# Tightly Coupled Systems

In distributed systems, tightly coupled systems are a type of system that should be handled with care. In general, tightly coupled systems are defined as systems that depend strongly on each other, such as through the characteristics of the exchanged data (eigenvalues and natural frequencies) or causal dependencies, mathematically speaking. This will be discussed in the following.

## Tightly Coupled through Frequencies

A subsystem that is sensitive to input signals, which often include fast dynamics, is said to be tightly coupled to the submodel environment through frequencies. Hence, additional care must be taken when distributing such systems. One example of such a tightly coupled system is a three phase generator powering an electrical motor, both modelled as submodels in a distributed system by using the abc-reference frame. Then, by assuming that the model interfaces are voltage and current, these interfacing signals oscillates with the operating frequency of the power system, typically 50 $Hz$ in steady state, in addition to other transient frequencies. According to the Nyquist sampling theorem, the two submodels should exchange data with a frequency of atleast 100 $Hz$ in order to avoid aliasing. However, in practice the frequency of the data exchange should be much higher in a distributed system in order to obtain stable and realistic simulation results. Thus, such a system is tightly coupled through frequencies and eigenvalues.

Typically, electrical systems modelled in the abc-reference frame are considered to be tightly coupled through frequencies. However, if another reference frame is used, such as the dq0-reference frame, the electrical system is not considered tightly coupled. An example of a marine power plant modelled in the dq0-reference frame is given in [1]. See Model interfaces for more information about the dq0-reference frame.

## Tightly Coupled through Causality

Some systems are said to be tightly coupled to causality, which means that they are closely connected through the differential equations, where only one of the systems can have a full state space implementation, whereas the other systems must have a differential causality, reducing the number of states in each system. This, in order to assure connectivity between the systems and to avoid iterations on a higher level in a distribute system.

One typical example of a system that is tightly coupled through causality in the field of maritime industry is a marine vessel with a deck crane as shown in the figure.

The vessel and the crane have in general large time constants, which means that they are not tightly coupled through frequencies, but they are still tightly coupled, thus through causality. This type of tightly coupled systems are illustrated in the following example where two mass-spring-damper systems are connected.

**Example 1** *Tightly Coupled Systems*

*A mass-damper spring system only affected by gravity can be expressed as*
\begin{equation}
\begin{split}
\dot{x}_{1,1}&=x_{1,2}\\
\dot{x}_{1,2}&=-\frac{b_1}{m_1}x_{1,2}-\frac{k_1}{m_1}x_{1,1}+g
\end{split}
\end{equation}
*where $x_{1,1}$ is the position of the mass, $x_{1,2}$ is the velocity of the mass, $m_1$ is the mass, $b_1$ is the damping coefficient, $k_1$ is the spring stiffness and $g$ is the acceleration of gravity. Now, assume that another mass-damper-spring system is added, as shown in the figure.*

*Now, the two masses has the same motion, and by assuming that we distribute the total system such that system 1 and system 2 are implemented as subsystems, the systems become tightly coupled through causality. If both systems are implemented with integral causality, both subsystems would give a velocity as an output and receive a force as an input. Thus, iteration on a higher level is necessary. Alternatively, one of the systems can be implemented with differential causality.
Mathematically speaking, by assuming integral causality for system 1, the two systems can be implemented as*
\begin{equation}
\begin{split}
\dot{x}_{1,1}&=x_{1,2}\\
\dot{x}_{1,2}&=-\frac{b_1}{m_1}x_{1,2}-\frac{k_1}{m_1}x_{1,1}+g+m_1\tau\\
&\\
\dot{x}_{2,1} &= v\\
\tau &= m_2\frac{dv}{dt}+b_2v+k_2x_{2,1}-m_2g\\
\end{split}
\end{equation}
*where $v=x_{1,2}$. Hence, one state is lost and the signal $v$ must be differentiated numerically inside subsystem 2. Since signals are held constant in between each global communication time step, numerical differentiation of $v$ is a bad idea. Hence, the distribution of tightly coupled systems must be handled with care.*

For the vessel-crane system, the best simulation results would be obtained when the vessel and the crane are implemented as one subsystem in a distributed system. Such an implementation has been thoroughly studied in [2]. However, there also exists other ways of implementing the system in order to overcome the problems caused by differential causality in distributed systems, such as Hybrid Causality.

^{↑}S. Skjong and E. Pedersen., 2017.

*A Real-Time Simulator Framework for Marine Power Plants with Weak Power Grids.*Elsevier - Mechatronics.

^{↑}B. Rokseth, S. Skjong and E. Pedersen., 2016. Modeling of Generic Offshore Vessel in Crane Operations With Focus on Strong Rigid Body Connections.

*IEEE Journal of Oceanic Engineering*.