**KEYWORDS:** hydroelectric station, 3^{rd}-type Pearson’s family of curves, A.Foster’s probability curve, valuation methods.

**ABSTRACT.** According to the recommendations of the EU commission, in member countries up to 12% of all energy consumption needs should be met by renewable energy resources in 2010. At the present time this part constitutes only 6%; it includes water-power and the heat energy which is obtained of wood. The share of water-power in the energy balance of the country is about 13%. In the meantime we use only 14% of the available water-power resources. If to use the whole water power, it would be possible to satisfy up to 15% of the country’s energy needs.

When solving energy problems in Lithuania, it should be improved first of all the management of this economy branch as well as a rational and favourable for the state privatisation performed.

The work offers and practically applies to the valuation of nine hydroelectric stations new valuation methods by using binomial 3^{rd}-type Pearson’s family curves that allow to determine the probabilistic values of some hydroelectric stations. Applying the method of use value in valuating hydroelectric stations, our approach reliably forecasts the water-power processes as well as values of hydroelectric stations for modelling investment strategies.

**1. INTRODUCTION**

Hydraulic power engineering, its development, its decay and rebirth, has already its own history. There were in Lithuania hundreds of water mills which were grinding not only grain, but also rags for producing paper; in workshops the mill-wheels were lifting hammers and forging iron, and they even helped threshing crops.

Hydraulic power engineering applied for producing electricity in Lithuania appeared at the beginning of the 20^{th} century. It is supposed that in 1900 an electric generator rotated by a mill-wheel was installed in the mill of Sukančiai village across the Virvyčia river. If it is true (the fact is being verified), the lifetime of our hydraulic power engineering is 100 years.

A description of Lithuanian electricity application and usage in 1935 shows, that in this year among 309 electricity producing plants there were 96 hydroelectric power stations (HE stations) and mixed power stations with 102 hydroelectric machinery units (of total 1432 kW capacity) producing electricity. In these stations 835.6 thousand kWh were produced a year. This amount included about 2.5% of all generated electric power (the same as at the present time). According to the data collected by Lithuanian institute of energetics, in 1958 there were 320 hydraulic power plants (hydroelectric stations, water mills, sawmills, etc.), the total power of which amounted to 17.6 thousand kW. Since 1954 it was allowed to adapt rural localities to electric power from the state electric network, therefore it was permitted to close smaller HE stations. Thus, 32 stations stopped to operate, the installed power was about 3.3 thousand kW and the production of electricity amounted to 12.9 mill per year [1].

**2. ****Production of elektricity in small hydro-power plants and their prospects**

At the beginning of 2000, 28 smaller HE stations were operating, their power has increased up 8.8 thousand kW and in 1998 the production of electricity has reached 26.3 mill kWh, i.e. it has been enlarged twice in comparison with that in 1993. It shows that building smaller HE stations is being enlarged when 2 or 3 new electric energy producing stations are erected every year. Skleipiai HE station across the Virvyčia river, Angiriai HE station (the Šušvė river), Jundeliškės HE station (the Verknė river), etc. are being built, projects for erecting 9 new HE stations have been approved, special conditions for designing 30 HE stations have been presented. The features of some of them are given in Table 1.

**Table 1.** Operating HE stations in Lithuania

No | HE name | River | Hydroelectric station | ||||||||

Name | Dis-tance to mou-th | Basin’s area | A-nnual ave-rage dis-char-ge | Ins- talled dis-char-ge m ^{3}/s | Head | Num- ber of turbi- nes | Ins- talled power | Electri-city produ-ced (thou-sand kW) | Built in | ||

The larger HE stations | |||||||||||

1. | Kaunas | Nemunas | 223,4 | 45800 | 287 | 640 | 20,5 | 4 | 100800 | 352000 | 1959 |

2. | Kruonis HAE | Nemunas | 248,2 | 44800 | 278 | 678 | 100 | 3 | 600000 | 266000 | 1992 1994 |

The smaller HE stations | |||||||||||

1. | B. Ančia | B. Ančia | 4,3 | 815 | 6,1 | 6,4 | 11,0 | 2 | 400 | 2130 | 1955 |

2. | Kapčiamiestis | Nieda | 0,9 | 312 | 2,3 | 4,3 | 5,0 | 2 | 140 | 380 | 1957 |

3. | Aukštadvaris | Verknė | 55,3 | 124 | 0,94 | 2,5 | 14,9 | 2 | 270 | 790 | 1960 |

4. | Bagdononys | Strėva | 60,5 | 152 | 1,4 | 1,5 | 10,8 | 1 | 110 | 500 | 1960 |

5. | Eišiškės | Verseka | 22,0 | 280 | 2,2 | 2,2 | 10,0 | 3 | 120 | 500 | 1952 |

6. | Antalieptė | Šventoji | 212,0 | 426 | 4,2 | 10,1 | 32,4 | 3 | 2460 | 4500 | 1959 |

7. | Motiejūnai | Širvinta | 86,5 | 410 | 3,0 | 4,5 | 5,3 | 2 | 170 | 340 | 1959 |

8. | Antanavas | Šešupė | 177,0 | 1923 | 10,0 | 12,0 | 4,7 | 2 | 400 | 1260 | 1957 |

9. | “Šešupė” (Dovinė) | Dovinė | 0,6 | 588 | 3,2 | 6,2 | 4,0 | 2 | 170 | 360 | 1952 |

10. | Gondinga | Babrungas | 15,5 | 228 | 3,0 | 4,0 | 26,9 | 1 | 800 | 3140 | 1961 |

11. | Sukančiai | Virvyčia | 29,8 | 973 | 10,1 | 8,4 | 4,0 | 2 | 230 | 840 | 1960 |

12. | Renavas | Varduva | 41,6 | 355 | 3,7 | 3,6 | 8,2 | 1 | 300 | 1400 | 1996 |

13. | Padysnis | Dysna | 166,0 | 280 | 2,2 | 6,0 | 3,0 | 2 | 150 | 190 | 1959 |

14. | Būbliai | Strėva | 9,0 | 731,0 | 3,98 | 5,28 | 6,8 | 2 (3) | 316 | 1912 | 1957 |

15. | Valtūnai | Siesartis | 10,7 | 622,0 | 4,5 | 4,5 | 3,5 | 1 | 200 | 1350 | 2001 |

16. | Kadrėnai | Mūšia | 1,6 | 215,0 | 1,63 | 3,5 | 5,5 | (1) | (80) | (500) | 2001 |

17. | Senoji Varėna | Varėnė | 2,1 | 408,0 | 3,48 | 3,08 | 5,0 | 1 | 132 | 620 | 2000 |

18. | Širvintų tvenk. | Širvinta | 84,0 | 478,0 | 3,29 | 2,5 | 3,64 | (2) | (180) | (677) | 2002 |

19. | Bartkuškis | Musė | 31,0 | 243,2 | 1,74 | 2,5 | 8,0 | (2) | (150) | (655) | 2002, 2003 |

20. | Krūminiai | Verseka | 7,5 | 339 | 2,92 | 5,0 | 6,6 | 1 | 160 | 460 | 2002 |

21. | Biržuvėnai | Virvyčia | 72,0 | 420 | 4,2 | 6,0 | 3,5 | 2 (3) | 210 | 685 | 2004 |

22. | Kuodžiai | Venta | 189,9 | 4060 | 21,9 | (24,0) | (3,0) | (4) | (600) | (3000) | – |

23. | Mūro Vokė | Vokė | 9,45 | 562 | 3,91 | (9,0) | (6,0) | (2) | (400) | (2000) | – |

24. | Juciai | Virvyčia | 66,0 | 436 | 4,66 | 6,0 | 4,4 | 1 | 100 | 600 | 1950, 2004 |

7598 | 28429 |

In the meantime smaller HE stations are being built at ponds, on sites of the former HE stations and water mills. The Ministry of Environment Protection is anxious about the damage of natural environment when erecting HE stations. By the order No. 411 of Minister of Lithuanian Environment Protection (Dec. 21, 1999), the territories have been defined where building new dams as well as rebuilding the old ones is forbidden [2]. For erecting new smaller hydropower stations or restoring the previous ones first of all the ponds of greater energetic potential should be chosen, as well as the sections of rivers where parts of the former hydropower stations have remained. This governmental document includes the list of 140 ponds and the former 49 hydropower stations, that should be restored first of all. Someone wishing to erect a HE station can make a selection in the list.

We have surveyed all the ponds larger than 5 *ha*. At present their number is 433. At 140 enlisted ponds that can be considered the most effective ones it is possible to erect a HE station not less than of 20 kW power. Their total amount of produced power would reach about 16 thousand kW. If we restore 49 hydropower stations enumerated in the list [2] and each them is of 20 kW, we shall obtain a total power of 9.8 thousand kW. Admitting that the power of smaller 28 HE stations is 9.8 thousand kW, we shall have the total power of all three groups of HE stations amounting to 34.5 thousand kW. Thus, it would remain for usage about 25.3 thousand kW (59.8-34.5) potential power of all rivers (the Nemunas and Neris rivers) that is expressed by 59.8 thousand kW.

No doubt, this calculation is rather approximate, because the HE station power is usually greater than the natural one, because the hydroelectric machinery units are installed in stations that are able to pass a greater discharge than the annual average one and the production of electricity is actually less because of an uneven annual flow of a river. In a dry weather, HE stations work usually at low power, but in spring, though at high waters, the water pressure decreases because of a risen afterbay. Therefore the installed power of a smaller HE station operates up to 4000 h per year. Thus, when using the remaining 25.3 thousand kW power, the smaller HE stations should be erected at the prospective river sections. The number of needed stations could be more than 120, if the average power of a HE station would be 200 kW. Thus in Lithuania the number of smaller HE stations could reach 300-350. The electricity production at smaller HE stations is presented in Table 2.

**Table 2.** Electricity production in the smaller HE stations in 1981-95 (mill kWh)

Name of HE station | Year | ||||||||||||||

1981 | 1982 | 1983 | 1984 | 1985 | 1986 | 1987 | 1988 | 1989 | 1990 | 1991 | 1992 | 1993 | 1994 | 1995 | |

B. Ančia | 1,69 | 1,84 | 1,80 | 1,49 | 2,04 | 2,11 | 1,95 | 2,28 | 2,20 | 1,79 | 1,36 | 1,05 | 3,02 | 2,37 | 2,36 |

Kapčia-miestis | 0,25 | 0,30 | 0,32 | 0,35 | 0,39 | 0,42 | 0,34 | 0,44 | 0,34 | 0,44 | 0,40 | 0,17 | 0,53 | 0,57 | 0,69 |

Aukšta-dvaris | 0,85 | 0,93 | 0,99 | 0,56 | 0,55 | 0,63 | 0,61 | 0,81 | 0,65 | 0,68 | 0,47 | 0,51 | 0,55 | 0,77 | 0,84 |

Bagdono-nys | 0,43 | 0,61 | 0,62 | 0,54 | 0,53 | 0,56 | 0,53 | 0,58 | 0,63 | 0,65 | 0,52 | 0,22 | 0,67 | 0,68 | 0,60 |

Eišiškės | 0,58 | 0,58 | 0,48 | 0,45 | 0,45 | 0,37 | 0,29 | 0,35 | 0,44 | 0,52 | 0,31 | 0,56 | 0,07 | – | 0,06 |

Antalieptė | 6,87 | 5,84 | 5,87 | 4,20 | 5,41 | 5,96 | 6,17 | 5,35 | 6,51 | 8,93 | 4,28 | 4,41 | 4,85 | 7,59 | 7,21 |

Motiejūnai | 0,20 | 0,27 | 0,24 | 0,16 | 0,20 | 0,33 | 0,26 | 0,16 | 0,28 | 0,39 | 0,09 | 0,29 | 0,29 | 0,28 | 0,39 |

Antanavas | 1,47 | 0,65 | 1,10 | 1,39 | 1,81 | 1,87 | 1,32 | 1,05 | 1,42 | 1,85 | 1,24 | 1,62 | 1,22 | 0,44 | 2,32 |

“Šešupė” | 0,42 | 0,30 | 0,28 | 0,14 | 0,41 | 0,50 | 0,43 | 0,33 | 0,32 | 0,37 | 0,30 | 0,31 | 0,54 | 0,48 | 0,48 |

Gondinga | 5,23 | 3,07 | 4,26 | 3,21 | 0,09 | 4,60 | 4,04 | 3,38 | 2,73 | 2,43 | 3,05 | 2,03 | 0,00 | 0,02 | 0,66 |

Sukančiai | 0,72 | 0,55 | 0,78 | 0,65 | 0,41 | 0,24 | 0,23 | 0,17 | 0,35 | 0,24 | 0,17 | 0,26 | 0,13 | – | – |

Padysnys | 0,13 | 0,09 | 0,09 | 0,05 | 0,09 | 0,04 | 0,05 | 0,04 | 0,07 | 0,11 | 0,07 | 0,06 | – | – | 0,05 |

Žiobiškis | – | – | – | – | – | – | – | – | – | – | – | – | – | 0,064 | 0,09 |

To find a suitable section it would be not an easy task, because about a third of river beds crosses the protected natural territories. Especially it refers to energetically effective rivers (Šventoji, Merkys, Minija, Dubysa, Žeimena, Šešuvis, Ūla-Pelesa, Luoba, etc.), which flow mainly through reservations. In this aspect many restrictions are to be overcome by the smaller hydraulic power engineering in our country.

Is it worth to develop hydraulic power? For instance, in 1998 there was produced 17.63 billion kW power in Lithuania. During this time span the smaller HE stations produced 26.3 mill kWh, the Kaunas HE station 391 mill kWh, all HE stations produced 417.3 mill kWh, and this constitutes 2.37%, including the smaller HE stations 0.15%, together with Kruonis HAE station 5.08%.

The smaller HE stations represents only a droplet in the country’s power generation, though they give somebody job and income. Up to 2000, 16.1c was paid for 1 kWh (since 2000, 17.8 c/kWh). Multiplying it by 26.3 mill kWh, we get 4.2 mill Lt. If at this price (16.1 c/kWh) the electric energy of 1998 of Kaunas HE station (391 mill kWh) were sold, we would obtain 63.0 mill Lt. This year its cost price was 1 c/kWh.

**3. ****valuation of** **hydro-power plants**

In 1999 we evaluated the Kaunas HE station [3]; in 2000, seven smaller HE stations were appraised [4] and cost prices of electricity production were analysed. The cost price of 1 kWh at the Kaunas HE station during the 1^{st} half-year was 0.80 c. At the Ignalina nuclear power station, the cost price of 1 kWh electricity to be sold was 7.848 c. The cost price of 1 kWh of the smaller HE stations evaluated by us, according to our data of 2000, was 6.0 c. According to the data of the joint-stock company “Lietuvos energija”, the price of the electric energy supplied to the network by the smaller HE stations was 18.04 c/kWh.

Diagram 1 shows that the change during years of average annual water flow *Q* (m3/s) predetermines the amount of annual production, *E*, of electricity at the Kaunas HE station, namely, 352 000 000 kWh per year.

**Diagram 1. Yearly dynamics of the average annual run-off of the river Nemunas at Kaunas city and annual production of elektricity**

**Diagram 2. Link between the average annual run-off and the elektricity production**

Besides, between the average annual flow *Q *(m3/s) and the electric energy, *E mill kWh,* produced in a year there is a strong correlation, more than 0.9.

The Būbliai HE station [5] is in the basin of the Nemunas river. It determines the fact that the hydrological processes are analogous to those characterised by us when evaluating the Kaunas HE station. We have analysed the electricity production of 13 operating HE stations (the data are given in Table 2) during the period of 1981-95 and performed calculations according to the presented methods which are acceptable only for evaluating hydraulic energy objects.

**4. ****methods for** **valuation of** **hydro-power plants**

The theoretical basis of these methods is binomial 3^{rd}– type Pearson’s family curves expressed by the dependence:

, (1)

where x is the investigated quantity, x is its arithmetic mean

; m ; l

parameters

e is natural logarithmic base,

(m+1) is gamma function symbol.

When calculating the curves of this type, one should know three parameters: distribution centre x, variation coefficient C_{v} and asymmetry coefficient C_{s} .

When C_{s} =0, the curve becomes normal.

The curve is limited from one side. Its origin x _{A} (the possible least value) is determined by the equation

1 – , (2)

If the asymmetry is positive x_{A }<0, when C_{s }<2 C and x_{A} >0, if C_{s }>2 C_{v}. Thus, when the C_{V} value is determined at the change of value of C_{S}, the origin of the curve must change.

All the hydrological values may be negative, therefore, when analysing them, a very interesting case occurs at x_{A} = 0, i.e. when the variable reaches its lowest limit. In this case C_{S} = 2C_{v}.

When analysing hydrological orders, not always C_{S} = 2C. If C_{S} 2C_{v} we obtain that the minimum of the investigated value is negative, but this contradicts to the physical essence of hydraulic energy phenomena. If C_{S} > 2C_{v} the value of the minimum defines formally the curve equation itself, but not the character of the hydrological phenomenon oscillation.

**4.1. ****Calculation of guarantee curve parameters***.*

Having a limited order of data, we can calculate approximate parameter values of the guarantee curve according to the formulas:

- Distribution centre or the arithmetic average

x , (3)

2. Standard or mean square deviation

, (4)

3. Coefficient of variation

C_{v } , (5)

4. Coefficient of skewness

C_{S } . (6)

If we had not only one order of data, consisting of* n* members, but many orders and calculate parameters of each of them, they would differ. These parameters may be regarded as variables and, applying the theory of probabilities, we can determine errors. The parameter errors are calculated according to the following formulas:

- Standard error of arithmetic mean

_{x } , (7)

- Dispersion standard error

_{}_{ } , (8)

3. Variation coefficient standard error

, (9)

4. Skewness coefficient standard error

. (10)

With the order of 50 data and using the binomial curve, where x_{A }=0, we obtain the following limit values of standard errors expressed by parts of the calculated parameters:

C_{v} | ||||

0.1 | 0.014 | 0.1 | 0.11 | 1.80 |

1.0 | 0.14 | 0.2 | 0.35 | 0.60 |

As we can see, for the most frequently occurring variation of the coefficient interval C_{v} = 0.1 -1.0 and having the order of 50 data, it is possible to calculate approximately only parametres x and C_{v}. The errors of other parameters may be very large.

Practically and by much shorter orders of years (15-20) we calculate all three parameters x, C_{v} and C_{s}. In this case we can take only the first two as more or less reliable, if their values do not differ considerably from the neighbouring rivers with parameter values based on long-term observations. There is no sense in calculating the skewness coefficient. It is better to take it as a mean value for the calculated hydrological region and the hydrological characteristic to be considered.

**4.2. ****Comparison of curves and actual data**

The theoretical curve must be compared to actual data, therefore for each member of data order we admit a certain guarantee. The guarantee expressed by a percentage can be calculated using the following formula:

p 100 , (11)

where *m *is the order number of statistical order member,n is the number of data.

The guarantee expressed in years and equal, say, to 100 years must be considered in such a manner that the flow rate obtained may occur for a long period, on average once per 100 years. In no way it means that the flow rate (discharge) obtained cannot occur several times during the next century and that it will happen only during next century.

The variable of every guarantee p is calculated by the formula

x_{p} x (F_{p} C_{v} +1) , (12)

where x_{p }is the variable with the guarantee percentage p,

x is arithmetic mean,

F_{p} is H.Foster’s function depending on p and C_{s},

C_{v} is variation coefficient.

We present the obtained theoretical curve in the network of A.Hazen as actual points, because there is a large curvature in a simple network, especially at the ends of the curve.

**5.****valuation of bubliai ****hydro-power plant**

In 2000, we performed the evaluation of the Bagdononys HE station, located in the middle of the Strėva river [4]. After the production of electric energy (Table 2) in 1981-95 by 7 smaller He stations we have calculated the H.Foster’s probabilistic lines for each station. The results are presented in Table 3.

**Table 3. Possibilities for electricity production in small hydro-plants**

Name of HE station | Electricity production (kWh) in years of 20% water abundance probability (moist) | Electricity production (kWh) in years of 50% water abundance probability (average) | Electricity production (kWh) in years of 80% water abundance probability (dry) |

B. Ančia | 2 400 000 | 1 800 000 | 1 450 000 |

Aukštadvaris | 800 000 | 650 000 | 510 000 |

Bagdononys | 640 000 | 590 000 | 515 000 |

Eišiškės | 545 000 | 460 000 | 365 000 |

Antalieptė | 6 800 000 | 5 800 000 | 4 500 000 |

Gondinga | 4 450 000 | 3 400 000 | 2 050 000 |

Sukančiai | 530 000 | 310 000 | 175 000 |

Būbliai | – | 1 912 000 | – |

There is a strong correlation between the amounts of electricity production in Bagdononys and Būbliai HE stations (correlation coefficient = 0.98), therefore the H.Foster’s probability curve of electrical energy production in Būbliai He station is analogous to that of Bagdononys. The technical and economical indices of Būbliai HE station are presented in Table 4.

**Table 4.** Technical and economical indices

No. | Name | Measurement unit | Quantity |

1. 1.1 1.2 1.3 1.4 2.1 2.2 3.1 3.2 4.1 | Turbine PO 300-BO 84 head max min discharge max min power max min rotational speed Max HE station discharge Max HE station power Consumption of electricity in a year of the average water abundance: 1HYDROELECTRIC STATION^{st} variant 2^{nd} variant 3^{rd} variant in a dry year of 95% probability 1^{st} variant 2^{nd} variant 3^{rd} variantPONDNormal retention level (NRL) Max water level (WL) Min WL Area at NRL Area at max WL Area at min WL Content at NRL Content at max WL Content at min WL Active content Layer of active content Type of flood appurtenance Calculated discharge Head Total cost of building, including the cost of turbines Pay-back of investments 1 ^{st} variant 2^{nd} variant 3^{rd} variant | unit m m m ^{3}/s m^{3}/s kW kW min m^{3}/s kWthousand kWh thousand kWh thousand kWh thousand kWh thousand kWh thousand kWh m m m ha ha ha thousand m ^{3} thousand m^{3} thousand m^{3} thousand m^{3} m reinforced concrete weir with flashboards m^{3}/s m thousand Lt thousand Ltyears years years | 2 7,0 2,5 2,64 1,58 158 33 152 – 254 5,28 316 1 912 2 327 2 135 1 105 1 648 1 389 60,00 60,50 59,40 21,0 25,5 15,5 245 360 135 110 0,6 79,4 6,8 770,0 569,0 15 11 13 |

According to the results of the USA company ,,The Money Group Inc.“ [7], the capitalisation rate for old, successfully operating and financially stable enterprises with a very reliable business is 12-15%.

Taking these data and factors into account, we conclude that the value of Būbliai HE station will be expressed by the value of 50% water abundance probability at the capitalisation rate of 13.0%.

We calculate the net profit, which Būbliai HE station could get for its electrical energy sold for market price. The calculations are given in Table 5.

No | Paragraphs of financial accounts | Calculations (Lt) |

1. | Cost price | 1 912 000 kWh x 6.0 ct [8]= 11 472 000 ct = 114 720 |

2. | Selling | 1 912 000 kWh x 18.04 ct = 34 492 480 ct = 344 925 |

3. | Net profit | 344 925 – 114 720 = 230 205 |

The use value of Būbliai HE station is calculated by dividing the annual net profit by the capitalisation rate:

230 205: 0.135 = 1 705 222 Lt.

- Though the meteorological conditions in Lithuania are not favourable for hydraulic power engineering and its possibilities would satisfy at the best only 15% of electric energy demand in the country, it is worth to take use of this type of engineering, as it is required by the Commission of the EU and other international documents pertaining to the environment protection.
- The present research suggests new evaluation methods for actually 9 HE stations applying binomial 3rd-type Pearson’s family curves enabling to determine probabilistic values for HE stations.
- It has been ascertained that the methods suggested by us for evaluating HE stations by the mode of the use value forecasts in a reliable way the processes of hydraulic power resources and at the same time the HE stations prices for investment strategy models.

**References**

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- L R Aplinkos ministro 1999 m. gruodžio 21 d. įsakymas Nr. 411 ,,Dėl užtvankų statybos (atstatymo) apribojimo aplinkosauginiu požiūriu svarbiose upėse ir jų atskiruose ruožuose“. ,,Žinios“, 1999, Nr. 112 – 3261.

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^{rd}edition. Prentice hall carerr & technology. 1994, 325 p.

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