Developing a Cost Minimizing Power Load Prediction Model for

3952

Ind. Eng. Chem. Res. 2010, 49, 3952–3956

Developing a Cost Minimizing Power Load Prediction Model for Steelwork Industries Jun-hyung Ryu,* Dong Joon Yoo, and In-Beum Lee Department of Chemical Engineering, POSTECH, 790-784, Korea, and Department of Energy & EnVironmental Systems, Dongguk UniVersity, Gyeongju, 780-814, Korea

The key issue of power systems is to match power demand and supply with the minimum gap and delay. In the steelmaking industry that is well-known for intensive power demand, multiple types of power supply sources are thus prepared. One is to use gas turbine generators consuming byproduct gases, another is to purchase power from external electricity companies, and yet another is to use in-house self-generators to prevent any interruptions. Because we have to prepare the power supply before demands are actually realized, the fuels for generators should be purchased based upon a relatively long-term plan, and redundant power should be consumed by being resold or in other ways. It is economically important to predict power load accurately for the profitability of the steelworks. A load prediction model is therefore mathematically formulated as a linear programming (LP) problem with a view to minimizing the overall power cost. A case of an actual steelmaking company in Korea is addressed to illustrate the applicability of the proposed model with some remarks. Introduction The recent volatility of fossil fuel prices illustrates difficult but typical challenges for heavy energy consuming industries like steelworks. While methods are sought to overcome the challenge, the abrupt changes of external conditions often make the performance of the power supply below their expectation. Steel companies are thus eager to minimize the impact of changes by installing multiple types of supply sources: One is to use gas turbine generators consuming by-produced gases. Another is to purchase power from external electricity companies. The other is to use self-generators for the sake of stable supply without any interruptions. Since the fuels for generators should be prepared based upon a relatively long-term, mostly monthly, basis, the redundant power should be consumed: either resold or in other ways. The primary objective of the power-system operation in steelworks lies in the real-time matching between total power demand and supply. If the total supply consists of either power generation only or purchase only, the problem remains straightforward as to predict the demand, for example, a score of minutes ahead and set the supply tracing the prediction. However, the power supply in most of steelworks usually includes both generation and purchase, an optimization problem emerges in terms of deciding how much power to be generated inside and how much purchased from outside; see ref 1. It is practically important to predict the power load for the maximum power utilization. Although several methods such as fuzzy logic, neural networks, and autoregression are used to predict the load in general situations2,3 as long as a steelworks is concerned, the load can be predicted only from the state to come since the stochastic component of the load is ignorable compared to the deterministic one. This work aims to focus on constructing a future load prediction model from the states to come as an optimization problem. The corresponding problem is formulated as a linear programming problem that is implemented by many commercial solvers.4 * To whom correspondence should be addressed. Tel.: +82-54-7702859. Fax: +82-54-770-2834. E-mail address: [email protected].

The rest of this paper is as follows. The load prediction model is presented in section 2 and section 3 focuses on the economic operation with the numerical illustration of using the proposed model in section 4. Conclusions are summarized in section 5. 2. Load Prediction 2.1. Overview of Steelworks. Before looking into a power problem, it would be better to briefly overview steelworks. It consists of three main processes: iron-making, steelmaking, and rolling. In the iron-making process, comprised of raw material treatment, sintering, coke baking, blast furnace, etc., iron ore and coke are deposited into a blast furnace where 1200 °C of superheated air is blasted, burning the coke, and melting the iron ore. In the steelmaking process, the melted iron with steel scraps and limestone are poured into an oxygen converter, where pure oxygen is injected, oxidizing the impurities and refining the steel. The melted steel is immediately run through continuous casting machines, and is turned into semifinished products such as slab or bloom. Subsequently, in the hot rolling process, the slab or bloom is reheated and put through rolling mills to be one of end-products including hot rolled coil, steel plate, and wire rod of required dimensions and quality. Hot rolled coil may be further processed by a cold rolling mill into cold rolled sheet, electrical steel, or other value-added products. Before describing a model, the assumptions made for the model can be listed as follows: • The state j of power consuming unit i at time interval t within a prediction horizon is known a priori. • The state j of each gas producing and/or consuming unit i at each time interval t within the prediction horizon is assumed to be known a priori, as well as the net gas production rate, which is the gas production rate minus gas consumption rate. • The atmospheric temperature and the states of selfgenerators at individual time intervals within the horizon are known a priori. • The unit cost of power purchase at each time interval within the horizon is given.

10.1021/ie900227e  2010 American Chemical Society Published on Web 03/23/2010

Ind. Eng. Chem. Res., Vol. 49, No. 8, 2010

• The unit price at each time interval within the horizon is given for resale. • Capacities, depending on the atmospheric temperature and the states of self-generators at time intervals within the horizon, are known a priori. 2.2. Prediction Model. In steelworks, there are a large number of power consuming units, which can be classified into two groups. One consumes power steady and its power consumption rate or the load depends only on its state like coke baking or a blast furnace. There is the other whose load depends not only on the state but also on the intensity of the operation in progress. For example, the load of a hot or cold rolling mill grows as the thickness of the coil decreases. To predict the load precisely over the time, both factors should be considered. In this paper, we are going to consider the first case since we have found from the discussion with people in the field that the load of unit is assumed to depend only on its state as long as the prediction of the average load over 15 min or an hour is available. The states in steelworks can be summarized into the following four categories: • “Operating” state indicates that the unit operates normally and consumes the full power. • “Hot reserve” state indicates that the unit is temporarily idle or is under a short-term maintenance and keeps itself ready for a quick restoration. • “Cold reserve” state indicates that the unit is under a longterm maintenance and consumes the minimum power required. • “Shutdown” state indicates that the unit is turned off and disconnected from the power line. The load of unit i in each state j is approximated by the average of the past data. Then, the total load at time interval t, c(t), can be stated as c(t) )

∑ ∑

cij(t)sij(t)

∀t ∈ T

(1)

∀i∈IC ∀j∈J

where cij(t) denotes the load of unit i in state j at time interval t and sij(t) is the binary variable that is 1 when a unit i is in state j at interval t or 0 otherwise. For the prediction of the generation load of gas turbines, the balance for each byproduct gas such as BFG (blast furnace gas), COG (coke oven gas), and LDG (LD-converter gas) must be considered. Here, the entire redundant by-produced gas is converted into electricity within the time interval. The production and consumption rates of gases depend on the state of the unit. Usually, they are positive only when the unit is in the “operating” state. The state j of each gas producing and/or consuming unit i at each time interval t within the prediction horizon is assumed to be known a priori, as well as the net gas production rate, which is the gas production rate minus gas consumption rate of gas h from unit i in state j is estimated from the mean of historical values. Then, the total generation load of gas turbines at time interval t, g(t), is represented as follows: g(t) )

∑ ∑ ∑

eh(t)ghijsij(t)

∀t ∈ T

(2)

∀h∈H ∀i∈IG ∀j∈J

where eh(t) denotes the internal generation per mass of gas h at time interval t and ghij denotes the net production rate of gas h from unit i in state j. Finally, the net total load of the steelworks at time interval t, l(t), is as follows:

l(t) ) c(t) - g(t)

∀t ∈ T

3953

(3)

where l(t) denotes the net total load at time interval t. This load must be compensated by the internal generation and/or purchasing from external electric power companies. If the internal generation load exceeds the net total load, the redundant power may be resold to regional distributors. 3. Economic Operation 3.1. Object Function. The purpose of the operation optimization in the power system of steelworks is to minimize the overall power cost consisting of the following four main parts: 3.1.1. Internal Generation Cost. Internal generation cost corresponds to the cost of the fuel consumed by self-generators. Since the efficiency of a self-generator depends on the generation output, the atmospheric temperature, and the state of the generator, the fuel consumption rate varies from time to time. Here, it is assumed that the atmospheric temperature and the states of self-generators at each time interval within the horizon are known a priori so that a linear function which correlates the fuel consumption rate with the generation output is given as φt( · ). 3.1.2. Power Purchase Cost. Power purchase cost increases in proportion to the amount of the energy received from regional distributors. However, the unit cost varies depending on the overall supply and demand of the market; for example, in summer when the demand in the afternoon is at peak values for cooling, and so is the unit cost. On the other hand, in winter, the cost hits a climax in the evening for heating, and so does the unit cost. Considering the recent rapid increase of fossil fuel prices and abrupt decrease, the power purchase is a complex issue that cannot be simplified. 3.1.3. Power Resale Profit. Power resale profit comes from reselling the redundant energy to regional distributors. The unit price of the resold power also varies depending on the supply and demand of the market, but the peaks and valleys are the reverse of those of the unit cost of power purchase. The unit price at each time interval within the horizon is assumed to be given. 3.1.4. Maximum Demand Cost. The maximum demand cost denotes a monthly cost which increases in proportion to the maximum demand recorded during a month. Normally there is a fixed capacity limited in a contract that customers should pay an additional penalty cost over the capacity. To reduce this cost, it is important to prevent the demand from exceeding the contracted capacity by holding the power intensive tasks off or increasing the internal generation load. Consequently, the overall objective function of the total cost can be formulated as follows: cost )

G(t) ∑ {c (t)φ ( ∆(t) )∆(t) + c (t)P(t) - c (t)R(t)} + G

P

R

t

∀t∈T

cBPB + cAPA (4) where ∆(t) denotes the time interval, and in this paper, the ∆t is 1 h. The above objective function is subject to the following constraints: 3.2. Constraints. 3.2.1. Load Balance. The total power load of the steelworks must be compensated by the internal generation and/or purchasing from external distributors. On the other hand, the redundant load can be resold. Therefore the load balance can be summarized as follows:

3954

Ind. Eng. Chem. Res., Vol. 49, No. 8, 2010

l(t) e

G(t) + P(t) - R(t) ∆(t)

∀t ∈ T

(5)

3.2.2. Internal Generation. The internal generation load cannot exceed the capacity of self-generators. It is assumed that the capacity are known a priori and depend on the atmospheric temperature and the states of self-generators at each time interval within the horizon. G(t) e gmax(t) ∆(t)

∀t ∈ T

(6)

The fuel for the internal generation is assigned according to a long-term, such as monthly, plan that the monthly allocated amount must be exhausted between upper and lower bounds. f min e f used +

∑

∀t∈T

φt

G(t) ( ∆(t) )∆(t) e f

max

(7)

3.2.3. Maximum Demand. In an energy cost model, electricity price is one of the most important parameters. But the parameter is not fixed in practice. The price outsourced from the external electricity suppliers can be adjusted based upon the agreement with electricity suppliers in terms of total consumption amount and other conditions. The contract for the price of the next month is made based on the next month’s maximum demand, pmax. The electricity price is determined in terms of the maximum instantaneous electricity, prec. When prec is less than pmax, the predetermined price is charged. Otherwise a much higher price should be paid. In order to formulate this, prec is initially assumed as PA + PB if prec is less than pmax with being the maximum demand below capacity, prec is PB. Otherwise with maximum demand above capacity, prec is PB + PA where PA denotes the additional charge. The maximum demand must be greater than the power purchase demand at time interval t in the horizon as well as the already recorded maximum demand of the month. P(t) e PB + PA ∆(t)

∀t ∈ T

(8)

Figure 1. Predicted and the measured total load of actual steelworks for 36 h.

alternative, the following “two-step LP scheme” that provides reasonably acceptable suboptimal solutions in a quite fast way is proposed: Step 1. Compute the solution of the above (model 1) problem composed of (4)-(10) to obtain G(t), P(t), R(t). Step 2. Add the following constraints, (11) and (12), into (model 1) as a new LP problem and resolve the new LP problem to compute G(t), P(t), R(t). If P(t) < R(t) in step 1, fix P(t) at 0 for that time period and compute the solution of (model 1). Otherwise when P(t) is bigger than R(t), fix R(t) at 0 for that time period and compute the LP problem again. P(t) ) 0

∀t ∈ TR

(11)

R(t) ) 0

∀t ∈ TP

(12)

In the next, an actual industrial case is presented to illustrate the applicability of the proposed model. 4. Case Study

prec e PB + PA

(9)

Of course, the maximum demand below the capacity cannot exceed the contracted capacity. PB e pmax

(10)

Therefore the following optimization problem, model 1, can be constructed: min cost )

G(t) ∑ {c (t)φ ( ∆(t) )∆(t) + c (t)P(t) - c (t)R(t)} + G

P

R

t

∀t∈T

cBPB + cAPA

(model 1)

Subject to (4)-(10). 3.3. Solution Procedure. Since the power purchase and resale are simultaneously considered in (model 1), there can be a time interval where both purchase and resale occur at the same time, and it may cause an error in evaluating cost and profit. This situation could be avoided by introducing a binary variable for each time interval, indicating the occurrence of either net purchase or net resale. Then the original (model 1) may be reformulated into a mixed integer linear programming (MILP) problem. But this can increase the computation time exponentially as the number of the time intervals grows. As an

The proposed methodology of predicting the total load and exploring the economic operation is implemented in an actual steelmaking plant in Korea. The result according to the proposed methodology is graphically summarized in Figure 1. The prediction is executed every 5 min over a 90 min horizon, and the operational intensity is also considered to obtain the prediction. To show the prediction accuracy, the predicted total load represented by a bold line is compared to the measured counterpart of a thin line in Figure 1. As can be seen in Figure 1, the predicted load seems to follow the general trend of the actual load though discrepancies are observed from time to time. These discrepancies are mainly due to the disagreement between the scheduled and actual period of operation in electric arc furnaces. On the basis of the prediction of the total load, an operation profile is obtained and graphically described in Figure 2. The total load as well as the corresponding gas generation load is evaluated every hour considering the state of unit. In the company, a couple of liquified natural gas (LNG) generators are implemented for the internal generation where the LNG is supplied according to a monthly plan. The unit cost of LNG in the horizon is assumed to be $0.3/Nm3, which implies that the unit cost of the internal generation corresponds to about $47.3 per MW h for the output of 500 MW, respectively. Since

Ind. Eng. Chem. Res., Vol. 49, No. 8, 2010

Figure 2. Economic power generation, purchase, and resale in actual steelworks for 5 days. Table 1. Unit Cost of Power Purchasing and Unit Price of Power Resalea

a

hour

cP(t)

cR(t)

0-8 8-18 18-22 22-0

24.6 38.7 47.2 24.6

24.8 40.0 48.7 24.8

Unit: dollars per megawatt hour.

the unit cost of the internal generation can be less than the power resale price in the evening (i.e., from 18:00 to 22:00) as shown in Table 1, the output of the internal generation must be kept at its maximum, if possible, to make a profit. Because the monthly allocated amount of LNG should be exhausted, the remainder must be consumed in the rest of time. However, the overall cost of power can be saved even more if it is consumed in the daytime (i.e., from 8:00 to 18:00) rather than at night or in the early morning (i.e., from 22:00 to 8:00) because the unit cost of power purchasing is higher in the daytime. These force the internal generation load to have a chair-shaped area in Figure 2. In terms of computational statistics, the solutions of the above LP problems are obtained using LINDO API5 at the computing time of 3.4 s on an Intel Pentium 2.4 GHz using 2GB of RAM. It would be better to mention a number of issues regarding the proposed methodology and case study. At first, one can argue that a model-based real-time optimization scheme such as model predictive control6 can be employed to address the load prediction problem. And, we also thought of using some autoregressive regressive moving average (ARMA) models; that is, a model for understanding and predicting future values in terms of a time series. When we were initially faced with the problem, we had also thought of introducing such advanced and complex methodologies and actually tried some of them. We could not obtain the desired level of results, but only a simple and straightforward method like the present work turned out to have an actual impact in terms of providing continuously accurate results. This is mainly caused by the integration with other computation systems of the company. Second, the time interval for predicting the load and optimal operation is fixed to 1 h in this paper. The time horizon represents the time period from the current time to the end of this month. For example, if the current time is August 1st, 00: 00, there are 744 time intervals until the end of August 31st, 23:00, by multiplying 31 by 24. This is the main difference

3955

between the case study and the rolling steps procedure in the literature in a sense that the last time interval is fixed in the present paper. The main reason of not using a rolling horizon control scheme is associated with the practice of the company. Third, the fuel consumption rate and the power generation output are correlated as a linear function in the model. This can be better explained by mentioning the details of the company’s in-house power generation situation. The company employs an LNG integrated generator consisting of two gas turbines and one steam turbine as in-house power generation. At first, as LNG is combusted in a gas turbine, its gas expansion power is transformed into rotation power and electricity. The heat of the gas is used for a steam generator to make steam. The amount of LNG fuel for unit electricity power can be different depending on many factors such as (i) the state of an integrated generator like no failure, steam turbine failure, steam turbine and gas turbine failure, etc. and (ii) the external temperature because of its impact on Carnot efficiency. Therefore it is generalized and expressed as a linear function, φt( · ) in the model. Fourth, the original model is constructed as a mixed integer programming (MIP) problem and there are many excellent MIP solvers available these days. The solutions of the MIP problems can be computed using commercially available solvers like CPLEX. From the perspective of a company, this requires additional time and cost to integrate it with other existing control and management systems that are modified in response to the progress of the company. Therefore, we proposed a heuristicbased modification that can be utilized in a cheap way. It would be a good further research topic to integrate load prediction models and other systems without any delays or inconsistency. 5. Concluding Remarks The economic operation of the power system can be implemented by following common sense: buy it when it falls and sell it when it rises. That is to say, energy should be produced more and longer by using self-generators when the external price of power is high. It is also important to keep the maximum demand under the contracted capacity by maintaining the minimum internal generation load continuously in order not to pay a penalty for high rates. In order to balance between the power demand and the supply in terms of profitability, accurate power load prediction is more important than any other factors. In this paper, an optimization model to predict the total power load is presented to explore the economic operation of the power system in steelworks. Assuming that the load of each power consuming unit depends only on its state, a mathematical optimization problem is proposed to predict the load. There could be further research opportunities associated with the present work. First, the exact balance of byproduct gases might be constructed to predict the generation load of gas turbines more precisely. Moreover, considering the environment under the deregulation of the electric industry as well as the competitive power market, a model for forecasting the future cost of power should be developed.7 Second, the present methodology applied to the steelmaking industry can be expanded to other heavy energy-consuming industries such as chemical complexes. The present work could be further incorporated with energy systems of other industries. Acknowledgment This work is the outcome of a Manpower Development Program for Energy & Resources supported by the Ministry of Knowledge and Economy (MKE).

3956

Ind. Eng. Chem. Res., Vol. 49, No. 8, 2010

Nomenclature Sets H ) set of gases IC ) set of power consuming units IG ) set of gas producing/consuming units J ) set of states T ) set of time intervals TP ) set of time intervals where power purchase dominates power resale in solution of step 1 TR ) set of time intervals where power resale dominates power purchase in solution of step 1 Indices h ) gas i ) unit j ) state t ) time interval Parameters cij ) load of unit i in state j [MW] cA ) charge for maximum demand above capacity [$/MW] cB ) charge for maximum demand below capacity [$/MW] cG(t) ) fuel cost for internal generation at time interval t [$/ton] cP(t) ) power purchase cost at time interval t [$/MWh] cR(t) ) power resale price at time interval t [$/MWh] eh(t) ) internal generation per mass of gas h at time interval t [MW h/ton] f max ) upper bound on monthly fuel consumption [ton] f min ) lower bound on monthly fuel consumption [ton] f used ) fuel already consumed [ton] ghij ) net production rate of gas h from unit i in state j [ton/h] gmax(t) ) capacity of self-generators at time interval t [MW] pmax ) contracted capacity of power purchase [MW] prec ) recorded maximum demand [MW]

sij(t) ) 1 if unit i is in state j at time interval t; 0 otherwise ∆(t) ) length of time interval t [h] Function φt( · ) ) linear function correlating fuel consumption rate [ton/h] with output of internal generation [MW] at time interval t Variables c(t) ) total load at time interval t [MW] G(t) ) internal generation at time interval t [MW h] l(t) ) net total load at time interval t [MW] P(t) ) power purchase at time interval t [MWh] PA ) maximum demand above capacity [MW] PB ) maximum demand below capacity [MW] R(t) ) power resale at time interval t [MWh]

Literature Cited (1) Debs, A. S. Modern Power Systems Control and Operation; Kluwer Academic Publishers: Norwell, MA, 1988. (2) Lie, K.; Subbarayan, S.; Shoults, R. R.; Manry, M. T.; Kwan, C.; Lewis, F. L.; Naccarino, J. Comparison of Very Short-Term Load Forecasting Techniques. IEEE Trans. Power Syst. 1996, 11 (2), 877. (3) Trudnowski, D. J.; Johnson, J. M. Real-Time Very Short-Term Load Prediction for Power-System Automatic Generation Control. IEEE Trans. Contr. Sys. Tech. 2001, 9 (2), 254. (4) Wood, A. J. Power Generation, Operation, and Control; Wiley: New York, 1996. (5) LINDO API User’s Guide; LINDO Systems Inc.: Chicago, 2001. (6) Morari, M.; Lee, J. Model predictive control: past, present and future. Comput. Chem. Eng. 1999, 23 (4-5), 667. (7) Weedy, B. M.; Cory, B. J. Electric Power Systems; Wiley: Chichester, England, 1998.

ReceiVed for reView February 10, 2009 ReVised manuscript receiVed March 9, 2010 Accepted March 13, 2010 IE900227E