Emerging Importance of EVs in the Green Grid Era
Fuhuo Li^{1}, S. Galhotra^{2}, S. Kanemitsu^{3}
^{1}Director of Sanmenxia SuDa Communication Group, Sanmenxia Economical Development Zone, Sanmenxia, P.R.China
^{2}Xerox Research Centre India, Bangalore, India
^{3}Department of Information Science, Kinki University, Iizuka, Japan
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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. 51, 2015, pp. 3845. 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 supercapacitors) only in small quantity.
The main ingredient of the cost of electricity is the cost of the highvoltage power lines, which compose a grid (system). In general, the operation cost of a grid depends highly on the PAR (peaktoaverage 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)
(2.1)
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
(2.2)
which we solve as
(2.3)
where
(2.4)
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
(2.5)
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
(2.6)
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 electromotive 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
(2.7)
The potential difference and are given respectively by
(2.8)
and
(2.9)
where .
Hence we conclude that
(2.10)
whence that
(2.11)
or
(2.12)
on using the relation .
Also for the grid filter consisting of , the potential difference is given by
(2.13)
so that from (2.9), we obtain
(2.14)
whence
(2.15)
Finally
(2.16)
We may express the control plant in [ZH, pp. 8384] in the form of (2.1). Let
(2.17)
be the state variable, the external input and the output, respectively. Hence, writing
,
, , ,
(2.18)
Then by viewing in (2.1) as block decompositions,
(2.19)
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.
,,
(2.20)
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
(2.21)
whence that
(2.22)
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.
(3.1)
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
(3.2)
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
(3.3)
Substituting the first equality in (3.3) into (3.2), we obtain
or
(3.4)
Substituting this in (3.3), we obtain
,
whence
(3.5)
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
(3.6)
We choose (condenser) and a resister with resistance . Then (3.6) becomes
(3.7)
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
(3.8)
describes the whole paradigm as either
(3.9)
or
(3.10)
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 .
Example 1. We consider the three cascade connections = 1, 2, 3 connected at a nodewith the flowingin currents and flowingout 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 .
Other two are similar as given Table 3.1 below.
In this case, (3.4) reads
(3.11)
Hence from Theorem 3.1 it follows that
(3.12)
，
.
We want to add one more state variable which is the electromotive force generated by the condenser . Since the current flowing the condenser is , we have
(3.13)
At the node , we have by the Kirchhoff law,
,
whence
(3.14)
Hence (3.13) amounts to
(3.15)
Substituting (3.11) in this, we deduce that
(3.16)
We put
(3.17)
where . Then, putting
(3.18)
，
,
，
， ，
，
，
(3.12) and (3.16) lead to
(3.19)
.
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
(4.1)
where
(4.2)
According to the embedding principle, this is to be thought of as corresponding to the second equality in (2.1). (4.1) means that
(4.3)
Suppose that is fed back to by
(4.4)
where is a controller. Multiplying the second equality in (4.3) by and incorporating (4.4), we find that
whence
.
Substituting this in (4.3), we find that
(4.5)
where is given by
(4.6)
and is referred to as the closedloop transfer function . (4.6) is sometimes referred to as a linear fractional transformation and denoted by .
controller problem
Find a controller such that the closedloop system is internally stable and the transfer function satisfies
(4.7)
for a positive constant .
4.2. Chain Scattering Representation
Assume that is a (square) regular matrix (whence ). Then from the second equality of (4.3), we obtain
(4.8)
Substituting (4.8) in the first equality of (4.3), we deduce that
(4.9)
Hence putting
(4.10)
which is usually referred to as a chain scattering representation of , we obtain an equivalent form of (4.1)
(4.11)
Substituting (4.4), (4.11) becomes
,
whence we deduce that the closedloop transfer function is expressed also as
(4.12)
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 nonconstant condition . Then . If is obtained from under the action of , then its composition with (4.12) yields i.e.
(4.13)
which is referred to as the cascade connection or the cascade structure of and .
Thus the chainscattering 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 halfspace 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 :
(4.14)
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
(4.16)
Theorem 4.1. For a controller living in the Siegel upper space, its rotation lies in the right halfspace RHS. i.e. stable having positive real parts. For the controller Z, the feedback connection
(4.17)
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 :
(4.18)
or ,where is subject to the condition
(4.19)
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 wellknown. 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 halfspace into the right halfspace RHS, which is a counter part of the righthalf 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 feedforward compensator defined as the operator
(4.20)
where
(4.21)
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
(4.22)
where indicate the Laplace transforms of , respectively and is the compensator continuous transfer function
(4.23)
The derivation of (4.23) from (4.21) depends on the following. The general fractional calculus operator is symbolically stated as
(4.24)
where and t are the lower and upper limits of integration and is the order of calculus.
More precisely, the definition of the fractional differointegral is given by the RiemannLiouville expression
(4.25)
where indicates the fractional part of , with [] the integral part of . Thus we are also led to the RiemannLiouville fractional integral transform:
(4.26)
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 graphbased dynamical systems [2], [9].
(5.1)
where indicates the state and an input, respectively. This is quite suited for the purpose of assessing the cyberattack impact on the smart grid. For there is a need of relating a cyberattack to physical consequences in the electrical network. A dynamical system paradigm gives a flexible framework to model the causeeffect relationships between the cyber data and electric grid states signals. The work is in progress.
Acknowledgement
The authors express their thanks for the support provided by International Cooperation Projects of Science and Technology Agency of Shaanxi Province (No.2015KW022)
References