Electromechanical Dynamics: Discrete Systems Pt. 1

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To verify the accuracy of the digital measuring chain, they have been installed in parallel to a conventional metering system. The installation described in [8], showed that after three years of operation the difference of the accumulated energy measured by the conventional and the digital system was at around 0. This is not the absolute accuracy but the difference of the two measuring systems, which could be up to 0. Even better results are presented in [7], which describes two installations with NCITs, process bus and grid meters.

Here the observed differences for active energy between conventional and non-conventional systems range from 0. Transient performance Transient performance classes of instrument transformers play important roles in dimensioning protection applications. The parts for conventional instrument transformers are already released, but for non-conventional or electronic CTs and VTs, still the old standard has to be used.

In both cases they describe the behavior at the secondary interface of the instrument transformers, which are the terminal blocks in case of conventional CTs and VTs and the communication interface in case of their non-conventional variants. Figure 3: Interfaces and standardization click to enlarge. Until the IEC standard part is ready, interoperability beyond communication of the stand-alone merging unit of one vendor with the protection relay of another vendor has to be carefully verified.

Complete system testing stressing the dynamic performance and transient response of the analog conversion is critical to ensure proper system operation. In this setup the protection zones of busbar and line protection as well as the protected zones by the line protection overlap. In case of air-insulated systems, the NCITs can be integrated in the bushings of the circuit breaker or in case of gas-insulated systems; they can be located at each side of the circuit breaker between breaker and disconnectors. In case of combined NCITs for current and voltage, more voltage measuring points than normal are available in a diameter, which results in biggest flexibility in choosing voltage sources for e.

Benefit of sensors not saturating The result of using a current sensor that does not saturate can have a profound effect on the setting, and thus the sensitivity of a relay. Take for example, the differential relay. A differential relay relies on current sensors to provide the exact reproduction of the primary currents to it for analysis.

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It then adds the current vectors together and computes a differential current. Then using an operating curve, as shown in Figure 8 determines whether to operate or not. If the differential current falls above the characteristic curve for a given restraint current, the relay operates. If not, it restrains. The slopes in red section, and green section are there to adjust for the performance of a conventional current transformer. As the restraint current goes up, the chances of two conventional current transformer operating exactly the same reduces. This output difference between the current transformers is compensated for by increasing the slope of the characteristic such that more differential current is needed to operate as the restraint current increases.

While this compensation is necessary for conventional current transformers to make it secure during current transformer saturation, it has the effect of decreasing the sensitivity of the differential scheme. With the use of non-conventional current sensors, these slopes can be set to close to zero which increases the sensitivity of the differential scheme during high current conditions.

He held several positions, from commissioning of substation automation systems, through technical support and project management. Today he is a global product manager for process bus solutions, where he coordinates the introduction IEC process bus in pilot and commercial projects. Stefan studied electrical science at the University of Applied Sciences Northwestern Switzerland, and holds a master degree in business administration from Edinburgh Business School of Heriot-Watt University, Scotland. Steve Kunsman is a recognized Substation Automation Specialist with over 32 years in substation automation, protection and control applications, communications technologies IEC and DNP , cyber security for substation automation, and the Relion product family of protection and control relays.

His ABB career began in as an Electrical Designer for the protective relay group and has held various engineering, technology and product management positions within the North American and global substation automation organizations. Steve holds a B.

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Models with increased mechanical node densities were also analysed using the dRDM approach with , , and. For the main results simulations presented in this paper, a times higher mechanical node density dRDM and was used with the dRDM model. Compared to these dRDM simulations previous PKN research used simulations with up to times lower node density with and [12].

If we compare the computational performance for a typical simulation in this paper e. Thus, the application of the PKN model for higher resolutions and larger system sizes is not computationally tractable for studying extended duration model simulations. It is possible to use advanced numerical techniques to improve the numerical performance of finite element methods such as the PKN approach, however that is beyond the scope of this study.

The primary aim of this study was to develop a simple and efficient alternative to the PKN approach for the study of basic effects of deformation on wave propagation in excitable media. The dRDM approach provides a computationally tractable method for studying large RDM systems with high temporal and spatial numerical resolutions. The usefulness of the dRDM approach is illustrated in the following results section.

We have introduced a discrete modeling framework to study the basic properties of RDM systems. We first show that the dRDM approach is able to reproduce some previously reported results on pacemaking activity, which were identified using the PKN model [12]. The RD model in [12] is identical to Eqs. In addition, no flux boundary conditions for the RD equations and fixed boundaries of the mechanical mesh were used in the present study, as reported in [12].

On the other hand, [12] uses a continuum mechanics formulation that follows the Mooney-Rivlin material relation.

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The Mooney-Rivlin relation shares similarities with the Seth material relation used in this dRDM model, because both constitutive relations describe isotropic elastic mechanical response. However, the Mooney-Rivlin material relation describes a nonlinear force-displacement relationship for finite deformations. Therefore, we did not seek an exact correspondence of the two approaches, but rather a qualitative agreement as a reflection of the underlying basic mechanisms determining pacemaker dynamics. In [12] , Panfilov et al. The main objective of this section is to test if the dRDM approach reproduces important mechanisms on self-organized pacemakers that were identified with the continuous PKN modeling framework [12].

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To begin, the phenomenon of self-organised pacemaking activity is described. It has been found that a single electrical or mechanical point stimulus can cause the formation of a pacemaker in a RDM medium with non-oscillating RD kinetics. Pacemaking activity occurs because the contraction of the medium that follows a radially propagating wave of excitation subsequently stretches the medium in the neighborhood of the initiation site.

This stretch induces a depolarizing stretch activated current Eq. The location of this pacemaker may drift over the course of time depending on the position of the initial stimulus [12].

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Two main drift directions were identified: to the center of the medium for larger medium sizes and to the boundary for smaller medium sizes with an intermediate regime involving multiple symmetric attracting points. In particular, the dRDM model reproduced the phenomena of pacemaking activity as well as the dependence of pacemaker drift on the location of the initial stimulus and the size of the medium. Figure 5A shows typical drift patterns for a large dRDM model. The pacemaker drifted to the center of the medium from all initialization locations. Figure 5B shows the same experiment performed with the PKN model.

Both approaches describe one spatial attractor in the center of the medium. Small black dots indicate positions of point stimuli and the arrows indicate drift directions and the estimated positions of sequential action potentials slow drift is indicated by short arrows. Attractors are indicated as big black dots. Figure 6A shows the drift patterns for a smaller system size in the dRDM model with peripheral attractors and attractors on the diagonals of the medium. It should be noted that the diagonal attractors were not previously reported in [12].

However, we performed the same experiment using the PKN model Figure 6B and found that these attractors indeed existed using the continuous PKN approach. Thus all spatial attractors were present in both modeling approaches. Notations are as in Figure 5. We also studied how the location of the peripheral attractors depended on the medium size. Figure 7 demonstrates the distance of the peripheral attractor from the center on a graph similar to that in [12]. Although the elastic properties of PKN and the dRDM model are not identical, the drift patterns showed qualitative agreement.

Both modeling approaches demonstrated that there is a shift of peripheral pacemaker attractor locations to the center of the medium as the size of the model is increased. Additionally, this transition occurs at comparable sizes of the medium:. Therefore, we conclude that the dRDM model reproduces the same phenomena on pacemaker activity as reported in [12].

Relative shift location of the peripheral attractors as a proportion of the distance from the center to the boundary of the medium against medium size.

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Computations with the dRDM model black symbols, continuous black lines were performed using , , and. The results from the PKN model red squares, dotted red line are from [12]. The increased resolution of the dRDM model compared to [12] allows one now to study this system in greater detail. In particular, we shift the focus now onto the following open issues: the effects of change of medium size on the pacemaker period; and the mechanisms underpinning pacemaker drift.

This section is devoted to the cases of pacemaking activity that result in a static pacemaker located at the center of the medium. The aim of this section is to understand the factors that determine the period of the pacemaker and its dependency on the medium size. This investigation commenced with the study of the spatial and temporal transient processes leading to the steady state configuration of a pacemaker with a constant period located at the center of the medium.

Figure 8 illustrates how the period of a pacemaker of the large system shown in Figure 5 evolves during the drift of the pacemaker to the center of the medium. The results of two simulations are shown: for a pacemaker that was initiated at the center of the medium the red line and for a pacemaker that was initiated at the boundary of the medium the black line. In both cases, the pacemakers initially had a long period that rapidly decreased over 3—5 cycles. Following this transition phase, the period of the centrally located pacemaker rapidly settled to the value of.

For the peripherally located pacemaker, its period rapidly decreased during the transition phase to and then the period slowly decreased further during the drift process. By the time the pacemaker had reached the center of the medium, its period had approached the same value of.

Therefore, the drift of a pacemaker to the center can be described as drift to a region of shorter period. Pacemaker period for a pacemaker drifting from the boundary of the medium initiated from the center for the medium size of to the center, in comparison to the period of a pacemaker that was initiated at the center. The results on the study of how the medium size affects the equilibrium period of a stationary pacemaker located in the center of the medium are shown in Figure 9A upper panel. Biphasic behavior was observed. For system sizes larger than , the equilibrium period decreased with a decrease in the medium size.

On the other hand, for system sizes smaller than , the steady-state period increased with a decrease in the medium size. This biphasic behavior is explained in the following. The first regime is the result of an increase in the maximal stretch of the medium. Figure 9A lower panel shows that the maximal stretch monotonically increased with a decreasing medium size. This observation was qualitatively reported in a previous study using the continuous PKN description [12].

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The larger stretch resulted in a larger stretch-activated current , which in turn resulted in a shorter period. The second regime occurred due to a different mechanism. The decrease in medium size also resulted in a decrease of the size of the pacemaker. Figure 9A middle panel shows the monotonic increase of the curvature of a new forming pulse of a pacemaker with decreasing medium size.

This resulted in an increasing influence of the curvature on wave propagation.

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Upper panel Pacemaker period vs medium size. Middle panel Curvature of a new forming pulse vs medium size. Lower panel Maximal stretch scaled using vs. B Excitation variable and diffusive current scaled by for the center point for one pulse of a pacemaker in a system of size and a system of size. Curvature effects are well known in the theory of excitable media [35] and can be explained using the following formal consideration. If a polar coordinate system is used to describe the dynamics of a radially expanding wave front, then the expression for the Laplacian will be given by:.

For an expanding wave front , and thus the curvature related term results in a negative diffusive current. This negative diffusive current reduces the velocity of wave propagation and for higher curvature results in the critical curvature phenomenon, i. However, for the wave back , which results in a positive diffusive current that tends to prolong the action potential. Both of these effects are important to understand the second branch of Figure 9A upper panel.

Indeed, comparing the shapes of action potentials for medium sizes and as shown in Figure 9B black lines , one sees that the upstroke of the action potential was slightly slower in the smaller medium compared to the larger medium due to negative curvature related current. The recovery process in the smaller medium was also slower due to the curvature effect on the wave back.

This is also illustrated in Figure 9B via the diffusive current red lines , which showed a larger amplitude for the smaller medium that in turn slowed down the upstroke and prolonged the action potential duration. This prolongation increased the period of a pacemaker see Figure 9A. When the medium size was decreased below , the firing area became smaller than the critical size and the pacemaker activity disappeared. Indeed, for the medium described with the dRDM model without deformation the critical curvature found was , which is close to the curvature below which a block of the pacemaking activity was observed.

This section focusses on pacemaker drift. Figure 10 demonstrates a representative example of pacemaker drift to the center of an RDM medium. It illustrates the formation of the 26th pulse after initiation of pacemaking activity near the boundary of the medium. The lower panel reveals the distribution of local dilatation in the medium and the upper panel illustrates the time course of the main variables of the dRDM model along the pacemaker drift line, which is indicated as a thick black horizontal line in the lower panel.

This line indicates the route of the pacemaker during its drift to the center of the medium. The formation of pulse in the tail of the previous th wave is shown. The following reasoning is based on the stretch distribution in the medium the green line generated by this wave. Initially, the stretch is reasonably symmetric around the new forming pulse see the green line near the arrow in Figure 10A. However, a clear gradient is evident with higher stretch directed to the center of the medium at a later stage of pulse formation see the green line near the arrow in Figure 10B.

As higher stretch produces a higher stretch activated current , this gradient in stretch leads to a slightly faster depolarization and subsequent excitation closer to the center of the former excitation point Figure 10C. As a result the subsequent pacemaker position is shifted towards the center of the medium and so on until the pacemaker ended up at the center of the medium.

From this, one can conclude that the main driving force of the drift in this case is the asymmetry of the stretch pattern. But why does this asymmetry occur? To study the influence of curvature on the stretch distribution and the pacemaker drift, we compared two cases: a pacemaker initiated by a point stimulus; and a pacemaker initiated by a line electrode. Figure 11 shows the stretch distribution and main variables along the drift line immediately prior to the first pacemaker pulse following the stimulus. Again, a gradient in stretch was evident following the point stimulus Figure 11A.

However, this asymmetry was not present for the line stimulus Figure 11B. This indicates, that indeed the curvature of the wavefront causes the spatial asymmetry in stretch. A emergence of the 26th pulse at.

B The same pulse after at and C after of its emergence at. The traces in the upper panels illustrate main state variables in the medium along the pacemaker drift lines thick black horizontal lines in the lower panels : the excitation variable black , recovery state scaled using red , and regional dilatation scaled using green. The locations of the emergence of the 1st and 26th pulse are marked by vertical dotted lines in the upper panels.

The maximum voltage is marked with an arrow. The lower panel indicates the regional dilatation in the medium scaled using by a color spectrum. The point of initial stimulation is indicated by a white dot. Medium size. Pacemaker stimulation with a point stimulus and a line electrode shown in white in the lower panels. Snapshots taken at A , and B. System size and notations are the same as in Figure To show that the stretch activated current is not important for the formation of the stretch gradient we did similar simulations in the absence of.

Figure 12A shows the stretch-contraction pattern in this situation. A formation of a gradient in stretch in the vicinity of the previous pacemaker position around the vertical dotted line in Figure 12A , without stretch activated current is shown. Since stretch is the elastic response to a spatial contraction pattern, we studied of how the shape of the wave front affects the formation of this gradient in stretch.

For this study, the wave was initiated by linear electrodes with increasing size, which resulted in the generation of waves with progressively decreased curvature Figure 12 , lower panel. A decrease in curvature decreased the stretch asymmetry until it disappeared for a plane wave stimulus Figure 12D.

Therefore, we conclude that a gradient in stretch in the studied system is formed by the curvature of the wave. This effect is different for points at different distances from the front, which generates a gradient in stretch. A detailed study of the effects of front shape on deformation patterns will be presented as a separate study.

The conclusions that can be drawn here is that the drift of a pacemaker to the center in the mechanical setup introduced in [12] is driven by the asymmetry of the stretch pattern, which in turn, is strongly influenced by the shape of the wave front. Waves of different curvature generated by electrodes of different shape shown in white in the lower panels. Snapshots are taken at A , B , C , and D. The stretch activated current Eq. As demonstrated in Figure 7 , pacemaker drift was directed towards the boundaries of smaller models.

In smaller media the influence of the diffusive current is increased, and it starts affecting the duration of the action potential by inducing a gradient in action potential duration towards the center of the medium. However, to date the authors were unable, to quantify the effects of the diffusive current in relation to drift direction and discriminate them from other observed factors, such as elliptical shape of the firing region, etc. This may serve as a good starting point for follow-up studies. In this paper, we introduce a discrete modeling framework for the study of reaction-diffusion-mechanics systems.

The model is based on the coupling of a mass-lattice model with reaction-diffusion equations. Mass-lattice models are widely used in various areas of computational mechanics research and application [16] , [20]. There are several advantages of the dRDM approach presented in this paper. Firstly, its implementation does not require finite element methods, but can be achieved using explicit methods, which allow for a more frequent update rate and higher spatial resolution of the mechanical mesh configuration.

Furthermore, the explicit numerical scheme used in this paper to solve the mechanics equations is very effective in studying large systems as the computational speed scales approximately linearly with the number of mass points in the system. The main disadvantage of this approach is that it can not easily be connected to known continuous material properties.

However, this does not pose a problem, as we have shown in this paper, that an isotropic Seth material can be used to study basic mechanisms of RDM systems. For more complex materials, it may be necessary to apply homogenization techniques to formulate their constitutive relations. An example of the application of homogenization techniques to derive constitutive relations for a mass-lattice model and its relation to cardiac tissue is given in [36]. It is important to note the possibility to relate discrete mechanics modeling to continuum mechanics by obtaining forces in the mass-lattice model directly from the corresponding constitutive relations [20].

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