Models of Risk Measurement and Control in Power Generation Investment

Systems Engineering Procedia

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Systems Engineering Procedia 00 (2011) 000–000

Systems Engineering Procedia 3 (2012) 125 – 131

www.elsevier.com/locate/procedia

The 2nd International Conference on Complexity Science & Information Engineering

Models of Risk Measurement and Control in Power Generation Investment Hong-xing Sun, Zhong-fu Tan North China Electric Power University, Beijing 102206, China

Abstract In the electricity market environment, Power generation companies are faced with much uncertainty and greater risks. It is a realistic problem for power generation companies to optimize their investment decisions and minimize risks in investment of engineering projects. In this paper, Conditional Value at Risk (CVaR) measurement techniques are used to establish models of risk assessment and risk control for power generation companies to invest under multimarkets and multi-types power units. Optimization models of risk control based on two criterions can be used by power generation companies in different electricity markets and different types of power generation units to distribute there investment in order to maximize benefits and minimize risk value. The case demonstrates the validity of these models.

©©2011 Ltd. Selection Selectionand andpeer-review peer-reviewunder underresponsibility responsibility Desheng Dash 2011Published Published by by Elsevier Elsevier Ltd. of of Desheng Dash WuWu. Keywords: power generation investment; Conditional Value at Risk; risk measurement; risk control; engineering projects

1. Introduction Value at Risk is a wide-used and calculable method for risk management. With respect to a specified probability α, the VaR loss value of a portfolio is the lowest amount such that, with probability α, the loss will not exceed у.The concept of Conditional Value at Risk (CVaR) was presented by Rockafeller R T and Uryasev S in 1999, mainly to solve the problem that VaR is not sub-additive. With a specified probability α, the loss value of CVaR is the conditional expectation of losses above the VaR[1]. CVaR overcomes several limitations of VaR and has good properties, especially its computability[2]. With development of electricity market, more and more techniques of risk management are used to solve problems about power generation investment. Liu Ya’an and Guan Xiaohong discusses the purchase allocation problem of two markets with risk management[3]. Feng Yi, Tian Kuo established economic evaluation models and corresponding risk measurement models based on interval net present value and interval internal rate of return respectively to provide investors with more reliable decision support[4]. WANG Fang and ZHOU Xiao-yang(2008) proposed a optimal portfolio decision-making flow for power producer, which takes the conditional value at risk as the risk measurement index and aims at both maximum avail and minimum risk value[5].HE Hong-bo(2008) analyzed the uncertainties

 Corresponding author. Tel.: +86-10-8172-1961; fax: +86-10-8079-6904. E-mail address: [email protected].

2211-3819 © 2011 Published by Elsevier Ltd. Selection and peer-review under responsibility of Desheng Dash Wu. doi:10.1016/j.sepro.2011.11.017

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and risks in power generation investment and proposed an approach of generation investment decision based on CVaR[6].On base of literatures [5] and [6], We have a research on the optimization models of risk measurement and control for power generation companies to distribute their investment of engineering projects on different markets and in different types of power generation units. 2. Model of Risk Measurement in Power Generation Investment We assume that the number of markets that a generation company invests on is M and the number of types of power units that a company invests on a market is N. The total investment distributed by a company on M markets and in N types of power units is X:

X

x11

x12

x21

x22

xm1

xm1

 x1n  x2 n   xmn

So we can get the cost function of unit portfolio of a power generation company: 1    Tij  T0 c(x, a, b,  p) aij xij   bij Hij xij (1)t (1) ij  t 0   i 1 j 1    

M N

(1)

Where: Hij is the annual operating hours of a power unit, Tij0 is the construction period of a power unit, Tij is the operating period of a power unit, θ is the discount rat, aij is the construction cost per KW, bij is the operating cost per KWH, pij is the generation price . The profit function of unit portfolio of a power generation company is: 1

  T1   1 0 M N    Tij M N  ij T  r ( x , a ,b , p )     pij H ij xij (1 )  t  (1 ) ij     aij xij    bij H ij xij (1 )  t   t 0 i 1 j 1 t 0 i 1 j 1    



M

N



  a x

i 1 j 1



ij

ij



1 Tij









   pij  bij H ij xij 1     (1   )  



t 0





t



0

 Tij

  

  T 0  (1 ) ij   

(2)

The loss function of unit portfolio of a power generation company is:

  T  t T f (x,a,b, p)  r(x, a,b, p)  a x    ij ij bij  pij  Hij xij 1 (1)  i 1 j 1   t 0   M N

1 ij

0 ij

(3)

So we can calculate the VaR value and the CVaR value of portfolio of a power generation company with CVaR to be risk measurement index. The computing formula of CVaR by Fβ (x,α), which is the equivalent function of CVaR is : 

1 J F (x,a,b, p)    f (x,ak,bk, pk ) (1)J K1 

(4)

     

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Where: a1,a2…aJ are sample values of a, b1,b2…bJ are sample values of b, p1,p2…pJ are sample values of p. They are historical or imitative data. α is the value of CVaR With specified probability and specified risk level, α∈R, [f(x, ak, bk, pk)-α]+ is equivalent with max[f(x, ak, bk, pk)-α,0]. We can calculate the risk value of engineering projects investment of power generation company on different markets and in different types of power units by formula (3) and (4) . 3. Models of Risk Control in Power Generation Investment 3.1. Optimization Model of Risk Control in Power Generation Investment under criterion of risk minimization By using the fictitious variable zk(k=1,2…J), We can turn the formula (4) into form as follow:  1 J F (x, a,b, p)   zk (1 )J K1 zk  0  st. . z  f x ak ,bk , pk )  ( , k

(5)

Then we can get the optimal model of risk control in power generation investment under criterion of minimizing portfolio risk: min F  ( x, a, b, p)  

J 1 zk  (1   ) J K 1

 r( x, ak , bk , pk )  R0 z  f ( x, a , b , p )   (6) k k k  k zk  0 s.t.  M N  x  1 i 1 j 1 ij  0  xij  1 Where: r(x, ak, bk, pk) is the income function of unit portfolio of generation projects, R0 is the profit level of portfolio that the investor expects.

3.2. Optimization Model of Risk Control in Power Generation Investment under criterion of profit maximization is:

According to formula (2), Profit expectation function of unit portfolio of a power generation company

 M N   ( x, a, b , p ) E[ r ( x, a, b, p )] E   aij xij i 1 j 1 

1  Tij    t     pij  bij  H ij xij 1         t 0   

1    Tij  T 0   E  aij   Hij    E( pij )  E(bij )  (1   )t  (1   ) ij xij t 0   i 1 j 1    

M

N

(7)

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So we can get the optimal model of risk control in power generation investment under criterion of profit maximization: 1  M N   Tij    max  E  aij   Hij   E(pij )  E(bij ) (1  θ)t xij  max μ(x,a,b,p) t 0  i 1 j 1     J 1   α  (1 β)J  zk  ω K1  zk  0  s.t. zk  f(x,a k ,bk ,pk )  α 0  x  1 ij  M N  xij  1 i 1 j 1

(8)

Where: ω is the allowed risk level. 4. Case Study The illustrative example assumes that there are six generation projects of a company which has three types of units, namely Thermal power, Hydro power, Nuclear power, on two markets. Then we can distribute the investment within six projects according to the optimization models of risk control described above. 4.1. Parameter assumption Table 1: Parameter values of three types of power units symbols

thermal power

hydro power

nuclear power

construction period

Tij0 (Years)

2

4

6

operating period

Tij1 (Years)

30

30

30

Annual average load utilization hours

Hij (Hours)

6000

3500

7000

Sample data of three parameters: construction cost per KW, operating cost per KWH, generation price are born randomly according to Gaussian distribution as follows.

Hong-xing Sun/ Systems Engineering Procedia 00 (2012) 000–000

Hong-xing Sun and Zhong-fu Tan / Systems Engineering Procedia 3 (2012) 125 – 131

Table 2: Gaussian distribution characters of parameters of three types of power units

construction cost per kW

thermal power

hydro power

nuclear power

symbols

a11(Yuan/MW)

a12(Yuan /MW)

a13(Yuan /MW)

Average value

4000000

8000000

1200000

Variance

100000

200000

400000

symbols

a21(Yuan /MW)

a22(Yuan /MW)

a23(Yuan /MW)

Average value

4500000

8500000

1200000

Variance

120000 b11

250000 b12

400000 b13

(Yuan / kWh)

(Yuan / kWh)

(Yuan / kWh) 280

symbols operating cost

Average value

250

80

per kWh

Variance

24

7

26

symbols

b21(Yuan / kWh)

b22(Yuan / kWh)

b23(Yuan / kWh)

Average value

260

85

Variance symbols

25 p11(Yuan / kWh)

8 p12(Yuan / kWh)

285 26

Average value

340

320

Variance

20

18

22

symbols

p21(Yuan / kWh)

p22(Yuan / kWh)

p23(Yuan / kWh)

Average value

350

325

380

Variance

21

19

22

generation price

p13(Yuan / kWh) 380

4.2. Calculation results Applying Lingo programs, we can calculate the proportion of portfolio and the value of CVaR according to formula (6) and (8).Results are shown in Table 3 and Table 4 respectively. Table 3: The proportion of portfolio and values of VaR and CVaR in Certain profit level

R0 (109)

Confidence level β=0.90

4.5 β=0.95 β=0.90 6.0 β=0.95 10.0

β=0.90

Xi1

Xi2

Xi3

0.1090

0.1296

0.3527

0.0734

0.0950

0.2403

0.0376

0.1543

0.3724

0.0123

0.1198

0.3145

0.5617

0.2306

0.0000

0.1846

0.0231

0.0000

0.4639

0.2672

0.0000

0.1503

0.1186

0.0000

0.8963

0.0000

0.0000

0.1037

0.0000

0.0000

VaR (109)

CVaR (109)

6.3918

6.8744

7.7106

9.9437

12.6304

17.3847

18.6401

25.1874

26.9502

31.9979

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We can conclude from Table 3 that: (1) According to formula (6), we can calculate the proportion of investment distribution in certain profit level under criterion of risk minimization. (2) The value of R0 reflects the profit level of generation portfolio that investors expect. If the value of R0 is raised, investors will raise the proportion of thermal power units. Namely, investors would like to invest to the projects with high profit level and high risk level. (3) If the value of Confidence level β is raised, Investors would like to raise the proportion of hydro power and nuclear power units. Namely, investors would incline to enlarge the reliability of profit and lower the risk level. Table 4: The proportion of portfolio and values of VaR and CVaR in Certain risk level Confidence level

ω (109) 5.5

β=0.90 11 5.5 β=0.95

11

Xi1

Xi2

Xi3

0.1124

0.1048

0.3856

0.0896

0.0673

0.2403

0.1654

0.2768

0.1071

0.1335

0.2314

0.0858

0.1347

0.1158

0.3469

0.0971

0.0823

0.2312

0.1743

0.2717

0.1242

0.1448

0.2006

0.0844

VaR (109)

CVaR (109)

5.1268

5.4998

8.2601

10.9913

4.8524

5.4978

8.1259

10.9998

We can conclude from Table 4 that: (1) According to formula (8), we can calculate the proportion of investment distribution in certain risk level under criterion of profit maximization. (2) The value of ω reflects the Probable Maximum Loss that investors expect to control. If the value of ω is raised with other conditions unchanged, investors would like to invest to projects with high profit level and high risk level. (3) The value of β reflects the degree that investors disgust risks. If the value of β is raised, investors will incline to projects with lower profit and lower risk level. 5. Conclusion In electricity market environment, Power generation companies are faced with much uncertainty and greater risks. This paper sets up a model of risk measurement for power generation companies to distribute their investment of engineering projects on different markets and in different types of power units. Two optimization models of portfolio risk control are deduced under criterions f profit maximization and risk minimization. These models can be used to distribute investment of engineering projects for power generation companies to maximize there profits and minimize risk values. The case demonstrates the validity of these models. We can calculate the proportion of projects in the whole investment and the value of CVaR according to models described above. On the other hand, the power generation companies will be faced with more risk factors in the real market environment such as coal price , policies of Energy saving and Emission reduction and so on. Next topic of our research is to analyze the effect on the models under these risk factors.

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Acknowledgements This research is financially supported by the National Natural Science Foundation Grant 71071053, “Generation Performance Replacement Trading Optimization Model and Method under Energy-saving and Emission-reducing targets.” References [1] ROCKAFELLAR R T, URYASEV S. Optimization of Conditional Value-at-Risk.The Journal of Risk,2000, 2(3):21-41. [2] LIN Hui, HE Jian-min. The Shortcomings of VaR in Portfolio Management and Improved CVaR Model .Finance & Trade Economics,2003(12):46-49. [3] LIU Ya′an, GUAN Xiaohong. Optimization of Purchase Allocation in Dual Electric Power Markets with Risk Management .Automation of Electric Power Systems,2002,26(9):41-44. [4] Feng Yi, Tian Kuo, Xiao Lin, Zeng Ming. Economic evaluation and risk measurement for generation investment based on interval numbers. 2007 International Power Engineering Conference,2007:714-18. [5] WANG Fang, ZHOU Xiao-yang. Portfolio Decision-Making Flow for Generation Company. Proceedings of the CSUEPSA,2008,20(1):33-38. [6] HE Hong-bo. Risk Decision and System Dynamics Modeling of Generation Investment in Electricity Market.Changsha University of Science & Technology,2008.

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