1
Generator Bidding Strategies in a Competitive Electricity Market with Derating and Bid-Segment Considerations Ning Lu, Joe H. Chow, and Alan A. Desrochers Electrical, Computer, and Systems Engineering Department Rensselaer Polytechnic Institute, Troy, NY 12180-3590, USA email addresses:
[email protected],
[email protected],
[email protected] Abstract—This paper develops optimal generator bidding strategies in a competitive electricity market. Starting from a generator’s cost curve, basic bidding concepts such as the break-even bid curve and the maximum profit bid curve can be readily derived. The maximum profit bid curve can be extended to account for generator availability and derating. In addition, multiple-segment block energy bids can be optimized based on the maximum profit curve and the probabilistic distribution of market clearing prices. Index Terms—Electricity market deregulation, generator bidding strategies, maximum profit bid curve, generator availability, generator derating, multiple-segment block bids
I. Introduction In a deregulated electricity market, the day-ahead unit commitment is evaluated on a price-merit order based on the bid prices submitted by the participating generators. For each hour, a market clearing price (MCP) is obtained, which is equal to the incremental cost of supplying the next unit of power. Thus, the dispatch of and, hence, the profit received by a generator depend on its bidding strategies. Many generator bidding strategies have been proposed and analyzed [1]-[4]. The purpose of this paper is to develop optimal bidding strategies for individual generators in a competitive electricity market. By a competitive electricity market, we refer to the case where there is an ample electricity supply allowing the MCP to be insensitive to the bid price variation of a single supplier, which is referred to as perfect competition in [2]. This assumption is appropriate in many deregulated power pools for most of the time, except possibly for peak load days when the demand approaches the available energy supply. We build the optimal bidding strategies [5] starting from the familiar generator cost curve, which we assume to be continuous and differentiable. From the cost curve, we develop basic bidding concepts of the break-even bid curve and the maximum profit bid curve. The maximum profit bid curve can be further extended to account for generator derating. In this context, an estimate of the ratio of the day-ahead and real time electricity prices is required. Analysis is also performed for generator bid curves consisting of multiple segments, which are required in most deregulated power pools. As a result, the continuous maximum profit bid curve needs to be approximated by blocks. We show that the optimal blocks can be obtained from the probabilistic distribution of energy prices based on the load forecast.
In an attempt to keep the derivation simple, we only address the case of a single hour dispatch. For multiple-hour dispatches, the generator ramp rates have to be taken into account. We also have not taken congestion pricing explicitly into account, although the electricity price forecast will implicitly include the effect of congestion. In addition, we have not included secondary effects such as losses into consideration. The paper is organized as follows. Section II develops basic bidding strategies from a generator cost curve. Section III describes a bid curve taking into account generator derating as a hedging mechanism against real time operation uncertainty. Section IV examines the optimization of multi-segment block bids based on the maximum profit strategy. II. Bidding Strategies Derived from Generator Cost Curves In a vertically integrated power system, generators are dispatched according to their cost curves. The cost curves vary according to the types of generators, such as fossil, hydro, nuclear, and gas turbines, and are generally well understood. A commonly used cost curve for a fossil steam unit is shown in Fig. 1 [6]. The cost C of running a steam unit consists of a start-up cost CS , a “min-gen” cost C0 associated with the cost of operating the unit at its minimum generation Pmin , and the variable cost of operating the generator at a power level P > Pmin , which is usually represented as a quadratic function C(P ) = CS + C0 + β1 (P − Pmin ) + β2 (P − Pmin )2
(1)
or a cubic function C(P ) = CS +C0 +β1 (P −Pmin )+β2 (P −Pmin )2 +β3 (P −Pmin )3 (2)
up to its maximum generation Pmax . The costs CS and C0 , and the coefficients β1 , β2 , and β3 are functions of the fuel cost. More specifically, β1 represents the fuel cost for generating beyond Pmin , and the quadratic and cubic terms model the decreasing efficiency of the generator and the increasing need of maintenance at higher power levels. In the sequel, we will use the quadratic cost curve (1) to illustrate the bidding strategies. The results can readily be extended for (2). In a regulated market, the cost curves (1) and (2) are used in unit commitment programs to determine the most economic dispatch. In a system with no congestion, the
2
C ($)
B(P) ($/MWH)
Maximum profit bid curve slope 2β2
slope β2 β1
C0+CS
0
Break-even bid curve
P
Pmin
0
Pmax P (MW)
Pmin
Pmax P (MW)
Fig. 1. A typical cost curve of a steam generator
Fig. 2. Bid curves of a steam generator
optimal solution occurs when the incremental cost is given by
B. Maximum Profit Bid Curve
λi =
dC(Pi ) dPi
(3)
where λi , the incremental cost for the ith unit, is the same for all units under dispatch. In a deregulated electricity market using the uniform pricing scheme [7], [8], the revenue for a generator is given by R(P ) = Rmin + B(P )(P − Pmin )
(4)
where Rmin is the revenue received for its startup and minimum generation cost, and B(P ) is the bid curve as a function of the generation level P . Therefore, the generator is profitable if R(P ) > C(P )
A. Break-even Bid Curve In the case where the generator is interested to only recover its cost, we have (6)
Using the quadratic cost curve (1) and the revenue equation (4), (6) becomes 2
Rmin + B(P )(P − Pmin ) = CS + C0 + β1 (P − Pmin ) + β2 (P − Pmin ) (7)
In this paper, we will always assume that Rmin = CS + C0 . Denoting Pc = P − Pmin , (7) simplifies to the break-even strategy BBE (P ) = β1 + β2 Pc
BMP (P ) =
dC(P ) = β1 + 2β2 Pc dP
(9)
This curve is also shown in Fig. 2. Note that the slope of this curve is twice that of the break-even bid curve. Assuming Rmin = CS + C0 , the maximum profit as a function of the generation level P is
(5)
Although a generator can submit any bid curve B(P ), in a competitive market, its objective is to ensure that not only (5) holds, but also its profit is maximized. Such bidding strategies should start from the cost curve (1). We will first examine two extreme cases: one to only recover the cost and the other to maximize profit.
R(P ) = C(P )
In practice, a generator has to not only recover cost, but also make a profit. To maximize profit given a fixed MCP, a generator needs to reach a generation level such that its incremental cost of generation would be equal to the MCP. Analytically, this maximum profit (MP) bid curve can be expressed as
(8)
which has a linear slope (Fig. 2), such that the profit πBE (P ) = 0. For a cubic cost curve (2), this analysis will result in a bid curve with a quadratic slope. The strategy (8) represents a competitive but unprofitable bidding strategy, whose parameters are determined from the traditional generator cost function. Furthermore, bidding below BBE (P ) would result in losing money.
πMP (P ) = BMP (P )Pc − (β1 Pc + β2 Pc2 ) = β2 Pc2
(10)
Thus, when all generators bid into the market based on their maximum profit curves, the unit commitment dispatch would be identical to the dispatch based on the generator cost curves. However, instead of the generators recovering their individual cost plus a fixed regulated profit, under the uniform MCP scheme, the maximum profit bid curves would potentially produce a larger return to the generators. We note that the maximum profit bid strategy has been derived in [9]. As our derivation is based on the perfect competition assumption, the solution of a multi-hour unit commitment program is not needed. C. High and Low Bid Curves Once the maximum profit curve has been established, any bid curve that deviates from it is non-optimal unless the generator has some degree of market power [10]. Two cases are possible: the “bid high” and “bid low” scenarios as shown in Fig. 3. For simplicity, linear bid curves are used. Assume that the unit is one of the units on the margin and submits the maximum profit bid curve BMP (P ), resulting in a market clearing price (MCP) of BMPo (Fig. 3). A bid-high curve BH (P ) results in less power allocated to the unit with a potential increase in the MCP. A bid-low curve BL (P ) results in more power allocated to the unit with a potential decrease in the MCP. If the unit receives a
3
3
B ($/MWH) equal profit curve
"bid high" bid curve Maximum profit bid curve
(BH* ,P*H)
"bid low" bid curve
700
2 ∆ BH($/MWH)
BMPo
Pc(MW) 800
β 2 = 0.025
2.5
600 500
*
1.5
β1
400 1
300 200
0.5
0
Pmin
Pmax P (MW)
P
100 0 0.05
Fig. 3. Effects of “bid-high” and “bid-low” bid curves
profit of πMP at a MCP of BMPo , we can obtain an equalprofit curve as shown in Fig. 3, from which we can observe the following: • The “bid-low” curve is intended to depress market prices. It drives some competitors from the market and reduces incentives for new generation investment. It is a poor short-term strategy because it reduces profitability. This strategy, however, can be used advantageously by units that must stay dispatched due to operational constraints. • The “bid-high” curve may be more profitable than the curve BMP (P ), depending on whether the increase of the MCP, ∆B = BH − BMPo , will more than compensate for the reduction in the dispatched energy. Pursuing the bid-high strategy further, let the bid-high curve be BH (PH ) = β1 + b2 PHc
(11)
with a slope of b2 > 2β2 and PHc = PH − Pmin . Thus, the profit at a dispatch of PH is 2 2 πH (PH ) = BH (PH )PHc − (β1 PHc + β2 PHc ) = (b2 − β2 )PHc
(12)
To receive the same profit (10) as the maximum profit bid, we set πH = πM P to obtain 2 (b2 − β2 )PHc = β2 Pc2
(13)
The solution to (13) is given by r ∗ PHc =
1 Pc b2 /β2 − 1
(14)
r
(15)
and so (11) becomes ∗ BH = β1 + b 2
1 Pc b2 /β2 − 1
Thus, the MCP margin required to achieve equal profit is r ∗ ∗ ∆BH = BH − BMPo =
b2
1 − 2β2 b2 /β2 − 1
Pc
(16)
For a generator using the bid-high curve, its profit will ∗ exceed the BMP strategy if the MCP is higher than BH , that is, a unit becoming more profitable when it exercises economic withholding. This analysis can serve as an alternative measure of market power.
0.055
0.06
0.065 b2
0.07
0.075
0.08
Fig. 4. The price margin for different slopes of the bid-high curve
As an illustration of this analysis, consider a unit with β2 = 0.025, Pmin = 100 MW, and Pmax = 800 MW. A plot ∗ of ∆BH for a range of b2 and P is shown in Fig. 4. For the specific values of b2 = 0.072 and P = 800 MW, the ∗ margin ∆BH needs to be greater than $2. In a competitive market, we would expect this value to be a fraction of $2, thus deeming this bid-high strategy to be non-optimal. III. Bid Curves Accounting for Generator Derating Although we have illustrated that the “bid-high” strategy is not profit maximizing, such a strategy may be optimal if some other factors are taken into account. Here we consider the impact of generator derating on the bidding strategy. Derating refers to a unit failing to deliver power at the committed level. An extreme case of derating is the loss of a unit. Consider a deregulated electricity market following the rules of the proposed standard market design [11] offering a 2-settlement system on the energy supply. Supply bids are submitted to the day ahead (DA) market (DAM) and the bids are evaluated using a bid-based unit commitment procedure. If a unit fails to generate in real time (RT), the amount of energy that it has committed in the DAM needs to be purchased by the unit. For example, a unit can only operate at 600 MW in real time while its committed power output at the DAM is 800 MW. This unit then needs to buy 200 MW in the RT market to compensate the discrepancy. Most likely, the RT energy supply tends to be more expensive than the DA prices, especially when the supply is tight on peak load days. Thus, a bidding strategy taking the probability of unit derating into account will provide some kind of insurance to the generator bidder for possible revenue losses. Let pa (P ) be the probability of the unit generating the amount of power P in RT after it has been committed in DAM and pu (i) the probability of the unit undergenerating an amount of PL (i) at a power output of P , where, pa (P ) +
n X
pu (i) = 1
(17)
i=1
Taking derating into account, the expected profit of a
4
unit given the DAM price Bd , the RT price BRT , and the committed power level P is πd (P ) = pa (P )(Bd Pc − (β1 Pc + β2 Pc2 )) +
Pn
i=1
[pu (i)(Bd Pc
−BRT PL (i) − (β1 (Pc − PL (i)) + β2 (Pc − PL (i))2 ))]
(18)
where Pc = P − Pmin and PL (i) is the derated power. The first term in (18) represents the expected profit from a full power delivery and the second term accounts for all the derating cases. Assume that the RT market price BRT is proportional to the DAM price Bd by a factor k(i) k(i) =
BRT (i) Bd (i)
(19)
PL (i) Pc
(20)
and denote kL (i) =
πd (P ) = pa (P )(Bd Pc − (β1 Pc + β2 Pc2 )) + [pu (i)(Bd Pc − k(i)Bd kL (i)Pc −
i=1
(β1 (Pc − kL (i)Pc ) + β2 (Pc − kL (i)Pc )2 ))]
Derating Power Output Level Power (MW) (MW) 100 200 300 400 500 600 100 0.005 0.01 0.01 0.01 0.01 0.02 200 0 0.005 0.01 0.01 0.01 0.01 300 0 0 0.005 0.001 0.001 0.005 400 0 0 0 0.005 0.001 0.005 500 0 0 0 0 0.01 0.005 600 0 0 0 0 0 0.02 700 0 0 0 0 0 0 800 0 0 0 0 0 0
700 0.03 0.03 0.02 0.01 0.005 0.005 0.03 0
800 0.05 0.05 0.01 0.01 0.01 0.01 0.01 0.05
TABLE II DAM/RT Price Ratio k at different power output level
as the percent of power undergenerated. Applying (19) and (20), (18) becomes n X
TABLE I Derating probabilities pu at different power output level
(21)
Derating Power (MW) 100 200 300 400 500 600 700 800
100 1 0 0 0 0 0 0 0
200 1.2 1.3 0 0 0 0 0 0
Power Output Level (MW) 300 400 500 600 1.3 1.3 1.3 1.5 1.4 1.4 1.4 1.6 1.5 1.5 1.5 1.6 0 2 1.6 2 0 0 2 2 0 0 0 2 0 0 0 0 0 0 0 0
700 2 2 2 2 2 3 3 0
800 3 3 3 3 3 3 3.5 3.5
To maximize the profit expressed in (21), we solve for dπd (P ) =0 dP
(22)
to obtain the maximum profit bid curve Bd =
B M P − pT u Bins 1 − pT u kins
(23)
where pTu is the vector representing the probabilities of derating events and
curve. A risk-averse bidder may tend to use the worst-case scenario which results in higher bids at high power outputs. If most of the market participants are risk averse, then applying insurance bids generally result in a higher MCP price in heavy load period than not taking derating into account. This, however, should not be considered as an exercise of market power but a necessary way to hedge the risk. 120
2 Bins = β1 kL (i) − 2β2 kL (i)Pc + 4β2 Pc kL (i)
(24)
kins (i) = kL (i)k(i)
(25)
B
Because
Bins 1 − pT u BM P Bd = BM P 1 − pT u kins
(26)
if Bins /BM P < kins , the Bd curve will be above the BM P curve. As an illustration, consider a unit with Pmin = 100MW and Pmax = 900MW, its cost coefficients are β1 = 5 and β2 = 0.025. The unavailabilities are represented by a vector pu shown in Table I, where the column represents the probability of the loss of a certain amount of power at the given output level Pc . For example, pu (3, 2) means that when Pc = 300MW, the probability of derating by 200 MW generation is 0.01. The k are shown in Table II. Fig. 5 illustrates the effect of different k factors and Fig. 6 illustrates the effect of different pu matrices. The uncertainties in determining the two matrices will cause the Bd curves to shift as shown in the figures. The knowledge of pu and k is therefore essential to determine the Bd
BMP ($/MWH)
100
1.2k 1.1k k
d
B
MP
80
60 B
BE
40
20
0 0
100
200
300
400 500 P (MW)
600
700
800
900
c
Fig. 5. Bd (P ) as a function of k
IV. Optimization of Block Bids The generator bid curves derived from the continuous cost curves will also be continuous functions of P . However, many unit commitment programs require bid curves to be in the form of either discrete points with straight-line interpolation or multi-segment blocks. Maximum profit
5
B
100
BMP ($/MWH)
1.2pu 1.1p u p
d
B3 = β1 + 2β2 (P1 + P2 + P3 ) = β1 + 2β2 (Pmax − Pmin )
u
Fig. 7 shows that the optimal strategy can be specified using only B1 and B2 . From (30) and (31), we obtain the energy blocks as
BMP
80
(32)
60
P1 =
B
BE
40
20
0 0
100
200
300
400 500 P (MW)
600
700
800
900
c
Fig. 6. Bd (P ) as a function of pu
bid curves that are affine can be handled without approximation by discrete points with linear interpolation. Multisegment block bids, however, require additional analysis including an assessment of the forecasted prices to ensure optimality. A block bid consists of blocks of energy at some fixed prices. For example, a 3-segment bid has, in addition to the min-gen bid, the form ( BB (P ) =
P1 P1 + P2 P1 + P2 + P3
if BMCP ≥ B1 if BMCP ≥ B2 if BMCP ≥ B3
(27)
where Pi is the amount in MW to be supplied when the MCP is at or above the bid price Bi (Fig. 7). Note that to be consistent P1 + P2 + P3 = Pmax − Pmin
B(P) ($/MWH) B3 B2
P1
P2
B 1 − β1 , 2β2
P1 + P2 =
B 2 − β1 2β2
(33)
The optimization of block bids with only a few segments can be challenging when the maximum profit bid curve has a steep slope. It has to rely on an estimate of the MCP, which is a function of demand (load) bids and supply (generation) bids. Furthermore, the demand bids may be priceresponsive, that is, they are also a function of MCP. In a deregulated market, a generator owner does not have access to the demand and supply bids submitted by the other market participants. Some aggregate information is available however. For example, the MCP versus the forecasted load is available immediately after unit commitment has been performed by the system dispatcher (known as the independent system operator in many electricity markets). In addition, historical MCPs versus actual loads are available [12]. Based on the MCP over a period of time, one can establish the statistics of the MCP for a specific forecasted load at a specific hour and construct the MCP probability pMCP (B),
for
Bmin ≤ B ≤ Bmax
(34)
where B is the MCP, and Z
(28)
Bmax
pMCP (B) dB = 1
(35)
Bmin
Such a probability may be a Gaussian or uniform distribution. The expected profit of a bidding strategy BB (P ) is obtained by integrating over the probability of the dispatch and the revenue minus the cost of each of the blocks:
P3
πB (B1 , B2 ) =
Z
B1 β1
B2
pMCP (B)[B B1
+
0 Pmin
Pmax
P (MW) +
Fig. 7. A 3-segment bid curve of a steam generator
Z
Z
B 1 − β1 B 1 − β1 B 1 − β1 2 − β1 − β2 ( ) ] dB 2β2 2β2 2β2
B3
pMCP (B)[B B2
B 2 − β1 B 2 − β1 B 2 − β1 2 − β1 − β2 ( ) ] dB 2β2 2β2 2β2
Bmax
pMCP (B)[B(Pmax − Pmin ) − β1 (Pmax − Pmin ) − B3
β2 (Pmax − Pmin )2 ] dB
From the discussion on the maximum profit bid, it follows that a necessary condition for a block bid curve BB (P ) to be optimal is that BB (P ) > BMP (P )
(29)
that is, BB (P ) is a bid-high strategy. If any part of BB (P ) is below BMP (P ), that part of BB (P ) would obviously not be profit maximizing. This strategy is shown in Fig. 7. Thus, it follows B1 = β1 + 2β2 P1
(30)
B2 = β1 + 2β2 (P1 + P2 )
(31)
(36)
To solve for the optimal values of B1 and B2 , we establish the first-order necessary conditions ∂πB = 0, ∂B1
∂πB =0 ∂B2
(37)
which are, in general, nonlinear if the probability function pMCP (B) is nonlinear. The equation (37) may admit many solutions. The optimal solution must satisfy B2 ≥ B1 . As an illustration, we develop 3-segment block bids for a unit with Pmin = 20 MW, Pmax = 120 MW, β1 = 5,
6
and β2 = 0.25 (such that B3 = $55/MWH), using a normal distribution pMCP (B) shown in Fig. 8a with the probability peaking at $30/MWH and the prices mostly between $20 and $40 per MWH. The expected profit for 10 ≤ B1 ≤ B2 ≤ 50 are plotted in Figure 8b. The optimal values are B1 = $21.0/MWH and B2 = $29.0/MWH, with an expected profit of πB = $614. The strategy is quite illuminating, namely, the lower two segments are bid in at below the expected MCP. This can be attributed to the fact that the unit is most profitable when the MCP is high and its cost of generation is low. Submitting a bid with low B1 and B2 will ensure that such opportunities are not missed. 0.08 0.07
Probability
0.05 0.04 0.03 0.02 0.01 0 0
10
20 30 MCP ($/MWH)
40
50
(a)
300
100
500
500
50
300
40
0
50
500
B2
35
0 60
30
0
60
25 20
B1>B2
500
15 10 10
Acknowledgment The research was supported in part by NSF grant ECS0085699 and in part by an EPRI/DoD CINSI grant administered by Carnegie–Mellon University. References
0.06
45
of the market clearing price. The optimization strategy indicates that blocks are bid to avoid lost opportunities. Several simplifying assumptions have been made in the paper. The minimum generation is bid in at cost. Its bidding can perhaps be optimized. The bidding analysis here is based on a single hour. Extensions to 24-hour periods need to be considered, which will require the generator ramp rates. Reserve prices are not included in the consideration. Potentially they may impact on optimal bidding strategies.
15
20
25
30 B1
35
40
45
50
(b)
Fig. 8. (a) Normal Distribution, (b) Equal profit contour versus B1 and B2
V. Conclusions We have investigated optimal bidding strategies for a generating unit in a competitive electricity market, in which the market clearing price is insensitive to the bid price of a single generator. Starting from the unit’s cost curve, the maximum profit bidding strategy has been developed, from which other optimal bidding strategies can be obtained. When unit derating is taken into account, the expected profit is optimized to develop a strategy which is a function of the estimated ratio of the real-time price versus the day-ahead price. Multi-segment block bids have also been investigated using the probabilistic distribution
[1] A. K. David, “Competitive Bidding in Electricity Supply,” IEE Proceedings-C, vol. 140, no. 5, 1993. [2] G. Gross, D. J. Finlay, and G. Deltas, “Strategic Bidding in Electricity Generation Supply Market,” Proc. 1999 Winter Power Meeting, vol. 1, pp. 309-315, 1999. [3] S. Hao, “A Study of Basic Bidding Strategy in Clearing Price Auctions,” Proc. 1999 PICA, pp. 55-60, 1999. [4] J. B. Park, et al., “A Continuous Strategy Game for Power Transactions Analysis in Competitive Electricity Markets,” IEEE Trans. on Power Systems, vol. 16, pp. 847-855, 2001. [5] N. Lu, J. H. Chow, and A. A. Desrochers, “Generator Bidding Strategies in a Competitive Deregulated Market Accounting for Availability and Bid Segments,” VIII Symposium of Specialists in Electric Operational and Expansion Planning, Brasilia, Brazil, 2002. [6] A. J. Wood and B. F. Wollenberg, Power Generation, Operation, and Control, 2nd ed, John Wiley, 1996. [7] NYISO, “Market Participant User Guide,” June 1, 2001 (availiable at http://www.nyiso.com/services/documents/manuals/ pdf/admin manuals/mpug 206 2001 version.pdf). [8] D. Gan and Q. Chen, “Locational Marginal Pricing - New England Perspective,” Proc. 2001 Winter Power Meeting, vol. 1, pp. 169-173, 2001. [9] R. Green and D. M. Newbery, “Competition in the British electricity spot market,” Journal of Political Economy, vol. 100, no. 6, 1989. [10] A. K. David and F. Wen, “Market Power in Electricity Supply,” Proceedings of 2002 Power Engineering Society Winter Meeting, vol. 1, pp. 452-457, 2002. [11] FERC Docket No. RM01-12-000, “Standard Electricity Market Design,” 2002. [12] NY ISO website, http://www.nyiso.com. Ning Lu received her B.S.E.E. degree from Harbin Institute of Technology in 1993, and her M.S. and Ph.D. degrees from Rensselaer Polytechnic Institute in 1999 and 2002, respectively. Her research interests include modeling and analyzing deregulated electricity markets. Joe H. Chow is a Professor of Electrical, Computer, and Systems Engineering at Rensselaer Polytechnic Institute. He received his B.S.E.E. degree from the University of Minnesota, and his M.S. and Ph.D. degrees from the University of Illinois. Before joining RPI, he worked in the power system business of General Electric Company. His research interests include modeling and control of power systems. Alan A. Desrochers attended the University of Massachusetts in Lowell, Massachusetts, where he received a B.S.E.E. degree in 1972. He received the M.S.E.E. degree and a Ph.D. in electrical engineering from Purdue University in 1973 and 1977, respectively. In 1980, he joined the Electrical, Computer, and Systems Engineering Department at Rensselaer Polytechnic Institute where he is currently Professor. He is a Fellow of the IEEE for contributions to manufacturing systems engineering and manufacturing systems education.
