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Application of photovoltaic power in systems involving intermittent loading requires an understanding of the dynamics of such systems. This paper explores this area by taking the vegetable slicing system as a typical example of intermittent loading. Bond graph methodology has been used in modelling. This method has appeared as a convenient tool in formulating dynamic models of electromechanical machinery, but has not yet been applied to study the dynamics of PV driven systems. This paper presents the first attempt in that direction. The results obtained by computer simulation are reported.

A bond graph is a graphical representation of a physical dynamic system. It allows the conversion of the system into a state-space representation. It is similar to a block diagram or signal-flow graph, with the major difference that the arcs in bond graphs represent bi-directional exchange of physical energy, while those in block diagrams and signal-flow graphs represent uni-directional flow of information. Bond graphs are multi-energy domain (e.g. mechanical, electrical, hydraulic, etc.) and domain neutral. This means a bond graph can incorporate multiple domains seamlessly.

The bond graph is composed of the "bonds" which link together "single-port", "double-port" and "multi-port" elements (see below for details). Each bond represents the instantaneous flow of energy (dE/dt) or power. The flow in each bond is denoted by a pair of variables called power variables, akin to conjugate variables, whose product is the instantaneous power of the bond. The power variables are broken into two parts: flow and effort. For example, for the bond of an electrical system, the flow is the current, while the effort is the voltage. By multiplying current and voltage in this example you can get the instantaneous power of the bond.

If the dynamics of the physical system to be modeled operate on widely varying time scales, fast continuous-time behaviors can be modeled as instantaneous phenomena by using a hybrid bond graph. Bond graphs were invented by Henry Paynter.[1]

Relationship between force and displacement, also known as Hooke's law. The negative sign is dropped in this equation because the sign is factored into the way the arrow is pointing in the bond graph.

Bond graphs have a notion of causality, indicating which side of a bond determines the instantaneous effort and which determines the instantaneous flow. In formulating the dynamic equations that describe the system, causality defines, for each modeling element, which variable is dependent and which is independent. By propagating the causation graphically from one modeling element to the other, analysis of large-scale models becomes easier. Completing causal assignment in a bond graph model will allow the detection of modeling situation where an algebraic loop exists; that is the situation when a variable is defined recursively as a function of itself.

In bond graph notation, a causal stroke may be added to one end of the power bond to indicate that this side is defining the flow. Consequently, the side opposite from the casual stroke controls the effort.

It is possible for a bond graph to have a causal bar on one of these elements in the non-preferred manner. In such a case a "causal conflict" is said to have occurred at that bond. The results of a causal conflict are only seen when writing the state-space equations for the graph. It is explained in more details in that section.

One of the main advantages of using bond graphs is that once you have a bond graph it doesn't matter the original energy domain. Below are some of the steps to apply when converting from the energy domain to a bond graph.

The next step is to pick a ground. The ground is simply an 0-junction that is going to be assumed to have no voltage. For this case, the ground will be chosen to be the lower left 0-junction, that is underlined above. The next step is to draw all of the arrows for the bond graph. The arrows on junctions should point towards ground (following a similar path to current). For resistance, inertance, and compliance elements, the arrows always point towards the elements. The result of drawing the arrows can be seen below, with the 0-junction marked with a star as the ground.

Next power flow is to be assigned. Like the electrical examples, power should flow towards ground, in this case the 1-junction of the wall. Exceptions to this are R,C, or I bond, which always point towards the element. The resulting bond graph is below.

Now that the bond graph has been generated, it can be simplified. Because the wall is grounded (has zero velocity), you can remove that junction. As such the 0-junction the C bond is on, can also be removed because it will then have less than three bonds. The simplified bond graph can be seen below.

Once a bond graph is complete, it can be utilized to generate the state-space representation equations of the system. State-space representation is especially powerful as it allows complex multi-order differential system to be solved as a system of first-order equations instead. The general form of the state equation is x ˙ ( t ) = A x ( t ) + B u ( t ) {\displaystyle {\dot {\mathbf {x} }}(t)=\mathbf {A} \mathbf {x} (t)+\mathbf {B} \mathbf {u} (t)} where x ( t ) {\textstyle \mathbf {x} (t)} is a column matrix of the state variables, or the unknowns of the system. x ˙ ( t ) {\textstyle {\dot {\mathbf {x} }}(t)} is the time derivative of the state variables. u ( t ) {\textstyle \mathbf {u} (t)} is a column matrix of the inputs of the system. And A {\textstyle \mathbf {A} } and B {\textstyle \mathbf {B} } are matrices of constants based on the system. The state variables of a system are q ( t ) {\textstyle q(t)} and p ( t ) {\textstyle p(t)} values for each C and I bond without a causal conflict. Each I bond gets a p ( t ) {\textstyle p(t)} while each C bond gets a q ( t ) {\textstyle q(t)} .

The matrices of A {\textstyle \mathbf {A} } and B {\textstyle \mathbf {B} } are solved by determining the relationship of the state variables and their respective elements, as was described in the tetrahedron of state. The first step to solve the state equations is to list all of the governing equations for the bond graph. The table below shows the relationship between bonds and their governing equations.

The concept of bond graphs was originated by Paynter. The idea was further developed by Karnopp and Rosenberg in their textbooks, such that it could be used in practice by Thoma, Van Dixhoorn, Breedveld and by others [1, 2, 3, 4].

The language of bond graphs aspires to express general class physical systems through power interactions. The factors of power i.e., effort and flow, have different interpretations in different physical domains. Yet, power can always be used as a generalized coordinate to model coupled systems residing in several energy domains.

A causal bond graph contains all information to derive the set of state equations. The procedure to derive these equations is covered by bond graph software like Enport (, 1974), MS1 (Lorenz, 1997), CAMP (Granda, 1985), and 20-SIM (Broenink, 1990, 1995, 1997, 1999; Broenink and Kleijn, 1999). Therefore, in practice, generation of equation need not be done by hand. However, we discuss the generation of equations to indicate what exactly has to be done.

The language of bond graphs aspires to express general class physical systems through power interactions. The factors of power i.e., effort and flow, have different interpretations in different physical domains. Yet, power can always be used as a generalized coordinate to model coupled systems residing in several energy domains. To demonstrate the bond graph methodology as an example an electrical model of RLC system is analyzed Figure 1a. An RLC circuit is an electrical circuit consisting of a resistor, an inductor, and a capacitor. The RLC part of the name is due to those letters being the usual electrical symbols for resistance, inductance and capacitance respectively [5].

We have discussed the basic bond-graph elements and the bond, so we can transform a domain-dependent ideal-physical model, written in domain-dependent symbols, into a bond graph. For this transformation, there is a systematic procedure, which is presented here. This electrical system contains a voltage source effort SE (SE:uz), a resistor R (R:R), an inductor I (I:L) and a capacitor C (C:1/C). In the step 1 we determine which physical domains exist in the system and identify all basic elements like C (capacitor), I (inductor), R (rezistor), SE (source of the effort), SF (source of the flow), TF (transformer) and GY (gyrator) [6, 7, 8].

In bond graphs, the inputs and the outputs are characterized by the causal stroke. The causal stroke indicates the direction in which the effort signal is directed (by implication, the end of the bond that does not have a causal stroke is the end towards which the flow signal is directed).

The causal bond graph of this system can be derived, in which the inputs and the outputs are characterized by the causal stroke. This is the starting point, from which we continue toward the differential equations describing the dynamics of the system.

Bond graphs represent a convenient tool for physical system analysis. We presented a method to systematically build a bond graph starting from an ideal physical model. Causal analysis gives, besides the computational direction of the signals at the bonds, also information about the correctness of the model. A practical example of an electrical model is presented as the application of this methodology.

The differential equations describing the dynamics of the system in terms of the states of the system were derived from a bond graph diagram of a simple electrical system. The results correspond with equations obtained using traditional method, where the equations for individual components are created first and then the simulation scheme is derived on their basis, although the described method uses the reverse procedure. 2b1af7f3a8