Research Article Minimizing Energy Cost in Electric Arc ...downloads.hindawi.com/journals/jen/2014/620695.pdf · Research Article Minimizing Energy Cost in Electric Arc Furnace Steel

Research ArticleMinimizing Energy Cost in Electric Arc Furnace SteelMaking by Optimal Control Designs

Er-wei Bai1,2

1 Department of Electrical and Computer Engineering, University of Iowa, Iowa City, LA 52242, USA2 School of Electronics, Electrical Engineering and Computer Science, Queen’s University, Belfast BT7 1NN , UK

Correspondence should be addressed to Er-wei Bai; [email protected]

Received 30 September 2013; Revised 23 December 2013; Accepted 14 January 2014; Published 20 February 2014

Academic Editor: Qilian Liang

Copyright © 2014 Er-wei Bai. This is an open access article distributed under the Creative Commons Attribution License, whichpermits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Production cost in steel industry is a challenge issue and energy optimization is an important part. This paper proposes an optimalcontrol design aiming at minimizing the production cost of the electric arc furnace steel making. In particular, it is shown that withthe structure of an electric arc furnace, the production cost which is a linear programming problem can be solved by the tools oflinear quadratic regulation control design that not only provides an optimal solution but also is in a feedback form. Modeling andcontrol designs are validated by the actual production data sets.

1. Introduction

Steel industries recycle scrap steel using the electric arcfurnace (EAF) by melting it and changing its chemicalcomposition to produce different product grades. Obviously,steel industry is one of the greatest energy consuming sectorsand there is a strong demand to decrease the use of electricityand other forms of energy in the EAF steel making.

In general, the current melting process control is manualand the production cost is not optimal. Steel industries havesome nominal automation for EAFs but are mostly operatordriven. Although operator intuition is invaluable for suchindustries, it rests on the recipes that have been effective inthe past and becomes difficult to take into account all possibleuncertainties unless a mathematical framework is developedfor the system and an automation process is in place. It hasbeen shown that a substantial part of the energy consumptionis wasted in melting the scrap [1–3].This illustrates that thereis a tremendous opportunity for control to play.

Unlike most existing papers in the literature, this paperconcerns the production cost which is related to energyconsumption for melting process and the way to achieve theoptimality. Therefore, the purpose of the model is differentfrom the large part of the existing literature [1–6]. Forexample, in [1], an adaptive model predictive control was

proposed to follow the preset trajectories. How to design thepreset trajectories was not discussed. In [2], a PID controllerwas proposed to have the electrodesmaintain constant powerconsumption. In [3], a model was developed but there was nodiscussion on how to control it. In [6], a model was proposedto achieving the maximum power input to the meltingprocess. In all these works, minimizing control consumptionwas not a concern. In the work reported here, a mathematicalmodel of EAFs is developed and its unknown parametersare then estimated. The goal of the model is to design acontrol algorithm that minimizes the energy consumption.The model adopted in this paper is a linear time invariantsystem which seems to work satisfactorily. More importantly,in the second step, optimal control inputs are calculatedaiming atminimizing the production cost. Because the actualproduction cost is linear in the consumptions of energy andgraphite electrode, the optimization is a linear programmingproblem. A problem of the linear programming is that itprovides an open loop input design. Though optimal in theabsence of model uncertainty and measurement noise, itsperformance in reality cannot be guaranteed under inevitablemodel uncertainty andmeasurement noise.The contributionof the paper is to show that with the particular structure of theEAF, the linear programming problem can in fact be solved bythe well-known linear quadratic regulation (LQR) problem

Hindawi Publishing CorporationJournal of EnergyVolume 2014, Article ID 620695, 9 pageshttp://dx.doi.org/10.1155/2014/620695

2 Journal of Energy

Figure 1: Electric arc furnace.

which is in a feedback form and thus is more robust than anopen loop solution.

All the data sets used in this study for modeling andcontrol validation are actual production data collected fromthe Gerdau Ameristeel mill facility in Wilton Iowa that iscomposed of a scrap shredding facility, an electric arc furnace(EAF) as shown in Figure 1, several continuous billet casters,and a rollingmill. Because of proprietary properties, no actualdata value is revealed in the paper, but the modeling resultsand final performance improvements are shown.

2. Problem Statement

In this section, we describe the system model and the goal ofoptimal control. Clearly, EAF models could be different fordifferent purposes. Our goal is to optimize the productioncost while the quality and efficiency are maintained. Tothis end, we focus on 7 key variables that provide a fairlyreasonable description of the dynamics of an EAF:

(i) kilowatt-hour consumption (KWH),(ii) electrode consumption (𝐼2ℎ),(iii) percentage of scrap melted (PM),(iv) average arc current (𝐼avg),(v) oxygen input (O

2),

(vi) gas input (Gas),(vii) carbon input (Car).

All 7 variables are eithermeasurable or computable in realtime. Among these variables, the kilowatt-hour consump-tion (KWH) is the consumption of electrical energy thatconstitutes a substantial part of the total cost, the electrodeconsumption (𝐼2ℎ) is the consumption of the graphite elec-trode that also contributes to the cost significantly, and thepercentage of scrape melted (PM) is the percentage of scrap

meltedwhich is a quality constraint on themelting control. Atthe end of eachmelting process, PMhas to be 100(%). Other 4variables are electrical energy input 𝐼avg and chemical energyinputs O

2(oxygen), Gas (gas), and Car (carbon), respectively.

All 4 energy inputs contribute to the total cost and amongthem 𝐼avg is the most expensive one.

Based on the physics, the increments of the kilowatt-hourconsumption (KWH) and the electrode consumption (𝐼2ℎ)from time 𝑘𝛿𝑡 to (𝑘 + 1)𝛿𝑡 are directly related to the current𝐼avg in the interval [𝑘𝛿𝑡, (𝑘 + 1)𝛿𝑡). On the other hand, thepercentage of scrape melted (PM) is related to all energyinputs during the same period including the electrical energyas well as chemical energy. The EAF assumes the followingstructure:

(

KWH [𝑘 + 1]

𝐼2ℎ [𝑘 + 1]

PM [𝑘 + 1]

) = (

1 0 0

0 1 0

0 0 1

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝐴

(

KWH [𝑘]

𝐼2ℎ [𝑘]

PM [𝑘]

)

+ (

𝜑1

0 0 0

𝜙1

0 0 0

𝑎1𝑎2𝑎3𝑎4

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝐵

(

𝐼avg [𝑘]

O2 [𝑘]

Gas [𝑘]Car [𝑘]

) ,

𝑘 = 1, . . . , 𝑁,

(1)

where (KWH[0]𝐼2ℎ[0]

PM[0]) = 0, 𝑘 = 𝑘𝛿𝑡 with 𝛿𝑡 = 5 (sec) being the

sampling interval and (𝑁 + 1)𝛿𝑡 is the total time by which100% of scrap should be melted. This model is not surprisingand has been used in the literature. To maintain the sameproductivity and efficiency, (𝑁 + 1)𝛿 is the total length ofthe melting process time and is predetermined by the currentpractice without optimal control. 𝐵 is a matrix whose entriesdepend on a particular EAF and the scrap loads. Note that𝐼avg[𝑘], O2[𝑘], Gas[𝑘], Car[𝑘] are the respective total energydelivered to the EAF during the period [𝑘𝛿𝑡, (𝑘 + 1)𝛿𝑡) andKWH[𝑘] and 𝐼2ℎ[𝑘] are the total electric energy and electrodeconsumption, respectively, at the end of the 𝑘th samplinginterval. The production cost is defined as

𝐽 = 𝑞1KWH [𝑁 + 1] + 𝑞

2𝐼2ℎ [𝑁 + 1] + 𝑞

3

𝑁

∑

𝑘=1

O2 [𝑘]

+ 𝑞4

𝑁

∑

𝑘=1

Car [𝑘] + 𝑞5

𝑁

∑

𝑘=1

Gas [𝑘] ,

(2)

where 𝑞1, 𝑞2, 𝑞3, 𝑞4, and 𝑞

5are the per unit cost of electric

energy, electrode consumption, and the chemical energyof oxygen, gas, and carbon, respectively. The unit cost 𝑞

𝑖’s

fluctuate according to themarket value and are adjustedwhenthe market price changes. The goal of control is to design aninput sequence 𝐼avg[𝑘], O2[𝑘], Gas[𝑘], Car[𝑘], 𝑘 = 1, 2, . . . , 𝑁

so that the above cost function (2) is minimized under theconstraint

PM (𝑁 + 1) = 100. (3)

Journal of Energy 3

In addition, because of hardware constraints, the input energymust satisfy for each 𝑘 = 1, 2, . . . , 𝑁

0 < 𝐼𝑙≤ 𝐼avg [𝑘] ≤ 𝐼

𝑢,

0 ≤ O2 [𝑘] ≤ 𝑂

𝑢,

0 ≤ Gas [𝑘] ≤ 𝐺𝑢,

0 ≤ Car [𝑘] ≤ 𝐶𝑢,

(4)

where 𝐼𝑙, 𝐼𝑢,𝑂𝑢,𝐺𝑢, and𝐶

𝑢denote the physical bounds on the

amount of electrical and chemical energy, respectively, thatcan be delivered to the EAF during each sampling interval 𝛿𝑡.

3. Optimal Control Calculation

To simplify notation, in the rest of the paper, we let

𝑥 [𝑘] = (

𝑥1 [𝑘]

𝑥2 [𝑘]

𝑥3 [𝑘]

) = (

KWH [𝑘]

𝐼2ℎ [𝑘]

PM [𝑘]

) ,

𝑢 [𝑘] = (

𝑢1 [𝑘]

𝑢2 [𝑘]

𝑢3 [𝑘]

𝑢4 [𝑘]

) = (

𝐼ave [𝑘]O2 [𝑘]

Gas [𝑘]Car [𝑘]

) .

(5)

Further, by some simple calculations, we notice that

KWH [𝑁 + 1] = 𝜑1

𝑁

∑

𝑘=1

𝐼avg [𝑘] ,

𝐼2ℎ [𝑁 + 1] = 𝜙

1

𝑁

∑

𝑘=1

𝐼avg [𝑘] ,

PM [𝑁 + 1] = 𝑎1

𝑁

∑

𝑘=1

𝐼avg [𝑘] + 𝑎2

𝑁

∑

𝑘=1

O2 [𝑘]

+ 𝑎3

𝑁

∑

𝑘=1

Gas [𝑘] + 𝑎4

𝑁

∑

𝑘=1

Car [𝑘] .

(6)

Therefore, the cost function of (2) can be rewritten as

𝐽 = 𝑝1

𝑁

∑

𝑘=1

𝐼avg [𝑘] + 𝑝2

𝑁

∑

𝑘=1

O2 [𝑘]

+ 𝑝3

𝑁

∑

𝑘=1

Car [𝑘] + 𝑝4

𝑁

∑

𝑘=1

Gas [𝑘] ,

(7)

where 𝑝1= 𝑞1𝜑1+ 𝑞2𝜙1, 𝑝2= 𝑞3, 𝑝3= 𝑞4, and 𝑝

4= 𝑞5.

To recap, the goal is to design an input sequence 𝑢𝑖[𝑘], 𝑖 =

1, 2, 3, 4, 𝑘 = 1, 2, . . . , 𝑁 for a given𝑁 so that the cost function

𝐽 = 𝑝1

𝑁

∑

𝑘=1

𝑢1 [𝑘] + 𝑝

2

𝑁

∑

𝑘=1

𝑢2 [𝑘] + 𝑝

3

𝑁

∑

𝑘=1

𝑢3 [𝑘]

+ 𝑝4

𝑁

∑

𝑘=1

𝑢4 [𝑘]

(8)

is minimized under the equality and inequality constraints,

𝑥 [𝑘 + 1] = (

1 0 0

0 1 0

0 0 1

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝐴

𝑥 [𝑘]

+ (

𝜑1

0 0 0

𝜙1

0 0 0

𝜃1𝜃2𝜃3𝜃4

)

⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

𝐵

𝑢 [𝑘] ,

𝑥 [0] = 0, 𝑘 = 1, 2, . . . , 𝑁,

0 < 𝐼𝑙≤ 𝑢1 [𝑘] ≤ 𝐼

𝑢,

0 ≤ 𝑢2 [𝑘] ≤ 𝑂

𝑢,

0 ≤ 𝑢3 [𝑘] ≤ 𝐺

𝑢,

0 ≤ 𝑢4 [𝑘] ≤ 𝐶

𝑢,

100 = 𝑎1

𝑁

∑

𝑘=1

𝑢1 [𝑘] + 𝑎

2

𝑁

∑

𝑘=1

𝑢2 [𝑘]

+ 𝑎3

𝑁

∑

𝑘=1

𝑢3 [𝑘] + 𝑎

4

𝑁

∑

𝑘=1

𝑢4 [𝑘] .

(9)

Here, KWH, 𝐼2ℎ, and PM describe the 3-dimensional statespace and 𝐼avg, O2, Car, and Gas describe the 4-dimensionalinput space. Matrices 𝐴 and 𝐵 are given as before.

3.1. Solution Using Linear Programming. The optimal controldesign problem described above is exactly a linear pro-gramming problem with a linear objective function andlinear equality and inequality constraints. This can be solvedby commercially available software packages. For real timeapplications, one issue is its computational complexity. Inreality, 𝑁 is approximately 1200 which implies that there are3⋅1200 = 3600 state variables, 4⋅1200 = 4800 input variables,2 ⋅ 4 ⋅ 1200 = 9600 inequality constraints, and 3 ⋅ 1200 = 3600

equality constraints. Simply put, it is a linear programmingproblem with 8400 variables and 14400 constraints. Thelinear programming has to be recalculated every time whenthe model experiences some changes. The computationalpower required for a real time application is not favorable.The second problem is that the linear programming providesan open loop control. Yes, the solution 𝑢∗[𝑘], 𝑘 = 1, 2, . . . , 𝑁

of the linear programming is optimal if everything is ideal. Inreality, model uncertainty and measurement noise are alwayspresent and the linear programming solution is open loop.It is impossible to guarantee a good performance of an openloop control system in the presence of uncertainty and noise.What we prefer is a solution that minimizes the cost function(8) under the constraints (9) but in a closed loop form. Tothis end, note that the linear quadratic regulation (LQR)control design has been extensively studied in the literature.The solution of LQR is in a feedback form which is morerobust than the open loop solution provided by the linearprogramming.Moreover, the LQRcontrol hasmany desirable

4 Journal of Energy

properties. We will show in the next section that by properchoices of the weights in the LQR calculation, the solutionof the LQR also solves the linear programming problem orequivalently the cost function (8) under constraints (9) butin a closed loop form.

Though the linear programming solution may not bepractically useful because of its open loop nature, it doesreveal some important insights. Let

𝑈𝑖=

𝑁

∑

𝑘=1

𝑢𝑖 [𝑘] , 𝑖 = 1, 2, 3, 4. (10)

Then, the cost function and the terminal constraint can berewritten, respectively, as

(𝐽 =) 𝐽LP = 𝑝1𝑈1+ 𝑝2𝑈2+ 𝑝3𝑈3+ 𝑝4𝑈4,

100 = 𝑎1𝑈1+ 𝑎2𝑈2+ 𝑎3𝑈3+ 𝑎4𝑈4.

(11)

Also from the state space equation, we have 𝑥1[𝑁+1] = 𝜑

1𝑈1

and 𝑥2[𝑁 + 1] = 𝜙𝑈

1. Now suppose 𝑢∗

𝑖[𝑘]’s, 𝑘 = 1, 2, . . . , 𝑁,

𝑖 = 1, 2, 3, 4 are the solution of the linear programming.Define

𝑢𝑖 [𝑘] =

1

𝑁

𝑁

∑

𝑚=1

𝑢∗

𝑖[𝑚] =

1

𝑁𝑈𝑖,

𝑖 = 1, 2, 3, 4, 𝑘 = 1, 2, . . . , 𝑁.

(12)

Clearly, ∑𝑁𝑘=1

𝑢𝑖[𝑘] = ∑

𝑁

𝑘=1𝑢∗

𝑖[𝑘] = 𝑈

𝑖and min

𝑚𝑢∗

𝑖[𝑚] ≤

𝑢𝑖[𝑘] ≤ max

𝑚𝑢∗

𝑖[𝑚] for each 𝑖 = 1, 2, 3, 4 and every

𝑘 = 1, 2, . . . , 𝑁. Therefore, the solutions 𝑢𝑖[𝑘]’s of (12) are

constants that minimize the cost function (8) and satisfyall the equality and inequality constraints (9). In short, thesolution of the linear programming may not be unique, butthere always exists a constant solution. We summarize thisobservation in a theorem form that will be used later.

Theorem 1. Consider the optimal control problem of (8) underthe constraints (9). Then, there always exist constant sequences𝑢𝑖[𝑘] ≡ 𝑢

𝑖[𝑚]’s, 𝑖 = 1, 2, 3, 4, and 1 ≤ 𝑘, 𝑚 ≤ 𝑁 that solve the

problem.

3.2. LQR Design. Linear quadratic regulator problem isextensively studied in the literature and its solution can bereadily found in a number of textbooks [7, 8]. In addition, itsproperties in terms of stability, robustness, and so forth, arewell documented and in fact are favorable [7, 8]. The differ-ence is however that LQRminimizes a quadratic performancefunction

𝐽LQR = 𝑥[𝑁 + 1]𝑇𝑆𝑥 [𝑁 + 1]

+

𝑁

∑

𝑘=1

(𝑥[𝑘]𝑇𝑄𝑥 [𝑘] + 𝑢[𝑘]

𝑇𝑅𝑢 [𝑘]) ,

(13)

where 𝑆 = 𝑆𝑇≥ 0, 𝑄 = 𝑄

𝑇≥ 0, and 𝑅 = 𝑅

𝑇> 0 are the

terminal state, state and input cost matrices, respectively. Ourgoal in this section is to show that by some properly chosen

matrixes 𝑆, 𝑄, and 𝑅, the LQR solution also solves the linearprogrammingproblemwhich is the goal of the control design.It is emphasized that the LQR is not onlymore robust but alsonumerically efficient. Note that the LQR is recursive and ateach time 𝑘𝑡, only a fixed dimension Riccati equation (in ourcase, 4 dimensional) needs to be calculated [7, 8].

To this end, we set

𝑆 = 0, 𝑄 = 0, 𝑅 = (

𝑟1

0 0 0

0 𝑟2

0 0

0 0 𝑟3

0

0 0 0 𝑟4

) (14)

for some 𝑟𝑖> 0 or equivalently

𝐽LQR (𝑢 (𝑟1, 𝑟2, 𝑟3, 𝑟4)) =𝑁

∑

𝑘=1

𝑢[𝑘]𝑇𝑅𝑢 [𝑘]

= 𝑟1

𝑁

∑

𝑘=1

𝑢2

1[𝑘] + 𝑟

2

𝑁

∑

𝑘=1

𝑢2

2[𝑘]

+ 𝑟3

𝑁

∑

𝑘=1

𝑢2

3[𝑘] + 𝑟

4

𝑁

∑

𝑘=1

𝑢2

4[𝑘] ,

(15)

where 𝐽LQR(𝑢(𝑟1, 𝑟2, 𝑟3, 𝑟4)) indicates that the cost depends onthe optimal input sequence 𝑢[𝑘] which in turn depends onthe weights 𝑟

𝑖’s. Together with the terminal constraint in (9),

the LQR solution assumes the form of [7, 8]

𝑢 [𝑘] = 𝐾 [𝑘] 𝑥 [𝑘] + 𝐻 [𝑘] 𝑥3 [𝑁 + 1] , (16)

where 𝐾[⋅] and 𝐻[⋅] are functions of 𝑟1, 𝑟2, 𝑟3, 𝑟4. Note that

the control sequence is in a feedback form. Interested readersmay find more details in many textbooks, for example, [7, 8].It is easy to verify the following theorem.

Theorem 2. Consider the above LQR problem of (15) underthe constraints (9). Then, the input sequences 𝑢∗

𝑖[𝑘]’s, 𝑖 =

1, 2, 3, 4, 𝑘 = 1, 2, . . . , 𝑁 that solve the LQR problem must beconstant sequences 𝑢∗

𝑖[𝑘] ≡ 𝑢

∗

𝑖[𝑚] for all 𝑖 = 1, 2, 3, 4 and

1 ≤ 𝑘,𝑚 ≤ 𝑁.

Proof. The proof is by contradiction. Suppose not, thereexists at least one 𝑘 so that 𝑢∗

𝑖[𝑘] = 𝑢

∗

𝑖[𝑘 + 1] for some 𝑖.

Let 𝑢𝑖[𝑚]’s, 𝑖 = 1, 2, 3, 4, and 𝑚 = 1, 2, . . . , 𝑁 be the exact

sequences of 𝑢∗𝑖[𝑚] except at 𝑘 and 𝑘 + 1,

𝑢𝑖 [𝑘] = 𝑢

𝑖 [𝑘 + 1] =1

2(𝑢∗

𝑖[𝑘] + 𝑢

∗

𝑖[𝑘 + 1]) . (17)

Journal of Energy 5

Then,

𝑟𝑖(𝑢2

𝑖[𝑘] + 𝑢

2

𝑖[𝑘 + 1])

= 𝑟𝑖((

(𝑢∗

𝑖[𝑘] + 𝑢

∗

𝑖[𝑘 + 1])

2)

2

+((𝑢∗

𝑖[𝑘] + 𝑢

∗

𝑖[𝑘 + 1])

2)

2

)

= 𝑟𝑖

(𝑢∗

𝑖[𝑘] + 𝑢

∗

𝑖[𝑘 + 1])

2

2

< 𝑟𝑖(𝑢∗2

𝑖[𝑘] + 𝑢

∗2

𝑖[𝑘 + 1]) .

(18)

In other words, all the constraints of (9) are satisfied bythe new sequences 𝑢

𝑖[𝑘], but the cost function generated by

𝑢𝑖[𝑘] is strictly smaller than the cost function generated by

𝑢∗

𝑖[𝑘]. Therefore, the optimal sequences have to be constant

sequences. This finishes the proof.

Now let the solution sequence of the LQR problem (15)under the constraints (9) be denoted by

𝑢LQR1

(𝑟𝑖) , 𝑢

LQR2

(𝑟𝑖) , 𝑢

LQR3

(𝑟𝑖) , 𝑢

LQR4

(𝑟𝑖) ,

𝑖 = 1, 2, 3, 4, 𝑘 = 1, 2, . . . , 𝑁.

(19)

Further, let

(𝑟∗

1, 𝑟∗

2, 𝑟∗

3, 𝑟∗

4) = argmin

𝑟𝑖

𝐽LQR (𝑢 (𝑟1, 𝑟2, 𝑟3, 𝑟4)) . (20)

Now we claim the following.

Theorem 3. The solution of the LQR problem𝑢LQR𝑖

(𝑟∗

1, 𝑟∗

2, 𝑟∗

3, 𝑟∗

4)’s, 𝑖 = 1, 2, 3, 4, defined above also

solves the linear programming problem of (8) and (9).

Proof. Since both the linear programming problem and theLQR problem assume constant solutions, the goal is to findconstant 𝑢

𝑖’s for all 𝑘 that solve, respectively, the linear

programming problem and the LQR problem as shown inTable 1.

For the linear programming problem, at least one of theoptimal solution lies at one of the corners formed by theintersections of the hyperplanes of the cost function, theterminal constraint, and the inequality constraints as shownin Figure 2 for a two-dimension case. On the other hand,the solution of the LQR problem is at the intersection ofthe same hyperplanes formed by the terminal and inequalityconstraints and an ellipsoid

1 =

(𝑢LQR1

(𝑟𝑖))2

(√𝐽LQR/𝑟1𝑁)

2+

(𝑢LQR2

(𝑟𝑖))2


2

+

(𝑢LQR3

(𝑟𝑖))2


2+

(𝑢LQR4

(𝑟𝑖))2


2

(21)

Table 1

LP LQR

s.t.

min𝑢𝑖

𝐽LP(𝑢𝑖) = 𝑁

4

∑

𝑖=1

𝑝𝑖𝑢𝑖

min𝑢𝑖

𝐽LQR(𝑢𝑖(𝑟1, 𝑟2, 𝑟3, 𝑟4)) =

𝑁

4

∑

𝑖=1

𝑟𝑖𝑢2

𝑖(𝑟1, 𝑟2, 𝑟3, 𝑟4)

𝑥1[𝑁 + 1] = 𝑁𝜑

1𝑢1

𝑥1[𝑁 + 1] = 𝑁𝜑

1𝑢1(𝑟𝑖)

𝑥2[𝑁 + 1] = 𝑁𝜙

1𝑢1

𝑥2[𝑁 + 1] = 𝑁𝜙

1𝑢1(𝑟𝑖)

𝑥3[𝑁 + 1] = 𝑁∑

𝑖=1

𝑎𝑖𝑢𝑖

𝑥3[𝑁 + 1] = 𝑁∑

𝑖=1

𝑎𝑖𝑢𝑖(𝑟1, 𝑟2, 𝑟3, 𝑟4)

0 < 𝐼𝑙≤ 𝑢1≤ 𝐼𝑢

0 < 𝐼𝑙≤ 𝑢1(𝑟𝑖) ≤ 𝐼𝑢

0 ≤ 𝑢2≤ 𝑂𝑢

0 ≤ 𝑢2(𝑟𝑖) ≤ 𝑂

𝑢

0 ≤ 𝑢3≤ 𝐺𝑢

0 ≤ 𝑢3(𝑟𝑖) ≤ 𝐺

𝑢

0 ≤ 𝑢4≤ 𝐶𝑢

0 ≤ 𝑢4(𝑟𝑖) ≤ 𝐶

𝑢

Terminal constraint

Inequality constraints

Cost function

X

Figure 2:𝑋 = solution of linear programming.

for a given set of 𝑟𝑖. By varying 𝑟

𝑖, the axes of the ellipsoid

can be adjusted arbitrarily and there always exist some 𝑟∗𝑖’s so

that the LQR solution is at one of the corners that is also asolution of the linear programming problem as illustrated inFigure 3 for a two-dimensional case. Denote such solutionsby 𝑢LQR1

(𝑟∗

𝑖), 𝑢LQR2

(𝑟∗

𝑖), 𝑢LQR3

(𝑟∗

𝑖), 𝑢LQR4

(𝑟∗

𝑖). This implies

𝐽LP (𝑢LQR1

(𝑟∗

𝑖) , 𝑢

LQR2

(𝑟∗

𝑖) , 𝑢

LQR3

(𝑟∗

𝑖) , 𝑢

LQR4

(𝑟∗

𝑖))

≤ 𝐽LP (𝑢1, 𝑢2, 𝑢3, 𝑢4)(22)

for any 𝑢𝑖or the linear programming cost function 𝐽LP

evaluated at the LQR solution 𝑢LQR1

(𝑟∗

𝑖), 𝑢LQR2

(𝑟∗

𝑖), 𝑢LQR3

(𝑟∗

𝑖),

𝑢LQR4

(𝑟∗

𝑖) is no larger than 𝐽LP(𝑢1, 𝑢2, 𝑢3, 𝑢4) for any 𝑢𝑖. On the

other hand,

min𝑢𝑖

𝐽LP (𝑢𝑖)

≤ 𝐽LP (𝑢LQR1

(𝑟∗

𝑖) , 𝑢

LQR2

(𝑟∗

𝑖) , 𝑢

LQR3

(𝑟∗

𝑖) , 𝑢

LQR4

(𝑟∗

𝑖)) .

(23)

Thus, the solution 𝑢LQR1

(𝑟∗

𝑖), 𝑢LQR2

(𝑟∗

𝑖), 𝑢LQR3

(𝑟∗

𝑖), 𝑢LQR4

(𝑟∗

𝑖) of

the LQR also solves the LP problem.This completes the proof.

6 Journal of Energy

Terminal constraint

Cost function

Inequality constraints

X

Figure 3:𝑋 = solution of LQR.

3.3. Computation and Validation of 𝑟∗𝑖’s. What the results of

Theorem 3 say is that there exist constants 𝑢𝑖’s that solve both

the LQR and LP problems. Furthermore, let 𝑢1(𝑟𝑖), 𝑢2(𝑟𝑖),

𝑢3(𝑟𝑖), 𝑢4(𝑟𝑖) be the solution of LQR for given 𝑟

1, 𝑟2, 𝑟3, and

𝑟4. Then,

(𝑟∗

1, 𝑟∗

2, 𝑟∗

3, 𝑟∗

4) = argmin

𝑟𝑖

𝐽LP (𝑢1 (𝑟𝑖) , 𝑢2 (𝑟𝑖) , 𝑢3 (𝑟𝑖) , 𝑢4 (𝑟𝑖))

(24)

and this provides a way to compute 𝑟∗𝑖.

For simplicity, let us leave out all the inequality constraintin (9) first. Consider the terminal constraint and the costfunction,

𝐽LQR = 𝑁(𝑟1𝑢2

1+ 𝑟2𝑢2

2+ 𝑟3𝑢2

3+ 𝑟4𝑢2

4) ,

100 = 𝑁 (𝑎1𝑢1+ 𝑎2𝑢2+ 𝑎3𝑢3+ 𝑎4𝑢4) .

(25)

To find the LQR solution, we introduce the Lagrange multi-plier

𝐹 = 𝑁(𝑟1𝑢2

1+ 𝑟2𝑢2

2+ 𝑟3𝑢2

3+ 𝑟4𝑢2

4)

+ 𝜆 (100 − 𝑁) (𝑎1𝑢1+ 𝑎2𝑢2+ 𝑎3𝑢3+ 𝑎4𝑢4) .

(26)

Setting the derivatives of 𝐹 with respect to 𝑢1, 𝑢2, 𝑢3, 𝑢4

and 𝜆 to zero implies that

𝑢1=𝜆𝑎1

2𝑟1

, 𝑢2=𝜆𝑎2

2𝑟2

,

𝑢3=𝜆𝑎3

2𝑟3

, 𝑢4=𝜆𝑎4

2𝑟4

,

𝜆 =200

𝑁[𝑎2

1

𝑟1

+𝑎2

2

𝑟2

+𝑎2

3

𝑟3

+𝑎2

4

𝑟4

]

−1

.

(27)

By solving these 5 equations, we have

𝑢1=100𝑎1

𝑟1𝑁

[𝑎2

1

𝑟1

+𝑎2

2

𝑟2

+𝑎2

3

𝑟3

+𝑎2

4

𝑟4

]

−1

,

𝑢2=100𝑎2

𝑟2𝑁

[𝑎2

1

𝑟1

+𝑎2

2

𝑟2

+𝑎2

3

𝑟3

+𝑎2

4

𝑟4

]

−1

,

𝑢3=100𝑎3

𝑟3𝑁

[𝑎2

1

𝑟1

+𝑎2

2

𝑟2

+𝑎2

3

𝑟3

+𝑎2

4

𝑟4

]

−1

,

𝑢4=100𝑎4

𝑟4𝑁

[𝑎2

1

𝑟1

+𝑎2

2

𝑟2

+𝑎2

3

𝑟3

+𝑎2

4

𝑟4

]

−1

.

(28)

Now putting back these values into 𝐽LP, we have

𝐽LP = 100𝑝1𝑎1𝑟2𝑟3𝑟4+ 𝑝2𝑎2𝑟1𝑟3𝑟4+ 𝑝3𝑎3𝑟1𝑟2𝑟4+ 𝑝4𝑎4𝑟1𝑟2𝑟3

𝑎2

1𝑟2𝑟3𝑟4+ 𝑎2

2𝑟1𝑟3𝑟4+ 𝑎2

3𝑟1𝑟2𝑟4+ 𝑎2

4𝑟1𝑟2𝑟3

.

(29)

We make two observations.

(i) If 𝑟1, 𝑟2, 𝑟3, 𝑟4

≥ 0, then the two sets of weights(𝑟1, 𝑟2, 𝑟3, 𝑟4) and 𝛼(𝑟

1, 𝑟2, 𝑟3, 𝑟4) would produce the

identical LQR optimal input sequences for any 𝛼 > 0.(ii) Practically, the usage of electric energy is always

positive which implies 𝑟∗1> 0.

Based on these two observations, we may set 𝑟1= 1 and

𝑟1= 1, 𝑟

2= 𝑟2𝑟1= 𝑟2,

𝑟3= 𝑟3𝑟1= 𝑟3, 𝑟

4= 𝑟4𝑟1= 𝑟4

(30)

which in turn gives rise to

𝐽LP = 100𝑝1𝑎1𝑟2𝑟3𝑟4+ 𝑝2𝑎2𝑟3𝑟4+ 𝑝3𝑎3𝑟2𝑟4+ 𝑝4𝑎4𝑟2𝑟3

𝑎2

1𝑟2𝑟3𝑟4+ 𝑎2

2𝑟3𝑟4+ 𝑎2

3𝑟2𝑟4+ 𝑎2

4𝑟2𝑟3

.

(31)

Suppose 𝑟∗𝑖’s minimize the above 𝐽LP. Then, the optimal

𝑢𝑖’s are those in (28) by replacing 𝑟

𝑖’s by 𝑟∗

𝑖’s. These are the

solutions without the inequality constraints (9). With (9), thebest 𝑟

𝑖’s are those that minimize 𝐽LP under the constraints

that the resultant 𝑢𝑖’s are in the box defined by the inequality

constraints (9).To support the above theoretical derivation, a large

number of actual production data sets (over 2 thousands) arecollected. Based on 30 randomly chosen data sets, we firstsolve the linear programming problem (2) and (4) directlyusing the command “linprog” embedded in MATLAB. Wethen solve the same problem by finding the optimal 𝑟∗

𝑖’s and

the corresponding optimal input sequences 𝑢∗𝑖[𝑘]’s through

the LQR optimization as discussed before. In each case, thetwo optimization results are not identical but very closeabout 1-2% difference in the cost functions. We believe thatthis small numerical difference of 1-2% is due to numericalimperfection of MATLAB for a large dimensional linearprogramming (8400 variables and 14000 constraints).

Journal of Energy 7

4. Parameter Estimation

In the previous sections, the entries of the matrix 𝐵 areassumed to be available. In reality, 𝐵 depends on the meltingprocess and in particular the loadweight of scraps. In general,the entries of 𝐵 have to be estimated. To put parameterestimation into a perspective, we first describe the meltingprocess. The melting process is a batch process. One batchof steel is called a heat. The time between taps is the amountof time required to produce a batch of liquid steel. Producedtons is measured using charged scrap tons. First, scrap isbrought into the melt shop and an overhead crane is usedto load the scrap into a charging bucket. When the chargebucket has been loaded, the loading crane picks up the chargebucket and places the scrap into the EAF.The EAF is typicallyoperated utilizing a 3-charge practice and a melting cycle iscomposed of the following stages.

(1) Furnace Turn Around—the EAF is inspected andprepared for the next heat.

(2) Charge—the first scrap is put into the EAF.

(3) Melting—power is turned on and the scrap is melteduntil the volume is sufficiently reduced to fit thesecond charge into the EAF.

(4) Charge and melting cycle is repeated for the secondand third scrap charges.

(5) Refine—alloys are added to the liquid steel as neces-sary and steel is 100% melted.

(6) Tap—the liquid steel is transferred from the EAF to aladle for casting.

Obviously depending on the weight of scrap in threecharges, the entry values of 𝐵 vary. To this end, we provide anidentifier, l = low, m = medium, and h = high to each chargeto reflect the amount of scrap loaded in each charge.

Therefore, there are totally 9 different scrap loadingscenarios as shown in Table 2. For instance, if three chargesfor a particular heat are 32 tons (1st charge), 32 tons (2ndcharge), and 18 tons (3rd charge), respectively, then it isidentified as (1l,2l,3l). If three charges are 32 tons, 34 tons, and23 tons, it is identified as (1l,2m,3h). For each scenario, theentries of 𝐵 are estimated separately. For actual estimation,let

Δ𝑥 [𝑘] = 𝑥 [𝑘 + 1] − 𝑥 [𝑘] . (32)

Then, we have for 𝑘 = 1, 2, . . . , 𝑁,

Δ𝑥1 [𝑘] = 𝑢

1 [𝑘] 𝜑𝑖, Δ𝑥2 [𝑘] = 𝑢

1 [𝑘] 𝜙1,

Δ𝑥3 [𝑘] = (𝑢

1 [𝑘] , 𝑢2 [𝑘] , 𝑢3 [𝑘] , 𝑢4 [𝑘])(

𝑎1

𝑎2

𝑎3

𝑎4

).

(33)

Table 2: Algorithm for choosing charge identifiers based on chargeweights.

Charge number CW (charge weight, in tons)l (low) m (medium) h (high)

1 CW < 33 33 ≤ CW ≤ 35 CW > 35

2 CW < 33 33 ≤ CW ≤ 35 CW > 35

3 CW < 19 19 ≤ CW ≤ 22 CW > 22

The estimates 𝜑1, 𝜙1, and 𝑎

𝑖, 𝑖 = 1, 2, 3, 4, of 𝜑

1, 𝜙1, 𝑎𝑖defined

as solutions of the least squares fits

𝜑1= argmin

𝜑

𝑁

∑

𝑘=1

(Δ𝑥1 [𝑘] − 𝑢

1 [𝑘] 𝜑)2

,

𝜙1= argmin

𝜙

𝑁

∑

𝑘=1

(Δ𝑥2 [𝑘] − 𝑢

1 [𝑘] 𝜙)2

,

(𝑎1, 𝑎2, 𝑎3, 𝑎4)

= argmin𝑎𝑖

𝑁

∑

𝑘=1

(Δ𝑥3 [𝑘] − (𝑢

1 [𝑘] , 𝑢2 [𝑘] , 𝑢3 [𝑘] , 𝑢4 [𝑘])

× (

𝑎1

𝑎2

𝑎3

𝑎4

))

2

,

(34)

can be readily solved as

𝜑1=∑𝑁

𝑘=1Δ𝑥1 [𝑘] 𝑢1 [𝑘]

∑𝑁

𝑘=1𝑢1[𝑘]2

,

𝜙1=∑𝑁

𝑘=1Δ𝑥2 [𝑘] 𝑢1 [𝑘]

∑𝑁

𝑘=1𝑢1[𝑘]2

,

(

𝑎1

𝑎2

𝑎3

𝑎4

) = (

𝑁

∑

𝑘=1

𝑢 [𝑘] 𝑢𝑇[𝑘])

−1𝑁

∑

𝑘=1

𝑢 [𝑘] Δ𝑥3 [𝑘] .

(35)

To validate the parameter estimation results, we let

Δ𝑥1 [𝑘] = 𝑢

1 [𝑘] 𝜑1, Δ𝑥2 [𝑘] = 𝑢

1 [𝑘] 𝜙1,

Δ𝑥3 [𝑘] = 𝑢

𝑇[𝑘](

𝑎1

𝑎2

𝑎3

𝑎4

)

(36)

and define the Goodness-of Fit as

GoF𝑖= 1 − √

𝑁

∑

𝑘=1

(Δ𝑥𝑖 [𝑘] − Δ𝑥

𝑖 [𝑘])2

(Δ𝑥𝑖 [𝑘] − (1/𝑁)∑

𝑁

𝑚=1Δ𝑥𝑖 [𝑚])

2,

𝑖 = 1, 2, 3.

(37)

8 Journal of Energy

Table 3: GoFs.

GoF of Δ𝑥1

(KWH)GoF of Δ𝑥

2

(𝐼2ℎ)GoF of Δ𝑥

3

(PM)Max GoF 0.9992 0.9995 0.9948Min GoF 0.9984 0.9992 0.9930Avg GoF 0.9990 0.9994 0.9941

Table 4: Production cost reduction of 15 heat files.

Heat filenumber

Ratio of the costs byoptimal control andmanual control

Cost saving

1 0.86 14%2 0.9 10%3 0.9 10%4 0.9 10%5 0.9 10%6 0.86 14%7 0.93 7%8 0.84 16%9 0.87 13%10 0.91 9%11 0.78 22%12 0.9 10%13 0.81 19%14 0.82 18%15 0.84 16%

The parameters are derived from one heat file and validatedby 30 other fresh heat files, which were not used for estima-tion but with the same identifier. The validation results areshown in Table 3.

Clearly from Table 3, the results are satisfactory and thevalues of 𝜑

1, 𝜙1, and 𝑎

𝑖will be used in the optimal control

design to replace the unknown 𝜑1, 𝜙1, and 𝑎

𝑖’s.

5. Control Design Validation

As discussed before, the control design involves two steps:(1) estimate the entries of 𝐵 for different scrap load weightsand (2) based on the estimates calculate the optimal 𝑟

𝑖for

the LQR problem and the corresponding optimal inputs𝑢∗

𝑖(𝑟∗

1, 𝑟∗

2, 𝑟∗

3, 𝑟∗

4). Recall that our goal is to design optimal

input sequences so that the production cost (2) is minimizedas compared to the current manual operation in which expe-riences play a key role. In theory, it has been demonstratedthat the LQR solution 𝑢

∗

𝑖(𝑟∗

1, 𝑟∗

2, 𝑟∗

3, 𝑟∗

4), 𝑖 = 1, 2, 3, 4, solves

the LP problem and consequently minimizes the productioncost (2) under the constraints. Whether it works in realityneeds to be validated. To this end, we randomly choose 15actual heat files with known production costs. For each heatfile, a fresh heat file with the same identifier but not in theoriginal 15 is chosen and used to estimate the entries of 𝐵 tobuild the EAFmodel and calculate the optimal LQR solutions

𝑢∗

𝑖(𝑟∗

1, 𝑟∗

2, 𝑟∗

3, 𝑟∗

4), 𝑖 = 1, 2, 3, 4. The optimal inputs are applied

back to the original file to compute the production cost bythe optimal control design. Based on the original productioncost (2), the ratios of the costs by the optimal control designsderived by the LQR in which the weights 𝑟

𝑖’s are adjusted to

minimize the LP cost and by the current manual practiceare shown in Table 4 along with their percentage of costsaving. In all 15 experiments (files), the optimal inputs meetthe constraints (9) and reduce the production cost rangingfrom 7% to 22%.This supports our theoretical derivation anddemonstrates that the production cost in which the energyconsumption is a main part can be reduced substantially bychanging the practice from manual to automatic.

6. Conclusion and Future Work

The method proposed for the optimal control strategy ofan arc furnace provides a closed loop robust scheme thatoptimizes the production cost of the EAF steel making. Inthis study, we focus on 7 key variables and a linear modeling.In the future, other minor variables and possibly nonlineareffects could and should be considered. It is expected that bycombining all the factors, the production cost can be reducedfurther. Also, the strategy developed in this paper can beextended to other applications in which the cost function islinear and the control design used is quadratic.

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper.

Acknowledgment

This work was supported in part by the Iowa Energy CenterGrant 09-01.

References

[1] R. Balan, V. Maties, O. Hancu, S. Stan, and L. Ciprian, “Model-ing and control of an electric arc furnace,” in Proceedings of the15 Mediterranean Conference on Control and Automation, pp. 1–6, Athens, Greece, July 2007.

[2] B. Boulet, G. Lalli, and M. Ajersch, “Modeling and control ofan electric arc furnace,” in Proceedings of the American ControlConference, pp. 3060–3064, Denver, Colo, USA, June 2003.

[3] U. Camdali and M. Tunc, “Modelling of electric energy con-sumption in the AC electric arc furnace,” International Journalof Energy Research, vol. 26, no. 10, pp. 935–947, 2002.

[4] Anuradha, K. B. Muni, and A. Kumar, “Modeling of EFAand control algorithms for voltage flicker mitigation usingDSTATCOM,” in Proceedings of the 16th National Power SystemsConference, Hyderabad, India, December 2010.

[5] M. Han and X. Huang, “Greedy kernel components acting onANFIS to predict BOF steelmaking endpoint,” in Proceedings ofthe 17th World Congress, International Federation of AutomaticControl (IFAC ’08), Seoul, South Korea, July 2008.

Journal of Energy 9

[6] L.Hocine,D. Yacine, K.M. Samira, andB. Kamel, “Closely para-metrical model for an electrical arc furnace,”World Academy ofScience, Engineering and Technology, vol. 40, pp. 96–100, 2008.

[7] F. Lewis and V. Syrmos, Optimal Control, John Wiley & Sons,New York, NY, USA, 1995.

[8] K. Zhou, J. Doyle, and K. Glover, Robust and Optimal Control,Prentice Hall, Upper Saddle River, NJ, USA, 1995.

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