DOI: 10.24193/subbmath.2019.2.02
On a stochastic arc furnace model
Hans-J¨ org Starkloff, Markus Dietz and Ganna Chekhanova
Abstract. One of the approaches in modeling of electric arc furnace is based on the power balance equation and results in a nonlinear ordinary differential equation. In reality it can be observed that the graph of the arc voltage varies randomly in time, in fact it oscillate with a random time-varying amplitude and a slight shiver. To get a more realistic model, at least one of the model parameters should be modeled as a stochastic process, which leads to a random differential equation.
We propose a stochastic model using the stationary Ornstein-Uhlenbeck process for modeling stochastic influences. Results, gained by applying Monte Carlo method and polynomial chaos expansion, are given here.
Mathematics Subject Classification (2010):34F05.
Keywords:Electric arc furnace, random differential equation, Ornstein-Uhlenbeck process, Monte Carlo method, polynomial chaos expansion.
1. Introduction
An electric arc furnace (EAF) is used for melting metals in steel industry. One deterministic model of the EAF energy system is based on the instantaneous power balance of the system, which results in the following nonlinear ordinary differential equation
k1rn(t) +k2r(t)dr(t)
dt = k3
rm+2(t)i2(t), t∈ I. (1.1) This equation describes how the arc radius rdepends on the arc current i, both are functions on a given time interval I (cf. [1]). The model coefficients k1, k2 and k3
are positive. The parametersmandnbelong to the set{0,1,2}and reflect different working conditions of the arc furnace (cf. [4] or [5]). Equation (1.1) suggests, that the arc radius function ris positive and should be prevented from getting zero. In case
This paper has been presented at the fourth edition of the International Conference on Numerical Analysis and Approximation Theory (NAAT 2018), Cluj-Napoca, Romania, September 6-9, 2018.
the arc radius takes on the value zero one has to deal with a differential equation with singularities which requires additional investigations.
Between the arc voltage u, the arc radius r and the arc currenti it holds the relationship
u(t) = k3
rm+2(t)i(t), t∈ I. (1.2)
In reality it can be observed that the arc voltage varies randomly in time, it oscillates with a random time-varying amplitude and the graph of the function shows a slight shivering (cf. [4] or [6]). To take this into account, it is better to model the arc voltage as a stochastic process. Here it is proposed to model the coefficient k3
of equation (1.1) as a stochastic process and then solve the corresponding random differential equation.
Section 3 presents the model in more detail and gives some first results gained by the Monte Carlo method. Section 4 describes how polynomial chaos expansions can be applied.
The stochastic model is based on a deterministic one, for which for certain parameters explicit solutions of (1.1) can be given. This deterministic model was investigated for example in [6] and is recapped here in section 2.
2. A deterministic model
In this section assume the time interval is I = R. In this paper we want to consider the casen= 2, for which (1.1) can be solved explicitly. For this parameter n the equation (1.1) with the substitution y =rm+4 results in the following linear ordinary differential equation
dy(t)
dt =−βy(t) +f(t), t∈ I (2.1) withf(t) := (m+4)kk 3
2 i2(t) andβ:= (m+4)kk 1
2 >0.
If we assume, that the arc currentiis a continuous function and that the initial value condition
y(t0) =y0>0 (2.2)
holds (t0∈R), then (2.1) has the unique continuously differentiable positive solution y(t) =y(t;t0, y0) =y0e−β(t−t0)+
Z t t0
f(s)e−β(t−s)ds, t∈ I. (2.3) If we additionally assume, that i is ap-periodic (with the period p), it is bounded and there exists a uniquep-periodic solution of the differential equation (2.1). We get a formula of thisp-periodic solutionyper also by applying the pullback limit oft0 on the initial value solution (2.3). It holds
yper(t) = lim
t0→−∞y(t;t0, y0) = Z ∞
0
e−βsf(t−s) ds, t∈ I. (2.4) In [6] this periodic solution is referred to as a steady state solution of the system.
Sometimes it is also called the equilibrium solution.
According to the real world data the graph of the arc current has a sinusoidal shape. To satisfy this behavior the arc current function is chosen as
i(t) =asin(ωt), t∈ I. (2.5)
Here a (amplitude) and ω (corresponding frequency) are positive parameters. If we apply (2.5), from (2.4) an explicit formula for the periodic solution can be derived.
(a) periodic solutionyper (b) arc radiusrper= (yper)m+41
(c) arc voltageuper (d) voltage-current characteristic
Figure 1. Graphs of characteristics of the deterministic model with the following parameters
a ω m k1 k2 k3
75 100π 0 2 000 5 35
Then also an explicit formula for the associated voltage function can be calculated.
It holds
yper(t) =b[1−csin(2ωt+ψ)], t∈ I (2.6) and
uper(t) =d[1−csin(2ωt+ψ)]−m+2m+4sin(ωt), t∈ I (2.7)
with the constants b=k3a2
2k1 , c= 1
r 1 +
2ωk2
(m+4)k1
2, (2.8)
d= 2m+2m+4k
m+2 m+4
1 k
2 m+4
3 a−m+4m , ψ= arctan
(m+ 4)k1 2ωk2
, (2.9) depending on the model parametersm,k1,k2, k3,aandω.
Figure 1 shows graphs of the periodic solution yper, the arc voltage uper, the arc radiusrper = (yper)m+41 and the voltage-current characteristic, i.e., of the curve ((i(t), uper(t)) :t∈ I).
3. A stochastic model
Here we want to propose a stochastic model, in which the coefficient k3 from the former deterministic equation (1.1) is now a stochastic process. One of the mod- eling challenges is to make sure, that the inhomogeneity in equation (2.1) stays pos- itive, which provides that the solutiony is always strictly positive and prevents the arc radius r from getting negative. We take this into account by considering a sto- chastic process (Xt)t∈I and multiplyingk3 with the non-negative stochastic process
(1 +Xt)2
t∈I. By inserting the stochastic process k3(1 +Xt)2
t∈I (3.1)
into equation (2.1) instead of the deterministic coefficientk3, the differential equation (2.1) turns into the random ordinary differential equation
dYt
dt =−βYt+Ft, t∈ I (3.2)
withFt=f(t) (1 +Xt)2. (Yt)t∈I and (Ft)t∈I are now stochastic processes.
LetI =Rbe the considered time interval and (Xt)t∈I be a stochastic process with continuous paths. If we assume additionally that holds Yt0 = y0 with a deter- ministic initial value y0 > 0 and a deterministic initial timet0 ∈ R, equation (3.2) has the unique pathwise random solution
Yt=y0e−β(t−t0)+ Z t
t0
e−β(t−s)Fsds, t∈ I. (3.3) As a stochastic process (Xt)t∈Rwe choose the stationary Ornstein-Uhlenbeck process (OUP). This is a Gaussian process with mean function constant zero and autocovari- ance function Cov(Xt, Xt+h) =σ2θ2exp(−θ|h|).σandθare positive parameters. If the time interval is restricted toI= [0,∞), the OUP can be also considered as a solution of the stochastic differential equation
dXt=−θXtdt+σdWt (3.4)
with a standard Wiener process W = (Wt)t≥0 and an initial value X0, which is independent of W and follows a centred normal distribution with variance σ2θ2. This also allows to consider the equations (3.3) and (3.4) as a coupled system of stochastic differential equations
d Yt
Xt
=
−βYt+f(t) (1 +Xt)2
−θXt
dt+
0 σ
dWt. (3.5)
Such coupled systems are investigated in many ways and special methods are devel- oped for them, but this path is not followed here. Instead the random differential equation (3.2) is investigated.
With the choice of Xt as Ornstein-Uhlenbeck process, there are explicit repre- sentations for certain characteristics of the solution (3.3) available. For example it holds for the mean function
E [Yt] =y0e−β(t−t0)+
1 +σ2 2θ
· Z t
t0
e−β(t−s)f(s) ds, t∈ I. (3.6) More challenging is the question of how to determine characteristics for the arc radius functionRand the arc voltage functionU. The reason lies in the nonlinear relationship between the functionsRandY, and respectivelyU,Y andX. So it holds
Rt=Y
1 m+4
t , Ut=k3(1 +Xt)2Y−
m+2 m+4
t i(t), t∈ I. (3.7)
The Monte Carlo method can be applied by simulating paths of the OUP and computing paths of the functionsU andR numerically. The red line in Figures 2(a) and 2(c) show the estimated mean function of U, which was gained by 1000 simula- tions of the OUP. The distance between the analytical mean function (3.6) and the estimated mean function gained by the Monte Carlo method in relation to the ana- lytical mean function is always less than 2.6% in the considered interval (see figure 3).
Figures 2(b) and 2(d) show a single path of the arc voltageU. The graph shows a slight trembling, but not as strong as it can be observed in real world data. Figure 2(b) shows a random time varying amplitude of the arc voltage, similar as it can be observed in reality (see, e.g. [4] or [6]).
4. Series expansions of the pathwise solution
In this section let the considered time interval be a bounded intervalI= [t0, T] with 0≤t0< T. According to the Karhunen-Lo`eve theorem the Ornstein-Uhlenbeck process restricted to a bounded time interval can be expanded as
Xt=X
k∈N
√
λkφk(t)ξk, t∈ I, (4.1) where (ξk)k∈
N is a sequence of independent standard normally distributed random variables, (λk)k∈
N are the eigenvalues and (φk)k∈
Nare the associated eigenfunctions of the covariance operator ofX (cf. e.g. [7], chapter 11). The eigenvalues (λk)k∈
Nare positive and converge to zero.
(a) (b)
(c) (d)
Figure 2. Left (a and c): estimated mean function of the arc voltage Ut from 1000 simulated paths using Monte-Carlo method (red line) and 50 paths of Ut (grey lines). Right (b and d): one single path of Utsimulations using Monte Carlo method. The following parameters were used.
a ω m k1 k2 k3 θ σ t0 y0
75 100π 0 2 000 5 35 0.5 0.5 0 6.58
The series (4.1) converges in the function space L2([t0, T]) as well as in the space C ([t0, T]) almost surely and in thep-th mean for all 1≤p <∞(cf. e.g. [2], chapter 3). For the OUP the eigenvalues can be calculated numerically and the eigenfunctions for given eigenvalues can be calculated analytically (cf. [3]).
The representation (4.1) can be applied to the random differential equation (3.2), which results in
dYt
dt =−βYt+f(t) 1 +X
k∈N
√
λkφk(t)ξk
!2
, t∈ I. (4.2)
Expanding the square leads to an equation with a simple linear structure. This can be used to get a representation of the pathwise solution Yt in terms of the random variables (ξk)k∈
Nand their products (ξkξj)k,j∈N.
Figure 3. Graph of the functiont7→ |E[YE[Yt]−E[Yˆ t]|
t] , where ˆE [Yt] is the estimated mean function estimated by the Monte Carlo method and E [Yt] is the analytical mean function (3.6) with the same parameters as in figure 2.
Notice that these random variables do not form an orthogonal system. It holds Yt=y0(t) +X
k∈N
yk(t)ξk+X
k∈N
ykk(t)ξk2+ X
k,j∈N, k<j
ykj(t)ξkξj, t∈ I (4.3)
where the deterministic functions yk and ykj are solutions of deterministic linear ordinary differential equations with corresponding initial values. It holds
dy0(t)
dt =−βy0(t) +f(t), y0(t0) =y0, (4.4) dyk(t)
dt =−βyk(t) + 2p
λkφk(t)f(t), yk(t0) = 0, (4.5) dykk(t)
dt =−βykk(t) +λkφ2k(t)f(t), ykk(t0) = 0 (4.6) fork∈Nand
dykj(t)
dt =−βykj(t) + 2p
λkλjφk(t)φj(t)f(t), ykj(t0) = 0 (4.7) fork, j∈Nwithk < j andt∈ I. Explicit solutions can be given for these equations.
It holds
y0(t) =y0e−β(t−t0)+ Z t
t0
f(s)e−β(t−s)ds, t∈ I, (4.8) yk(t) = 2p
λk
Z t t0
φk(s)f(s)e−β(t−s)ds, t∈ I, (4.9) ykk(t) =λk
Z t t0
φ2k(s)f(s)e−β(t−s)ds, t∈ I (4.10) fork∈Nand
ykj(t) = 2p λkλj
Z t t0
φk(s)φj(s)f(s)e−β(t−s)ds, t∈ I, (4.11)
fork, j∈Nwithk < j. For numerical computations the sums in equation (4.3) have to be truncated, which gives an approximation of the solutionY. The representation (4.3) could be achieved due to the simple structure of the random differential equation (3.2).
In general one can use an expansion of an already approximated solution in terms of orthogonal random variables, which is also known as polynomial chaos expansion.
Therefore, let YN denote the pathwise solution (3.3) of our initial value problem, where the OUP (Xt)t∈I is replaced by the truncated sum
XtN :=
N
X
k=1
√
λkφk(t)ξk, t∈ I, (4.12) withN ∈N. XtN
t∈I can be considered as a smoothed version of the OUP (Xt)t∈I. ThenYN can be represented as polynomial chaos expansion
YtN =
M
X
k=0
˜
yk(t)Ψk, t∈ I, (4.13)
with
M =(N+ 2) (N+ 1)
2 −1.
The (˜yk)0≤k≤M are deterministic functions and the (Ψk)0≤k≤M are orthogonal ran- dom variables, for which applies that for every Ψk exists aN-variate polynomialpk, such that
Ψk=pk(ξ1, ξ2, . . . , ξN), (4.14) in particular Ψ0 = 1 and Ψk = ξk for k ∈ {1, . . . , N}. In general the polynomial chaos expansion does not consist of a finite number of summands. But from the representation (4.3) follows, that one has to consider only polynomials up to degree two. It is convenient to choose the sequence (Ψk)0≤k≤M, such that the degrees of the associated polynomials (pk)0≤k≤M are increasing. In the considered case the sequence of polynomials can be determined by using the Hermite polynomials (cf. e.g. [7]). From the orthogonality of the random variables (Ψk)0≤k≤M follows, that holds
E
YN(t)Ψk
= ˜yk(t)·E Ψ2k
for allk∈ {0,1, . . . , M}andt∈[t0, T]. This results in representations of the coefficient functions (˜yk)0≤k≤M. It holds
˜ yk(t) =
(y0e−β(t−t0)+Rt
t0 e−β(t−s)f(s) ds+Rt
t0 e−β(t−s)g0(s) ds, ifk= 0 Rt
t0 e−β(t−s)gk(s) ds, ifk≥1 (4.15) withk∈ {0,1, . . . , M}, where the functions (gk)0≤k≤M are defined by
gk(t) :=
f(t)PN
j=1x2j(t) ifk= 0
2f(t)xk(t) if 1≤k≤N
f(t)PN i=1
PN
j=1xi(t)xj(t)Mijk else
(4.16)
with
xj(t) :=p
λjφj(t) forj∈ {1,2, . . . , N} and
Mijk :=E [ΨiΨjΨk]
E [Ψ2k] fori, j∈ {0,1, . . . , M}.
5. Conclusion
One modeling approach of the power system of electric arc furnaces leads to a nonlinear ordinary differential equation, which in some important cases can be solved with a linear differential equation for an auxiliary quantity. Real data show partly an irregular behaviour so that a stochastic modeling seems to be advisable.
In the present work one such stochastic model is investigated. Thereby one co- efficient of the differential equation is replaced by a stochastic process, leading to a random differential equation and hence also to a stochastic voltage process. The in- put stochastic process is modelled with the help of a stationary Ornstein-Uhlenbeck process, for which many properties and results are known. The random differential equation is investigated with the help of Monte Carlo method, but also the usage of polynomial chaos expansions is explained.
The results show a relatively good agreement with real data. In the future we plan to investigate further methods and models and we also plan to investigate methods for statistical inference from real data for the considered models.
References
[1] Acha, E., Semlyen, A., Rajakovi´c, N.,A harmonic domain computational package for nonlinear problems and its application to electric arcs, IEEE Transactions on Power Delivery,5(1990), no. 3, 1390-1397.
[2] Adler, R.J., Taylor, J.E.,Random Fields and Geometry, Springer, New York, 2007.
[3] Corlay, S., Pag`es, G.,Functional quantization-based stratified sampling methods, Monte Carlo Methods Appl., De Gruyter,21(2015), no. 1, 1-32.
[4] Grabowski, D., Selected Applications of stochastic approach in circuit theory, Wydawnictwo Politechniki ´Slaskiej, Gliwice, 2015.
[5] Grabowski, D., Walczak, J.,Analysis of deterministic model of electric arc furnace, 10th International Conference on Environment and Electrical Engineering, 1-4.
[6] Grabowski, D., Walczak, J., Klimas, M., Electric arc furnace power quality analysis based on stochastic arc model, 2018 IEEE International Conference on Environment and Electrical Engineering and 2018 IEEE Industrial and Commercial Power Systems Europe, 1-6.
[7] Sullivan, T.J.,Introduction to Uncertainty Quantification, Springer, Cham, 2012.
Hans-J¨org Starkloff
Technische Universit¨at Bergakademie Freiberg Faculty of Mathematics and Computer Science Pr¨uferstraße 9, 09599 Freiberg, Germany
e-mail:[email protected] Markus Dietz
Technische Universit¨at Bergakademie Freiberg Faculty of Mathematics and Computer Science Pr¨uferstraße 9, 09599 Freiberg, Germany e-mail:[email protected] Ganna Chekhanova
Technische Universit¨at Bergakademie Freiberg Faculty of Mathematics and Computer Science Pr¨uferstraße 9, 09599 Freiberg, Germany e-mail:[email protected]
