Frequency Stability with MPC-based Inverter Power Control in Low-Inertia Power Systems

Various control strategies that utilizes the IBRs in providing frequency regulation services has been proposed. The goal of these strategies is to design the active power response of the IBRs to changes in frequency, such that some frequency response objective is minimized. For example, standard objectives of interests are the magnitude of the frequency deviation, the rate of change of frequency (ROCOF) and the settling time. A unique challenge in the control of IBRs is that they tend to face much tighter limits than conventional machines. For example, solar and wind resources cannot increase their power output beyond the maximum power tracking point, which introduces a hard (and asymmetrical) constraint on the action of the inverters. For a storage unit, it has only a limited amount of energy that can be used to respond to a disturbance.

Of the varying control strategies proposed for IBRs, Droop Control and Virtual Synchronous Machines (VSMs) are the most popular as they function by mimicking the frequency-power dynamic response of a synchronous machine. As suggested by their names, droop control injects/absorbs an amount of active power in proportion to the frequency deviation, and VSM, in its basic configuration, acts as a second order oscillator to provide inertia and damping to the grid. The parameters (droop slope, inertia and damping constants) used in these strategies can be optimized using a number of techniques .

The structural simplicity of VSMs also leads to a fundamental limitation . Since there are only two parameters to tune (inertia and damping) in the basic VSM configuration, there is an inherent trade-off between different objectives and there is no choice of parameters that will make the frequency deviation, ROCOF, and settling time small at the same time . While the performance of the VSM can be enhanced by incorporating virtual governors, virtual exciters and other power systems controllers in their virtual form , it difficult to tune the multiple parameters of the combined virtual controller simultaneously, since the performance of one might negatively affect the other. In addition, it is also difficult to include hard constraints, since simply thresholding the output once the constraints are reached tend to lead to very poor performances . Adaptive rules can be used to alleviate this drawback somewhat, and works in change the parameter based on the measured frequency deviation and ROCOF values. However, it is difficult to find an optimal rule to update these parameters in real-time.

In this work, we propose a novel control strategy, based on model predictive control (MPC), called the MPC-based Inverter Power Control (MIPC). We explicitly formulate the problem of finding the optimal active power set-point of an IBR to minimize the frequency deviation and the ROCOF. It turns out that this formulation also implicitly minimizes the systems settling time. More specifically, at any timestep, we simulate the dynamics of the systems for a finite horizon, then find the best set-points that optimizes the objective over that horizon. The first action is then adopted for the current timestep, and the process repeats. Our approach is similar in spirit to the ones in  since an objective is optimized in an online fashion. However, instead of optimizing the parameters, we directly find the best power set-points. This approach turns out to provide both an easier optimization problem and better control performances. Namely, the hard constraints on the IBRs are explicitly included in the optimization process.

A requirement of MPC is that the IBR must have a model of the system to be optimized. If wide-area measurements are available, then the system states can be obtained from these measurements . In some systems, only a limited buses are equipped with these measurement devices (e.g. PMUs). We show that our proposed MIPC framework is still applicable to these systems by building an observer to estimate unmeasured disturbances and states. Through simulation studies, we show that the MIPC strictly outperforms optimally tuned VSMs for the IEEE 39-bus system, even under limited communication and large measurement noises.

This proposed controller finds practical application by enhancing the capability of the IBRs to participate in providing frequency regulation services. The additional power required can be obtained by running solar below its maximum power point to create sufficient headroom, utilizing the inertia from the decoupled rotating wind turbine and, leveraging on the stored energy in a battery. By explicitly considering hard constraints and costs on energy and power in the MPC formulation, economic considerations can be accounted for.

The remainder of this paper is organized as follows: Section 12 defines the models used in this paper. Section 13 presents the design and formulation of the MIPC algorithm. Section 14 presents the state and disturbance observer design. Section 15 compares the performances of MIPC to VSMs in a standard test system. Section 16 concludes the paper.

We denote the real line by $`\mathbb{R}`$, the cardinality of a set $`\mathcal{S}`$ as $`\vert \mathcal{S} \vert`$, the $`n \times n`$ identity and zero matrices as $`\vect{I}_n`$ and $`\vect{0}_n`$, respectively. Matrices and vectors are denoted by a bold-faced variables.

System Structure

Steady state conditions in a power systems are achieved when there is a balance between the power produced by the generating sources and the power consumed by loads and lossy components. For stability analysis, the entire system can be reduced to an equivalent network via Kron reduction . This eliminates passive and non-dynamic load buses and leaves only buses with at least one generating source connected. With this in place, frequency stability analysis can be carried out, with the frequency dynamics governed by the reactions of buses to active power imbalances in the system.

In this work, we assume the availability of state variables and network information for control purposes. In a later section, we will relax this assumption to partial availability of state variables from some generators.

Because the generators and IBRs had different dynamics, we denote their sets by $`\mathcal{G}`$ and $`\mathcal{I}`$, respectively. Note that the total number of generating sources in the network is $`\mathcal{N} := \mathcal{G} \cup \mathcal{I}`$.

Synchronous Machines

The rotor dynamics of each synchronous generator in a given power system is governed by the well-known swing equation . Here we adopt a discretized version of the equations, which in per unit (p.u.) system is:

\begin{equation}
 \label{eqn:swing_rotor_dis}
\begin{aligned}
\omega^{t+1}_i & = \omega^{t}_i + \frac{h}{m_i} \left(P_{\text{m},i}^t - P_{\text{e},i}^t - d_i \omega^{t}_i \right), \\
\delta^{t+1}_i & = \omega_{\text{b}} \left(\delta^{t}_i + h \ \omega^{t+1}_i\right),
\end{aligned}
\end{equation}

$`\forall i \in \mathcal{G}`$ where $`h`$ is the step size for the discrete simulation, $`\delta_i`$ (rad) is the rotor angle, $`\omega = \bar{\omega}_i - \omega_0`$ is the rotor speed deviation, $`\omega_{\text{b}}`$ is the base speed of the system, $`m_i`$ is the inertia constant, $`d_i`$ is the damping constant, $`P_{\text{m},i}`$ is the mechanical input power and $`P_{\text{e},i}`$ is the electric power output of the $`i^{th}`$ machine.

The electrical output power $`P_{\text{e},i}`$ is given by the AC power flow equation in terms of the internal emf $`\vert E_i \vert`$ and rotor angle $`\delta_i`$:

\begin{align}
 \label{eqn:pf_dis}
P_{\text{e},i}^t = \sum_{i \sim j} \vert E_i E_j \vert [g_{ij} \cos(\delta_i^t - \delta_j^t) + b_{ij}\sin(\delta_i^t - \delta_j^t)],
\end{align}

$`\forall i,j \in \mathcal{G}`$, where $`g_{ij} + j b_{ij}`$ is the reduced admittance between nodes $`i`$ and $`j`$. We assume the internal emf are constant because of the actions of the exciter systems.

The nonlinearity of the AC power flow in [eqn:pf_dis] makes [eqn:swing_rotor_dis] difficult to use for control applications. Linearizing [eqn:swing_rotor_dis] around the nominal point and using the DC power flow approximation  , the bus dynamics become:

\begin{equation}
 \label{eqn:swing_lin_dis}
\begin{aligned}
\bigtriangleup  \omega^{t+1}_i & = \bigtriangleup  \omega^{t}_i + \frac{h}{m_i} \left(\bigtriangleup  P_{\text{m},i}^t - \bigtriangleup P_{\text{e},i}^t - d_i \bigtriangleup  \omega^{t}_i \right), \\
\bigtriangleup \delta^{t+1}_i & = \omega_{\text{b}} \left(\bigtriangleup \delta^{t}_i + h \ \bigtriangleup \omega^{t+1}_i \right),
\end{aligned}
\end{equation}

where $`\bigtriangleup P_{\text{e},i}^t = \sum_{i \sim j} b_{ij} \bigtriangleup \delta^t_{ij}`$ is the dc power flow between 2 buses. We model changes to the mechanical input power $`\bigtriangleup P_{\text{m},i}^t`$ by a combination of droop and automatic governor control (AGC) actions .

Virtual Synchronous Machine (VSM)

From the network point of view, the grid-connected IBR is seen as producing a constant power according to its predetermined set-point and fast dynamics governed by closed-loop controls actions . When configured in the grid-following mode, these controls help maintain the output power of the IBRs while remaining synchronized to the terminal voltage set by the grid. For system analysis, the inverter can be modeled as a voltage source behind a reactance, much like a synchronous machine.

In the event of a power imbalance in the network reflected by a frequency deviation, an inverter does not have a “natural” response to frequency deviation as synchronous machines does since they are made of power electronics components and have no rotating mass. To elicit some response, an additional control loop is therefore needed to enable the inverters to participate in frequency control by changing the power set-point of the inverter based on frequency measurements. The concept of virtual synchronous machine (VSM) has been proposed in literature to provide this additional control loop and it comes in different configurations . The basic idea is to mimic the behavior of a synchronous machine’s response to a frequency deviation by choosing appropriate gains corresponding to the inertia and damping of the machines and producing power proportional to the ROCOF and frequency deviation. Since the response of the inverter is entirely digital, it can be programmed with almost arbitrary functions.

In this work, we adopt the VSM configuration in , where the additional power required to combat a frequency deviation is computed using:

\begin{equation}
 \label{eqn:p_add}
    \begin{aligned}
\bigtriangleup P = \bigtriangleup P_{\text{km}} + \bigtriangleup P_{\text{kd}}= K_\text{m} \frac{d \bigtriangleup \omega_{\text{ibr}}}{dt} + K_\text{d} \bigtriangleup \omega_{\text{ibr}}.
\end{aligned}
\end{equation}

The local frequency at the IBR node $`\bigtriangleup \omega_{\text{ibr}}`$ is approximated by the center of inertia (COI) frequency , which is an inertia-weighted average frequency given by:

\begin{equation}
 \label{eqn:local_freq}
    \begin{aligned}
        \bigtriangleup \omega_{ibr} = \frac{\sum_{i=1}^n m_i \bigtriangleup \omega_i}{\sum_{i=1}^n m_i}
    \end{aligned}
\end{equation}

where $`n = \vert \mathcal{G} \vert`$, $`\bigtriangleup \omega_i`$ is the rotor speed deviation, and $`m_i`$ is the inertia constant of the $`i^{th}`$ synchronous generators in the network. The gains $`K_\text{m}`$ and $`K_\text{d}`$ in [eqn:p_add] represent the virtual inertia and damping constants respectively. In contrast to synchronous machines where the constants are decided by the physical parameters, these constants of the VSM can be optimized over .

In the next section, we fully leverage the flexibility of the power electronic interfaces using a MPC framework.