Pure and Applied Mathematics Journal
Volume 4, Issue 5-1, October 2015, Pages: 38-45

Emerging Importance of EVs in the Green Grid Era

Fuhuo Li1, S. Galhotra2, S. Kanemitsu3

1Director of Sanmenxia SuDa Communication Group, Sanmenxia Economical Development Zone, Sanmenxia, P.R.China

2Xerox Research Centre India, Bangalore, India

3Department of Information Science, Kinki University, Iizuka, Japan

Email address:

(Fuhuo Li)
(S. Kanemitsu)

To cite this article:

Fuhuo Li, S. Galhotra, S. Kanemitsu. Emerging Importance of EVs in the Green Grid Era. Pure and Applied Mathematics Journal. Special Issue: Mathematical Aspects of Engineering Disciplines. Vol. 4, No. 5-1, 2015, pp. 38-45. doi: 10.11648/j.pamj.s.2015040501.18

Abstract: A very plausible picture of the near future–some decades from now–would be the Green Grid Era and the emerging importance of EVs in the spectrum, where by Green Grid we mean the advanced Smart Grid in which essential part of the whole electric power is supplied by distributed generation, i.e. it is generated by renewable energy including solar energy, biogas, geothermal energy, etc. The crucial ingredients in this grid system are enhanced batteries and the participation of EVs in the grid as a pool of electricity. Our primary concern in this paper is the control of EV participation in the Green Grid, and in particular the study of linear systems and elucidation of various state equations in [HHP] etc.

Keywords: Green Grid, Distributed generation, EV participation, renewable energy, linear systems

1. Introduction and Preliminaries

The biggest defect of electricity is that it cannot be stored in large quantity and therefore must be kept producing, which is one of two main excuses of big power companies for their bulldozing plans of building new power stations including Nukes. The only major storage unit in most power stations is the pumped storage systems, i.e. in the form of potential energy. Electricity can be stored in batteries (or super-capacitors) only in small quantity.

The main ingredient of the cost of electricity is the cost of the high-voltage power lines, which compose a grid (system). In general, the operation cost of a grid depends highly on the PAR (peak-to-average ratio) in aggregate load demand. For example, there is usually at least one major peak in a daily (residential as well as industrial) load demand profile and some 10 hours peak during the whole year.

Here arises the second of two main excuses of big power companies for building new power stations: To assure reliable service including the supply at peak hours, the companies must produce power superseding the peak value. This makes the value of PAR higher and can significantly increase the generation cost since the grid will be highly underutilized most of time.

2. State Space Representation and the Visualization Principle

Let and  be the state function, input function and output function, respectively. We write  for .

Then a state equation for a linear system is usually given as the system of DEs (differential equations)


where are given constant matrices.

It is usually the case that if we work in the frequency domain, the things are much easier. The Laplace transform has the effect of shifting from the time domain to frequency domain and as a version of the Fourier transform, it is invertible, i.e. it restores the information in the frequency domain into the time domain. Taking the Laplace transform of (2.1) with , we obtain


which we solve as




and where  indicates the identity matrix, which is sometimes denoted  to show its size.

In general, supposing that the initial values of all the signals in a system are 0, we call the ratio of output/input of the signal, the transfer function, and denote it by etc. We may suppose so because if the system is in equilibrium, then we may take the values of parameters at that moment as standard and may suppose the initial values to be 0.

(2.4) is called the state space representation (form, realization, description, characterization) of the transfer function of the system (2.1), and is written as


2.1. Preliminaries

The three main ingredients in (electrical) circuits are coil (), condenser () and resistance (). The inverse electromotive force generated by these component is given respectively by


where =(t) and =(t) with subscript indicates the current and the voltage of the prescribed component resp. Hereafter we write for the voltage .

The governing law of the circuits is the Kirchhoff laws which have two versions. The first law is the one used for node analysis to the effect that the sum of currents flowing into a node is 0 while the second law is the one used for loop analysis which is to the effect that the sum of all electro-motive forces in a closed circuit is 0.

2.2. Control Plant I

Figure 1. Control plant.

Here PWM=Power Width Modulation and IB=Inverter Bridge.

First consider the  filter consisting of  in Fig. 1. As a consequence of the Kirchhoff law we have


The potential difference and are given respectively by




where .

Hence we conclude that


whence that




on using the relation .

Also for the grid filter consisting of , the potential difference  is given by


so that from (2.9), we obtain






We may express the control plant  in [ZH, pp. 83-84] in the form of (2.1). Let


be the state variable, the external input and the output, respectively. Hence, writing


, , ,


Then by viewing in (2.1) as block decompositions,


amounts to (2.1).

2.3. Control Plant II

It is a challenge to keep both the THD (Total Harmonic Distortion) low of the inverter local load voltage  and the gird current (the current  flowing through the grid interface inductor). The inverter LCL plant (control plant ) may be thought of as a cascaded control structure consisting of an inner loop voltage controller and an outer loop of current controller.

Figure 2. The inner loop voltage controller.

Figure 3. The outer loop current controller.



be the state variable, the external input and the output signal (which is the tracking error), respectively.

The reasoning is similar to the one given for (2.1). Hence we conclude that


whence that


on using the relation .

3. Unification

In this section we shall unify linear systems (2.1) given above. For this we view (2.6) as impedance operators  with current  flowing through it, i.e.


where , and  indicate the coil, condenser and resister, respectively. We consider the cascade connection  of two impedances with the potential  and , with the current I flowing from  to , thus the voltage difference is . Then we have


where the addition of impedances is in the sense of addition of operators, i.e. .

Suppose  is a parallel connection of the coil with inductance  and a resister with resistance  and that the current flowing the coil is  and that  is a resistance with resistance .Then


Substituting the first equality in (3.3) into (3.2), we obtain



Substituting this in (3.3), we obtain




More generally, we consider the combination of two such cascade connections at node . Two impedances  with the potential  and  and with the current  flowing from  to . Then we have


We choose  (condenser) and  a resister with resistance . Then (3.6) becomes


where  is the current flowing the impedances  in (3.2). Hence substituting (3.7) in (3.5), we conclude

Theorem 3.1. If two cascade connections of two impedances ..., 4 are connected at the node  with voltage difference  and  flowing through it, then (3.3) in the form


describes the whole paradigm as either




where  is described in Corollary 3.1.

Corollary 3.1. The cascade connection (3.5) of two impedances  (resister) and  (the parallel connection of a coil and a resister) is a special case of (3.9) with and (therefore) . (2.11) is a special case with .

Figure 4. The three-phase inverter system (represented as a single one).

Example 1. We consider the three cascade connections = 1, 2, 3 connected at a nodewith the flowing-in currents  and flowing-out currents  ( is for harmonic distortion). And the configurations of each  are similar.  indicates the filter inductor with  a parallel connection of the coil with inductance  and a resistor with resistance  and that the current flowing the coil is  and thatis a resister with resistance .

Figure 5. Cascade connection.

Other two are similar as given Table 3.1 below.

In this case, (3.4) reads


Hence from Theorem 3.1 it follows that



We want to add one more state variable  which is the electro-motive force  generated by the condenser . Since the current flowing the condenser is , we have


At the node , we have by the Kirchhoff law,




Hence (3.13) amounts to


Substituting (3.11) in this, we deduce that


We put


where . Then, putting



(3.12) and (3.16) lead to



Table 3.1. Components of .

Connection Coil C.resist C.current Resistance Current

4. H-Controllers

4.1. H-Control Problem

Following [Kim, p. 7, p. 67], we first give the definition of a chain scattering representation of a system. Suppose and  denote errors to be corrected, observation output, exogenous input, and control input, respectively and that they are related by




According to the embedding principle, this is to be thought of as corresponding to the second equality in (2.1). (4.1) means that


Suppose that  is fed back to  by


where  is a controller. Multiplying the second equality in (4.3) by  and incorporating (4.4), we find that



Substituting this in (4.3), we find that


where  is given by


and is referred to as the closed-loop transfer function . (4.6) is sometimes referred to as a linear fractional transformation and denoted by .

-controller problem

Find a controller  such that the closed-loop system is internally stable and the transfer function  satisfies


for a positive constant .

Figure 6. Control plant

4.2. Chain Scattering Representation

Assume that  is a (square) regular matrix (whence ). Then from the second equality of (4.3), we obtain


Substituting (4.8) in the first equality of (4.3), we deduce that


Hence putting


which is usually referred to as a chain scattering representation of , we obtain an equivalent form of (4.1)


Substituting (4.4), (4.11) becomes


whence we deduce that the closed-loop transfer function is expressed also as


the linear fractional transformation (which is referred to as a homographic transformation and denoted by, where in the last equality we mean the action of  on the variable . We must impose the non-constant condition . Then . If  is obtained from  under the action of , then its composition  with (4.12) yields  i.e.


which is referred to as the cascade connection or the cascade structure of and .

Thus the chain-scattering representation of a system allows us to treat the feedback connection as a cascade connection.

4.3. Siegel Upper Space

Let * denote the conjugate transpose of a square matrix:  and let the imaginary part of  defined by . Let  be the Siegel upper half-space consisting of all the matrices  (recall Eq. (4.4)) whose imaginary parts are positive definite (Im> 0—imaginary parts of all eigen values are positive) and satisfies :


and let Sp(,R) denote the symplectic group of order :

Sp(,R) =.                   (4.15)

The action of Sp(,R) on  is defined by (4.12) which we restate as


Theorem 4.1. For a controller  living in the Siegel upper space, its rotation  lies in the right half-space RHS. i.e. stable having positive real parts. For the controller Z, the feedback connection


is accommodated in the cascade connection of the chain scattering representation  (4.13), which is then viewed as the action (4.13) of Sp(,R) on :


or ,where  is subject to the condition


with . controller (see below), being a unity feedback connection, is also accommodated in this framework.

Remark 4.1. With action, we may introduce the orbit decomposition of  and whence the fundamental domain. We note that in the special case of , we have  and Sp(1;R) = SL(R) and the theory of modular forms of one variable is well-known. Siegel modular forms are a generalization of the one variable case into several variables. As in the case of the sushmna principle in [7], there is a need to rotate the upper half-space into the right half-space RHS, which is a counter part of the right-half plane RHP. In the case of Siegel modular forms, the matrices are constant, while in control theory, they are analytic functions (mostly rational functions analytic in RHP). A general theory would be useful for controlling theory.

4.4. FOPID

"FO" means "Fractional order and ""PID" refers to "Proportional, Integral, Differential", whence "Proportional" means just constant times the input function , "Integral" means the fractional order integration  of  (> 0), and "Differential" the fractional order differentiation of  ( > 0). The FO  controller (control signal in the time domain) is one of the most refined feed-forward compensator defined as the operator




where  is the input function,  is the deviation and  are constant parameters which are to be specified ( the position feedback gain,  the velocity feedback gain). DE (4.21) translates into the state equation


where  indicate the Laplace transforms of , respectively and  is the compensator continuous transfer function


The derivation of (4.23) from (4.21) depends on the following. The general fractional calculus operator  is symbolically stated as


where  and t are the lower and upper limits of integration and  is the order of calculus.

More precisely, the definition of the fractional differo-integral is given by the Riemann-Liouville expression


where  indicates the fractional part of , with [] the integral part of . Thus we are also led to the Riemann-Liouville fractional integral transform:


For more details we refer to [10].

5. Cyber Attack Impact on Smart Grid

However, wherever there is light, there is shadow. Since the smart gird highly depends on the information technologies based on communications systems, it has the same vulnerabilities as the present Internet has. The complexity of integration (of information technology in traditional power grid), diversity of system vendors, urge for timely solutions etc. all lead to increased risk of cyber attack.

We apply the theory of graph-based dynamical systems [2], [9].


where  indicates the state and an input, respectively. This is quite suited for the purpose of assessing the cyber-attack impact on the smart grid. For there is a need of relating a cyber-attack to physical consequences in the electrical network. A dynamical system paradigm gives a flexible framework to model the cause-effect relationships between the cyber data and electric grid states signals. The work is in progress.


The authors express their thanks for the support provided by International Cooperation Projects of Science and Technology Agency of Shaanxi Province (No.2015KW-022)


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