# Optimal Observer-Based Power Imbalance Allocation for Frequency Regulation in Shipboard Microgrids

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Main Contribution

#### 1.2. Notation and Power Sign Convention

#### 1.3. Structure of the Paper

## 2. System Description

#### 2.1. SMG Architecture

#### 2.2. Modes of Operation

#### 2.3. State-Space Representation

#### 2.3.1. ESS Modelling

**Assumption?1.**?

**Remark?1.**?

#### 2.3.2. SMG Synchronous Machine Modelling

**Remark?2.**?

#### 2.3.3. Compact Representation

**SMG Model:**

## 3. Problem Formulation

**Objective?1**?

**.**Estimate in a finite time?${t}_{f}$ the unknown load demand ${d}_{f}$, where ${t}_{f}$ is a known positive constant.

**Objective?2**?

**.**Determine the optimal continuous-time reference ${x}_{g}^{\u2605}\left(t\right)$ for the output power for each ESS by solving an optimisation problem based on the principle of the receding horizon [25].

**Objective?3**?

**.**Enforce the condition

**Remark?3.**?

**Assumption?2.**?

- (A1)?
- The first time derivative of ${d}_{{g}_{i}}\left(t\right)$ is bounded with an a priori known bound, that is, $|{\dot{d}}_{{g}_{i}}\left(t\right)|<{\Delta}_{{d}_{g}}$.
- (A2)?
- The signal ${d}_{f}\left(t\right)$ remains constant ${d}_{f}\left(t\right)={d}_{f}$ where ${d}_{f}$ is an unknown positive constant. It is common practise to assume that the power load requirement remains constant when designing control strategies for power systems and microgrids [18]. This is required to guarantee the reach of the (optimal) equilibrium point.
- (A3)?
- To ensure that the optimal power imbalance allocation is feasible, we assume that$$max\left({d}_{f}\right)<{\mathbf{1}}^{?}{p}_{M},$$$${\mathbf{1}}^{?}{x}_{s0}<{\mathbf{1}}^{?}{p}_{M}{T}_{s}$$
- (A4)?

## 4. Problem Solution and Stability Analysis

**Low-Level STA Controller:**$$\begin{array}{ccc}\hfill {\sigma}_{g}\left(t\right)& :=& {x}_{g}\left(t\right)?{x}_{g}^{\u2605}\left(t\right)\hfill \end{array}$$$$\begin{array}{ccc}\hfill {u}_{g}\left(t\right)& :=& {u}_{g1}\left(t\right)+{u}_{g2}\left(t\right)\hfill \end{array}$$$$\begin{array}{ccc}\hfill {u}_{g1}\left(t\right)& :=& \mathrm{Col}(?{\alpha}_{{1}_{i}}|{\sigma}_{{g}_{i}}\left(t\right){|}^{\frac{1}{2}}\mathrm{sign}\left({\sigma}_{{g}_{i}}\left(t\right)\right))\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\dot{u}}_{g2}\left(t\right)& :=& \mathrm{Col}(?{\alpha}_{{2}_{i}}\mathrm{sign}\left({\sigma}_{{g}_{i}}\left(t\right)\right))\hfill \end{array}$$**Low-Level STA Observer**$${e}_{f}\left(t\right):={\widehat{x}}_{f}\left(t\right)?{x}_{f}\left(t\right)$$$$\begin{array}{ccc}\hfill {\dot{\widehat{x}}}_{f}\left(t\right)& =& ?{r}_{f}{x}_{f}\left(t\right)+{\mathbf{1}}^{?}{x}_{g}\left(t\right)\hfill \\ & & ?{\beta}_{1}{\left|{e}_{f}\left(t\right)\right|}^{\frac{1}{2}}\mathrm{sign}\left({e}_{f}\left(t\right)\right)+{w}_{f}\left(t\right)\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\dot{w}}_{f}\left(t\right)& =& ?{\beta}_{2}\mathrm{sign}\left({e}_{f}\left(t\right)\right)\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\widehat{d}}_{f}& =& ?{w}_{f}\left(t\right)\hfill \end{array}$$**High-Level Optimal Power Imbalance Allocator:**$$\begin{array}{c}\hfill {\left(\right)}_{{{x}_{g}}^{\u2605}}^{\left[j\phantom{\rule{3.33333pt}{0ex}}\right|\phantom{\rule{3.33333pt}{0ex}}k]}j=1{N}_{D}\\ :=& {\mathcal{S}}^{\u2605}\left[k\right]\hfill \end{array}$$$$\begin{array}{ccc}\hfill {\mathcal{S}}^{\u2605}\left[k\right]& =& \underset{{\mathcal{S}}^{\u2605}\left[k\right]}{\mathrm{argmin}}\sum _{j=1}^{{N}_{D}}{c}_{c}^{?}{\widehat{x}}_{g}\left[j\phantom{\rule{3.33333pt}{0ex}}\right|\phantom{\rule{3.33333pt}{0ex}}k]\hfill \\ \hfill {x}_{g}^{\u2605}\left(t\right)& :=& F\left(\right)open="("\; close=")">{\mathcal{S}}^{\u2605}\left[k\right],t,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{t}_{k}\le t{t}_{k}+{\tau}_{O}\hfill \end{array}$$$$\left(\right)open\; close="\}">\begin{array}{ccc}\hfill \phantom{\rule{1.em}{0ex}}& \mathbf{s}.\mathbf{t}.& \\ \hfill {\widehat{x}}_{s}\left[0\phantom{\rule{3.33333pt}{0ex}}\right|\phantom{\rule{3.33333pt}{0ex}}k]& =& {x}_{s}\left[k\right]\hfill \\ \hfill {\widehat{x}}_{s}[j+1\phantom{\rule{3.33333pt}{0ex}}|\phantom{\rule{3.33333pt}{0ex}}k]& =& {\widehat{x}}_{s}\left[j\phantom{\rule{3.33333pt}{0ex}}\right|\phantom{\rule{3.33333pt}{0ex}}k]?{\widehat{x}}_{g}\left[j\phantom{\rule{3.33333pt}{0ex}}\right|\phantom{\rule{3.33333pt}{0ex}}k]{\tau}_{O}\hfill \\ \hfill {\mathbf{1}}^{?}{\widehat{x}}_{g}\left[j\phantom{\rule{3.33333pt}{0ex}}\right|\phantom{\rule{3.33333pt}{0ex}}k]& =& {\widehat{d}}_{f}\hfill \\ \hfill {\widehat{x}}_{g}\left[j\phantom{\rule{3.33333pt}{0ex}}\right|\phantom{\rule{3.33333pt}{0ex}}k]& \in & {\mathcal{X}}_{g}\hfill \\ \hfill {\widehat{x}}_{s}\left[j\phantom{\rule{3.33333pt}{0ex}}\right|\phantom{\rule{3.33333pt}{0ex}}k]& \in & {\mathcal{X}}_{s}\hfill \end{array}$$

**Theorem?1.**?

**(I)**?- The low-level STA observer is capable of estimating the unknown load power demand ${d}_{f}$ in a finite time ${t}_{f}$.
**(II)**?- Provided that each entry of the unit cost vector ${c}_{c}$ satisfies$$\begin{array}{ccc}\hfill {c}_{ci}& \ne & {c}_{{c}_{n}}\hfill \\ \hfill ?i& =& 1,\dots ,N,\hfill \\ \hfill ?n& =& 1,\dots ,N,\hfill \\ \hfill \mathrm{s}.\mathrm{t}.\phantom{\rule{3.33333pt}{0ex}}i& \ne & n\hfill \end{array}$$
**(III)**?- The low-level STA controllers are capable of driving ${x}_{g}\left(t\right)$ to ${x}_{g}^{\u2605}\left(t\right)$ in a finite time ${t}_{g}$ and of dynamically tracking its smooth evolution over time.

**Proof.**?

**Proof of Part (I)—Solving Objective 1:**We subtract (5) from (18b), and the STA observer error dynamics hold:

**Proof of Part (II)—Solving Objective 2**

- (a)
- If during the time horizon of ${\tau}_{D}$ seconds, the evolution of ${\widehat{x}}_{s}\left[j\phantom{\rule{3.33333pt}{0ex}}\right|\phantom{\rule{3.33333pt}{0ex}}k]$ does not breach any of its associated constraints as per (19c), then the minimum ${x}_{g}^{\u2605}\left[j\phantom{\rule{3.33333pt}{0ex}}\right|\phantom{\rule{3.33333pt}{0ex}}k]$ for $J\left({\widehat{x}}_{g}\left[j\phantom{\rule{3.33333pt}{0ex}}\right|\phantom{\rule{3.33333pt}{0ex}}k]\right)$ will also minimise the overall cost function (19b). A series ${\mathcal{S}}^{\u2605}\left[k\right]$ composed of ${N}_{D}$ identical references will be generated and interpolated via the interpolator (19c).
- (b)
- If at a generic m-th step, the boundaries for the energy storage ${\widehat{x}}_{s}\left[m\phantom{\rule{3.33333pt}{0ex}}\right|\phantom{\rule{3.33333pt}{0ex}}k]$ are reached, these can be reflected by constraining the associated output powers to be equal to zero, hence obtaining a different hyper-rectangle redefining the boundaries of ${\widehat{x}}_{g}\left[m\phantom{\rule{3.33333pt}{0ex}}\right|\phantom{\rule{3.33333pt}{0ex}}k]$ and finding another single minimum for the cost function. A series ${\mathcal{S}}^{\u2605}\left[k\right]$ composed of ${N}_{D}$ nonidentical references will be generated and interpolated via the interpolator (19c).

**Proof of Part (III)—Solving Objective 3**

**Remark?4.**?

**Remark?5.**?

**Remark?6.**?

## 5. Simulation

_{1}), which belongs to the subset ${\mathcal{N}}_{r}$ and an FC (numbered ESS

_{2}) which belongs to the subset ${\mathcal{N}}_{c}$. We consider an SMG of 1 (MW) rated power, which is also set to be the base power for the per unit (p.u) parameters. We set ${p}_{{m}_{1}}={p}_{{M}_{1}}=0.575\phantom{\rule{0.166667em}{0ex}}(\mathrm{p}.\mathrm{u}.),\phantom{\rule{3.33333pt}{0ex}}{p}_{{M}_{2}}=0.425\phantom{\rule{0.166667em}{0ex}}(\mathrm{p}.\mathrm{u}.)$, ${x}_{{s}_{1}}^{M}=0.575\phantom{\rule{0.166667em}{0ex}}(\mathrm{p}.\mathrm{u}.\mathrm{h}),\phantom{\rule{3.33333pt}{0ex}}{x}_{{s}_{2}}^{M}=0.44\phantom{\rule{0.166667em}{0ex}}(\mathrm{p}.\mathrm{u}.\mathrm{h})$. These parameters are selected in accordance with the data made available via the acknowledged Innovate UK grant with industry partners. We employ widely accepted model parameters found in the existing literature [2] for the representation of the SMG state space, which are ${a}_{{g}_{1}}=?10,\phantom{\rule{3.33333pt}{0ex}}{a}_{{g}_{2}}=?3.87,\phantom{\rule{3.33333pt}{0ex}}{b}_{{g}_{1}}=10$, ${b}_{{g}_{2}}=3.87,\phantom{\rule{3.33333pt}{0ex}}{r}_{f}=0.60$. We consider ${\Delta}_{h}={\Delta}_{f}=10$, and we set the parameters of the STA controllers as ${\alpha}_{{1}_{1}}={\alpha}_{{1}_{2}}=4.74,\phantom{\rule{3.33333pt}{0ex}}{\alpha}_{{2}_{1}}={\alpha}_{{2}_{2}}=11.00$. The unit costs of consumption are selected as ${c}_{{c}_{1}}=0.60,\phantom{\rule{3.33333pt}{0ex}}{c}_{{c}_{2}}=0.40$. The STA observer design constants are set as ${\beta}_{1}=4.74,\phantom{\rule{3.33333pt}{0ex}}{\beta}_{2}=11.00$. We numerically estimate the values for ${t}_{g}$ and ${t}_{f}$ following the methodology presented in [27], obtaining ${t}_{g}={t}_{f}=0.15$ s. The STA observer and controllers are implemented in a MATLAB-Simulink environment using the Euler method with an integration step of 0.1 ms. The simulations run for a duration of ${T}_{sim}=1200$ s. If we consider (14), $max({t}_{g},{t}_{f})=0.15$ s, then ${\tau}_{O}?0.15$. Therefore, the power imbalance allocator scheme is executed with a sampling time of ${\tau}_{O}=10$ s. The optimisation problem (19a)–(19c) is implemented using the dedicated MATLAB Optimisation Toolbox and the algorithm Fmincon Sequential Quadratic Programming (SQP). Figure 3 shows an extract of the implementation of the MATLAB-Simulink code of the strategy proposed in this paper, following the architecture in Figure 1. In particular, the technical values of the considered SMG are given, along with the MATLAB R2023b functions architecture of the two-level control strategy.

**Scenario PI**: an arbitrarily defined power imbalance allocator is imposed to determine the power reference for each ESS, i.e., ${x}_{g}^{\u2605}:=\kappa {\widehat{d}}_{f},\phantom{\rule{3.33333pt}{0ex}}\kappa :={[{\kappa}_{1},\phantom{\rule{3.33333pt}{0ex}}{\kappa}_{2}]}^{?},\phantom{\rule{3.33333pt}{0ex}}{\kappa}_{1}={p}_{{M}_{1}},\phantom{\rule{3.33333pt}{0ex}}{\kappa}_{2}={p}_{{M}_{1}}$ Furthermore, during this scenario, each ESS is regulated via conventional PI controller.**Scenario PIO**: during which our optimal power imbalance allocator is utilised, and each ESS is regulated via PI controllers. The proportional and integral gains for the PI controllers are set equal to ?1.**Scenario SM**: the arbitrary power allocator defined in the scenario PI is used and each ESS is regulated via STA controllers.**Scenario SMO**: the proposal of this paper, where the optimal power imbalance allocator is used in conjunction with STA controllers.

_{2}is used to charge ESS

_{1}. Note that in scenario SM, as there is no high-level scheme with the associated interpolation architecture communicating changes in load demand, the frequency deviation always remains equal to zero. On the other hand, in the SMO scenario, we observe a small frequency deviation only when the unknown load power demand ${d}_{f}$ varies over time. These frequency deviations are much smaller than the one obtained in Scenario PIO, which demonstrated the better performance of the STA algorithm compared to the PI algorithm. To further compare the scenarios analysed, it is worth noting that Scenario PIO and Scenario SMO are characterised by identical cost $\mathcal{J}$. Nevertheless, the low-level STA controllers adopted in Scenario SMO better track the optimal reference when compared to standard PI controllers. This fact can be observed by analysing the frequency deviation ${x}_{f}$ in scenario PIO and SMO.

#### 5.1. Sensitivity Analysis

#### 5.1.1. Sensitivity Analysis 1

^{?}denotes the scaled gain, and the scaling factor $\chi $ ranges from $0.8$ to $1.2$. Following the insights in [27], the convergence time ${T}_{\mathrm{STA}}$—the time of the STA algorithm for both controllers and observer—achieves sliding motion proportional to $\sqrt{\chi}$, as is also numerically demonstrated in Figure 6. Notably, the proof of Theorem 1 confirms that sliding motion is maintained for gain adjustments within the specified range, thus ensuring system stability under both under- and over-tuning conditions.

#### 5.1.2. Sensitivity Analysis 2

**Remark?7.**?

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

STA | Super-Twisting Algorithm |

ESS | Energy Storage System |

SMG | Shipboard Microgrid |

LFC | Load Frequency Control |

BESS | Battery Energy Storage System |

FC | Fuel Cell |

EMS | Energy Management System |

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**Figure 1.**The considered architecture for the SMG, with the depiction of the low-level STA observer and the local low-level STA controllers, along with the high-Level optimal power imbalance allocator.

**Figure 2.**A visual interpretation of the optimised interpolated reference ${x}_{g}^{\u2605}\left(t\right)$ generated iteratively using the high-level optimal power imbalance allocator. The representation focusses on the i-th scalar component of ${x}_{g}^{\u2605}\left(t\right)$.

**Figure 3.**A schematic of the considered SMG composed of a BESS and a FC. The technical details of the nominal power-energy storage capacity of the SMG are also reported. An extract of the MATLAB-based code implementation of the low-level STA observer, of the optimal power imbalance allocator, and of the low-level STA controllers are also illustrated.

**Figure 4.**From Left to right: Time histories of the power balance with ${x}_{{g}_{1}},\phantom{\rule{3.33333pt}{0ex}}{x}_{{g}_{2}},\phantom{\rule{3.33333pt}{0ex}}{x}_{{g}_{3}}$; the frequency deviation ${x}_{f}$; the consumption variables ${x}_{{c}_{1}}$ and ${x}_{{c}_{2}}$; and the cost metric $\mathcal{J}$ for the four scenarios PI, PIO, SM, and SMO in each row of the figure.

**Figure 5.**(

**Top**): Time histories of the power load demand ${d}_{f}$ and its estimate ${\widehat{d}}_{f}$ obtained via the proposed STA observer, with a zoomed view during the first 0.25 s to show the convergence in finite time. (

**Bottom**): Time histories of $\left|\right|{\sigma}_{g}{\left|\right|}_{2}$ during the first 0.25 s and time histories of the ESS output power ${x}_{g}^{\u2605}\left(t\right)$ and its actual value ${x}_{g}\left(t\right)$.

**Figure 6.**Two sensitivity analyses of the algorithm proposed in this paper. (

**Sensitivity Analysis 1**): Time histories of ${d}_{f}$ and their estimates when the gains $bet{a}_{1},\phantom{\rule{3.33333pt}{0ex}}{\beta}_{2}$ are scaled and the numerical evaluation of the finite-time convergence. Time histories of ${\left|\right|\sigma \left|\right|}_{2}$ of the low-level STA controllers when the gains ${\alpha}_{{1}_{i}},{\alpha}_{{2}_{i}}$ are scaled and the impact of the finite-time convergence. (

**Sensitivity Analysis 2**): Time histories of the frequency deviation ${x}_{f}$ when the power demand ${d}_{f}$ is scaled. Time histories of the ESS output power ${x}_{g}\left(t\right)$ when the power demand is scaled.

Symbol | Physical Meaning and Measurement Unit |
---|---|

${x}_{{g}_{i}}\left(t\right)$ | ESS output power ??(p.u.) |

${x}_{{g}_{i}}^{\u2605}\left(t\right)$ | ESS output power optimal reference ??(p.u.) |

${x}_{{s}_{i}}\left(t\right),\phantom{\rule{3.33333pt}{0ex}}{\widehat{x}}_{{s}_{i}}\left[k\right]$ | ESS energy storage level and its discrete time prediction ??(p.u. s) |

${u}_{{g}_{i}}\left(t\right)$ | ESS low-level control ??(p.u.) |

${d}_{f},\phantom{\rule{3.33333pt}{0ex}}{\widehat{d}}_{f}$ ?? | Power load demand and its estimate ??(p.u.) |

${x}_{f}\left(t\right)$ | Frequency deviation ??(p.u.) |

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## Share and Cite

**MDPI and ACS Style**

Rinaldi, G.; Baby, D.K.; Menon, P.P.
Optimal Observer-Based Power Imbalance Allocation for Frequency Regulation in Shipboard Microgrids. *Energies* **2024**, *17*, 1703.
https://doi.org/10.3390/en17071703

**AMA Style**

Rinaldi G, Baby DK, Menon PP.
Optimal Observer-Based Power Imbalance Allocation for Frequency Regulation in Shipboard Microgrids. *Energies*. 2024; 17(7):1703.
https://doi.org/10.3390/en17071703

**Chicago/Turabian Style**

Rinaldi, Gianmario, Devika K. Baby, and Prathyush P. Menon.
2024. "Optimal Observer-Based Power Imbalance Allocation for Frequency Regulation in Shipboard Microgrids" *Energies* 17, no. 7: 1703.
https://doi.org/10.3390/en17071703