}_u+v_u(K_i{\tilde{\mu }}_i+K_p(u_n-v_b)),$$

(23)

respectively. Through optimization of values, it was determined that $K_p=2.5·{10}^{-3}$ and $K_i=0$. These values were selected to ensure that the flywheel had significant enough bandwidth to properly respond to commanded control efforts. The full flywheel mitigation system block diagram is given in Figure 5.

4. Results

This section presents the specifications required for the neutral-blocking $E_3$ mitigation device.

Table 2. Table of flywheel system requirements.

	Power Draw	Energy Draw/Storage	Bandwidth
Control Law Requirement	2.93 kW	12.3 Wh	4.9 Hz
System Flywheel	100 kW	9.75 kWh	80 Hz

states the required specifications for the flywheel. If the device exceeds the presented specifications, then it is a viable option for neutral path mitigation.

Figure 6, Figure 7, Figure 8, Figure 9, Figure 10 and Figure 11 illustrate the transformer’s magnetic core flux, control effort, error in tracking the commanded voltage, instantaneous power draw, energy storage state, and bandwidth. Additionally, this section details the performance of the flywheel energy system itself, with Figure 12, Figure 13 and Figure 14 depicting the flywheel’s Bode plot, angular velocity over time, and converter ratio to provide sufficient bus voltage.

5. Discussion

Given the specified parameters, the system successfully meets the performance criteria. Throughout the duration of the HEMP $E_3$ insult, the values of ${\lambda }_m$ remain consistently near zero, significantly exceeding the required criterion of $-1.2\mathrm{pu}\le {\lambda }_m\le 1.2\mathrm{pu}$. The neutral blocking device necessary for mitigating the HEMP insult demands a control effort of up to approximately $28\mathrm{kV}$. The required angular velocity and actual angular velocity are shown in Figure 13. It can be seen that the actual angular velocity of the spinning mass remains above the required angular velocity at all times during the insult. This suggests that the energy of the energy storage system meets the requirements. The Hamiltonian control law itself requires approximately $13\mathrm{Wh}$ of energy to operate. Any additional energy loss is due to bearing friction in the flywheel system and from the converter. From Figure 11, the cutoff frequency is at around $5\mathrm{Hz}$. An ideal actuator would maintain unity gain to approximately 10 times the required bandwidth from the controller. The flywheel in this paper has approximately 20 times the required bandwidth. Thus, bandwidth requirements are also met. Lastly, the maximum required power output is less than $3\mathrm{kW}$. This power output level is also within reason and can be met with today’s devices [32]. An example of a flywheel device that more than exceeds system requirements is the VYCON Flywheel Energy Storage System, Option Level 1. The output voltage would need to be stepped up, but the power output of such a device is $300\mathrm{kW}$. This device is capable of storing $\approx 1.44\mathrm{kWh}$ of energy, meeting system specifications by two orders of magnitude. Additionally, these flywheels are rated to last over 20 years, a significant improvement over chemical-based energy storage (i.e., batteries and capacitors). The cost for this flywheel system ranges from USD 65–100 k, depending on the vendor. This is a fraction of the cost of ABB SolidGround, a capacitor neutral blocking device. Adjusted for inflation, the cost of the blocking capacitor is approximately USD 500 k.

A recurring claim throughout this paper has been that the Hamiltonian controller acts as an energy storage optimal controller. To provide evidence for this claim, an additional simulation of a feedforward energy optimal control law, derived following the procedure laid out in [27], was generated for the transformer model. The control law aims to minimize the cost function

$$J={\displaystyle \frac{1}{2}}{\int }_0^{t_f}(\eta {\lambda }_m^2+W^2)d\tau ,$$

(24)

where ${\lambda }_m$ is the transformer core flux, W is the energy stored in the transformers, and $\eta$ is a weighting constant to weigh both terms equally. Figure 15, Figure 16 and Figure 17 depict the ${\lambda }_m$, control effort, and energy state of the transformer, following the energy-storage-optimal feedforward control law. Comparing these figures against Figure 6, Figure 7, and Figure 10 reveals that the Hamiltonian feedback control law in this paper is indeed an energy storage optimal controller.

6. Conclusions

HEMP insults pose significant challenges to the power grids that have yet to be comprehensively addressed. Numerous solutions exist for HEMP mitigation, and it is essential to explore and characterize these options before implementing them at the grid scale. This paper investigates the feasibility of one such mitigation strategy aimed at responding to $E_3$ HEMP insults. A neutral-blocking Hamiltonian controller was developed, which minimizes the energy storage requirements for mitigation. The optimality of the feedback control law was successfully verified by a feedforward energy optimal controller. The control effort necessary for the controller was generated using a flywheel energy storage system. The optimal control law formulated in this study facilitated the derivation of the specification requirements for the flywheel energy storage system. It was determined that, given the specified parameters for the flywheel energy storage system, effective mitigation of the HEMP insult is achievable. Many devices commercially available today exceed the requirements of the controller presented here. It should be noted that this is just one of many strategies for the mitigation of HEMP $E_3$ insults on the power grid.

Future research should focus on evaluating other energy storage devices suitable for use in neutral blocking applications. Alternative mitigation strategies, such as transmission line blocking devices and load-side exogenous power drain controllers, should be explored. There may exist devices that offer comparable performance to neutral blocking devices at a lower cost. Other future work on this topic will include hardware in the loop validation and a scaled, system level analysis for minimizing device placement costs. Additionally, further work on redundancy and reliability at a large scale needs to be carried out using the feasibility and specifications analysis provided in this study.