Balanced Truncation-Based Model Order Reduction for MIMO Microgrid Systems

Microgrids play an important role in the context of energy transition and smart operation since they integrate distributed generation units, storage devices, and local loads into controllable electrical networks that can operate in islanded mode. When several converters interact through a common network the overall dynamics often take the form of a multiple input multiple output structure with several coupled state variables. Such a structure is central to voltage regulation, power sharing, and stability assessment in modern microgrids because it links local control actions with system level performance under changing operating conditions and disturbances [1].

The increasing penetration of renewable sources, advanced storage technologies, and hierarchical controllers drives microgrid models toward higher orders. Detailed representations that capture inner control loops, virtual impedance, and network dynamics can involve many states and several control inputs and outputs [1, 2]. These features improve accuracy but make analysis, simulation, and controller design more demanding. Time domain studies with many scenarios or frequency domain evaluations over a wide range of operating points require repeated evaluations of models that may contain more than ten states and several control channels [2, 3]. This situation motivates the use of reduced order models that preserve the dominant dynamic behavior while easing computational effort.

Model order reduction has become a widely used tool in this context since it provides simpler models that approximate the response of high order systems in a controlled way [3]. Examples include reduced order representations of converters with droop and virtual impedance [1], soft robotic structures governed by finite element models [2], and linear active disturbance rejection controllers that benefit from more compact dynamic descriptions [3, 4]. Other works present techniques that combine structural methods and performance indices to reduce system order while keeping stability and limiting integral errors such as Integral of Time-weighted Absolute Error (ITAE), Integral of Absolute Error (IAE), and Integral of Squared Error (ISE) [4, 5]. Studies on electromagnetic field problems and delayed wind energy frequency control also employ reduced order models to decrease the gap between detailed and simplified dynamics in a systematic way [5, 6]. Model order reduction has further been applied to high pressure heavy water reactor models and parameterized large scale circuit descriptions for VLSI applications where the number of states can be very large [7, 8]. Hybrid schemes based on classical approximations and moment-based approaches have been proposed to generate stable reduced models for complex mechanical systems [9, 10]. In power electronics and control, several contributions exploit time domain parameters, Markov data, or meta heuristic optimization to tune reduced order models and to minimize integral error criteria for converters and related systems [11, 12].

Within this broad family of methods, balanced truncation stands out as a reliable and conceptually clear approach for linear time invariant systems [13]. The method relies on controllability and observability Gramians to transform the state space model into a balanced coordinate frame in which each state has an associated Hankel singular value that measures its joint contribution to input to state and state to output transmission [13, 14]. States with small Hankel singular values can then be discarded to form a reduced model while stability is preserved and error bounds are available [14, 15]. Balanced truncation has been applied to phase locked loop models in wind farm synchronization studies, to dynamic systems with special forms of outputs, and to switching systems with known switching sequences where time varying Gramians help construct reduced switching models [14-16]. In digital signal processing the method supports dedicated algorithms for finite impulse response multiple input multiple output filters and helps reduce their computational cost while keeping high accuracy [17]. Further works investigate adaptive order selection, compression of deep learning models on embedded platforms, and combinations with meta heuristic procedures in order to handle complex electrical networks and to extract informative error features for grid connected power electronic systems [18-21]. These studies confirm that balanced truncation can reduce model complexity in many application domains while keeping the essential dynamic characteristics of the original systems [13, 21].

Despite this progress the literature still provides limited insight into how balanced truncation behaves when applied to multi-inverter islanded microgrids that are modeled as coupled multiple input multiple output systems. Existing works that use model order reduction in converter control or networked power electronic systems often focus on specific controllers or device level models and rarely present a systematic scan of the reduced order together with a channel wise error analysis [1, 3, 4]. In particular the sensitivity of different output buses to truncation has not been quantified in a way that can guide practitioners in the selection of a suitable reduced order for microgrid simulation and control studies.

The present paper addresses this gap by applying balanced truncation to a linearized dq frame model of an islanded microgrid benchmark that includes two micro sources, two inverters, and two RLC loads [22]. The benchmark model in the study [22] has fourteen states, four control inputs, and four output voltages at a nominal frequency of fifty hertz. Balanced truncation is used to construct reduced models with orders from one to thirteen. The analysis focuses on orders four and eight and evaluates H infinity norms for each input output channel as well as impulse responses and Bode plots. The results show that some output voltages are much more sensitive to truncation and that the eight-state model offers a good compromise between simplicity and fidelity. These findings provide a quantitative guideline for selecting reduced order models in multi-inverter microgrid applications where accurate dynamic behavior and manageable computational effort are both required.

Balanced truncation is a classical model order reduction method in control engineering that produces a balanced realization of the state space model [13]. In this realization, the controllability Gramian and the observability Gramian are equal and diagonal and the diagonal entries are the Hankel singular values. States with small Hankel singular values have a limited impact on the input to output behavior so they can be removed while the main dynamics remain. The present work follows the standard balanced truncation procedure and the Lyapunov equations, Gramian factorizations, singular value decomposition, state transformation and truncation steps are given in Eqs. (17) to (24):

$P A^T+B B^T+A P=0$    (17)

$Q A+C^T C+A^T Q=0$    (18)

$P=\operatorname{chol}(R)^T$    (19)

$Q=\operatorname{chol}(L)^T$    (20)

$L^T R=V S U^T$    (21)

$T=S^{-1 / 2} V^T L^T ; T^{-1}=R U S^{-1 / 2}$    (22)

$S=\left[\begin{array}{cc}S_r & 0 \\ 0 & S_{r e m}\end{array}\right]$    (23)

$\tilde{A}=\left[\begin{array}{ll}A_{11} & A_{12} \\ A_{21} & A_{22}\end{array}\right], \tilde{B}=\left[\begin{array}{l}B_1 \\ B_2\end{array}\right], \tilde{C}=\left[\begin{array}{ll}C_1 & C_2\end{array}\right]$    (24)

The reduced-order model is then obtained by retaining the blocks $A_{11}, B_1, \mathrm{C}_1$.

By truncating the states with negligible contributions, balanced truncation ensures that the reduced-order model retains the key characteristics of the original system while minimizing the error between their outputs.

This study examined balanced truncation applied to a fourteen-state multiple-input multiple-output microgrid model and showed how the choice of reduced order shapes accuracy across individual input-output channels. The analysis of H infinity errors, impulse responses, and Bode plots indicates that a four-state model captures the dominant dynamics at the buses associated with the first two outputs but fails to represent satisfactorily the behavior at the more remote buses that define the third and fourth outputs. An eight-state model preserves the main modes that govern all four output voltages and achieves small and nearly uniform errors, so it offers a practical balance between model simplicity and fidelity.

The results highlight an important qualitative insight for microgrid applications. Voltage nodes located farther from the inverters and strongly influenced by RLC elements are more sensitive to state reduction than nodes that lie closer to the sources. Reduced models that are acceptable for studies focused on local buses may not be adequate when the analysis or controller design requires accurate behavior across the full network. The work provides a quantitative guideline for selecting reduced orders in such cases since it links order choice, channel-wise H infinity errors, and time and frequency domain responses.

The present model relies on a linear time-invariant representation obtained by linearizing the microgrid dynamics around a single operating point, and the study considers only a standard balanced truncation procedure. Nonlinear effects, operating point variation, and parameter uncertainty are not captured explicitly, so the conclusions apply to a restricted yet practically relevant class of operating conditions. The evaluation is based on simulation rather than experimental data and does not include implementation aspects in embedded platforms.

Future research can extend this framework in several directions. One promising line is to explore balanced truncation within nonlinear or parameter-dependent formulations that track changes in loading and generation. Another is to combine balanced truncation with other model order reduction techniques in hybrid schemes tailored to microgrids, such as positive real variants or mixed energy-based approaches. Real-time capable reduced models integrated in hardware in the loop setups and experimental validation on laboratory microgrids would further test the robustness and practical value of the proposed guideline for order selection.