ADMISSIBILITY OF DEVIATING FROM THE THOMA CRITERION WHEN DESIGNATING

THE CROSS-SECTIONAL AREA OF SURGE TANKS

G. I. Krivchenko UDC 621.221:627.846

According t o the existing practice of designing surge tanks of hydroelectric stations it is required that their cross-sectional area F t within the limits of the level for all steady operating regimes of the station satisfies the condition

F,>~k~Fr,, (I)

in which Fm is the Thoma critical area corresponding to the boundary of the region of oscil- latory stability of the system, and k d is the safety factor, amounting to 1.05-1.1 [i] (abroad up to 1.6-2.0). As is known, the Thoma formula is derived from the condition of constancy of power, and in a generalized form it can be represented as

Fr~=Fe/(~Ll-lo). (2 )

Here F d is the cross-sectional area of the diversion pressure conduit; H 0 is the head; and [dL is the linear coefficient of hydraulic resistance of the diversion system

~aL-=-2g/C'R + (1/L)E (~a~+-. 0.5), ( 3 )

where C and R are the Chezy coefficient and hydraulic radius of the diversion system; ~ai is the coefficients of local losses (entry, turns, exit, etc.); L is the length of the diversion system, the minus sign is for the downstream tank.

Since the second term in (3) is small in comparison with the first, it follows from (2) and (3) that Frn depends weakly on the length of the diversion system. This means that the relative volume of the tank increases with decrease of length of the diversion system. This characteristic is expressed most substantially at hydrostations having short diversion con- duits and a large range of variations of the pool level (upper pool for the upstream tanks and lower pool for the downstream tanks). A characteristic example of such stations is the Hoa Binh hydrostation in Vietnam [2]. In connection with this, the admissibility of designa- ting a smaller area of the tank than condition (I) requires can effect a perceptible economy in a number of cases.

Formula (2) was obtained for ideal conditions N - const and stabilization of the system here is provided just due to the effect of hydraulic losses. If the real structure of an automatic regulating system is taken into account, then it is necessary to take into account also the action of the stabilizing devices of the automatic regulators. Such investigations have been carried out (for example, [3]) and they showed their positive effect on stability of a system with a surge tank. However, since in this case automatic frequency regulators were examined, under conditions of operation in a power system with the distribution of disturbances on many simultaneously operating stations the problem of providing stability lost its urgency ("the power system prevails"). Furthermore, to provide the required charac- teristics of the stabilizing elements, for example, floating controls of fluid mechanical regulators, was a problem difficult to solve.

At present the situation has radically changed, since at large hydrostations electrical central regulators able to automatically maintain the prescribed power of the station indepen- dently of the frequency and load of other stations are widely used. -Thus the real operating conditions are approaching the ideal conditions adopted when deriving formula (2), which makes the problem of stability more acute. At the same time, flexibility of the adjustments of stabilizing devices of modern electrical regulators offers the real possibility of their use for reducing the required cross-sectional area of surge tanks. The problem consists in determining the characteristics of the stabilizing devices with consideration of the existing

Translated from Gidrotekhnicheskoe Stroitei'stvo, No. 7, pp. 27-30, July, 1988.

0018-8220/88/2207-0403512.50 �9 1989 Plenum Publishing Corporation 403

I Auto- matic reE.

y ~Central L I~ I power I_ I

regulato1~ in7 i

~ 2 G e n e r a - ~ --~ h i . ~ ~-i I ~ . ~ I - C-b. I syste~l f~ ,~h+ ~ I ..... ' '

# conduits tio. L]

Diversion I ~ " I System ,= I_ I

I Su.cge-tank I

Fig. I. Block diagram of the system regula- ting the power of the hydrostation.

constraints. A block diagram of a closed system automatically regulating the power of a hydrostation is shownin Fig. i. It consists of the following element: Power system. The most severe conditions of constancy of the rotational speed f - const are taken. Generator. It is considered an ideal component (the powers m t at the input and output are equal), which is completely admissible for the slow fluid mechanical processes under consideration. Tur- bine. Two actions arrive at the input: a change in opening ~ and a change in head Ah/~ due to a pressure surge and deviation of the level in the tank from the equilibrium value (minus for the downstream tank). A change in power (moment) on the shaft m t and a change in the station discharge qp are received at the output. The relationships between a, Ah~f and n%, qp are determined from the universal characteristic of the turbine according to its operating regime. The turbine is encompassed by two feedback elements through qp: this is 4 - the station pressure conduit (pressure surge Ah) - and 5 - the pressure diversion system with the surge tank (deviation of the level ~). Power regulator. The input actions for it are the total power of the units m and the specified value of the power m,, and at the output it creates a control signal y dependent on the difference m A - m - m.. The automatic regulators of the turbines receive at the input the signal y, in conformity with which the opening of the turbine = is changed.

For evaluating the stability of the system it is necessary to know the dynamic charac- teristics of all elements giving the relation between the input and output parameters, Since small deviations from the equilibrium regime are being considered, the dynamic characteristics of the elements are represented in the form of linear differential equations, which are given below.

Turbines, It is taken into account that regulation of the entire group ofoperating turbines is carried out synchronously with respect to opening. Under these conditions, for a turbine and for the entire group the relative deviations (subscript 4) from a steady regime (subscript e), m~=m,--m,~ and %~=q~--q~ are:

moment (power when f - const)

station discharge

rnta=BaA +C(Ah• (4)

qpA=A~LA + O,5q,~ (hh - ~). (5)

In (4) and (5) in conformity with Fig. 2 ~ - a/a n is the relative opening (a n is the maximum opening); %=~--a~, q,~ the relative reduced discharge q,,=Q',,./Q,,~Ah, and ~ are referred to the head of the turbines H 0.

A, B, and C are dimensionless coefficients determined from the universal characteristic of the turbine in conformity with its operating regime (ae, n'1, )

A= (OQ',lOa) (~. IQ',- e) ; ( 6 )

B= (JM" dOa) (a~lm',~ , ) ;

C=m,~+O,5b,. (7)

where ,n,~=M',JM',~. is the moment in a steady regime, and b is the coefficient of self-regu- lation

404

b=--(dM',/On" O (n'~,/M'la ,, ). (8)

Usually b - 0.8-1.6.

In (6) aM'i/aa is taken for n' 1 - n'le - const and in (8) aM'i/an' I is taken for a = a e - const. Their values are calculated with respect to finite increments (M' I - 93,700 Q' ~./n' D .

The station pressure conduit determines the pressure surge

Ah=--Twpqpa. (9 )

which is taken without consideration of elastic deformations, which is completely admissible for the slow processes under consideration. The constant of inertia of the pressure conduit T, is represented by the expression

Here Qn is the maximum discharge corresponding to Q'~c and n'I, ; the sum is taken over the entire station pressure conduit, including the turbine.

The pressure diversion system with the surge tank is represented by the system

Tdpq~ =~ ~-- (2? + ~)q~-- ~qp~; / TtP~ ~ ~ (q__qp~j. , (Ii)

in which T d is the constant of inertia of the diversion system

Td=LQ~/gF~Ho; (12)

T t is the time constant of the tank

T,=F,Ho/Q~. (13)

Here F a is the cross-sectional area of the diversion system, F t is the cross-sectional area of the tank. In (II) qA is the deviation of the relative discharge of the diversion system from steady, ~ is the relative hydraulic losses in the diversion system, and ~ is the relative velocity head at the transition of the tank with the diversion system (unlike the usual, it is assumed that 50% of the velocity head depends on the discharge of the diversion system and 50% on the station discharge). The values of ~ and @ are found for the station discharge Q,.

The central regulator receives the value of the total load m, and m ~ 1 ccrresponds to the installed capacity of the station. Then the relation between the load of the turbines m t and the total load is determined by the relationship

m=m,c, (14)

where

c=(M',~/M',.~)(Ho/H.)(s/s~). ( 1 5 )

H e r e M' ie a n d M'~n n a r e a c c o r d i n g t o F i g . 2 , H n i s t h e d e s i g n h e a d , s i s t h e n u m b e r o f operating units, s 5 is the total number of units. Accordingly, for the deviation we have

m~=m~Ac. ( 1 6 )

The power regulating unit, which includes the central regulator 6 and automatic regula- tors of the turbines 7 (Fig. i), is examined under the most difficult conditions, when float- ing control of the power is accomplished according to the proportional-plus-integral control law. Two structures of the given unit are possible.

I. Element 6 represents a proportional controller, and regulators 7 realize the PI control law. In this case, a device equalizing the openings of all operating turbines should be provided. For this case, the equations of the elements have the form

y = bn A;

~T~p% = -- (TiP-i- 1) Y.

Here k is the gain, 6 is the temporary offset behavior, the floating control and they are adjustable parameters.

(17)

and Ti-is the time constant of

405

I I 1 1 ,

a; % ,qof

Fig. 2. Positions of the regime points on the universal characteristic of the tur- bines.

2. Element 6 represents a floating PI controller, and regulators 7 operate as servo devices. For this case the equations have the form

a ~ T i N p y = ( r t N p + l ) m a ; t (18) ~o=a = -- y. J

Here the adjustable parameters are: 6 N, the temporary offset behavior of the central regulator; TiN , time constant of its integrating element; and 60 the off set behavior of the turbine regulators (for a sufficiently large value of 60 the required equality of openings of the operating turbines is provided).

It is easy to prove that dynamically both structures are identical under the condition

= ~o; T~ ----- TiA,.

(19)

For evaluating the dynamic indices of the system of group regulation of the power of the hydrostation, using (4), (5), (ii), (16), and (17), we set up the equation of free motion of the closed system, which will represent a linear differential equation of fourth order. Its characteristic equation has the form

a~ (20)

The coefficients a include the values of the time constants T i, Td, Tt, and Tw, parame- ters characterizing the regime of the turbine A, B, C, ql., gain of the regulator kc, and complexes

r = 2~ • + / and / (21)

D = 8 + Bkc.

In (21) the minus sign is for the downstream surge tank.

For evaluating the stability of the closed system described by Eq. to use the Hurwitz criterion having the form

(20), it is convenient

a~>O; } a s (ala:--aoa3)--aL~a4>O.

(22)

As a result, a rather general solution of the problem is obtained and it makes it pos- sible to find for the pressure system of the hydrostation and for the prescribed operating regime of the turbines the parameters of adjusting the regulators which provide stable opera- tion of the system with the upstream or downstream surge tank~ By means of them we can also

406

analyze the effect of various factors, including the cross-sectional area of the tank Ft, on the stability of the system. Examples of using the solution obtained are given below.

First of all we find the value of the criteria for the Thoma conditions. Ideal regula- tors corresponds to them, i.e., T i - 0 and T w - 0. In this case a 0 - 0 and a I - 0. Conse- quently, the motion of the closed system is described by an equation of the second order.

It is interesting to note that the gain kc is reduced, which corresponds to the adopted ideal regulators. For the second-order system the condition of stability is positivity of the coefficients a > O. Hence

TLr.~Ta[(CA--O,SqI~B)/Br]~ (23)

r+_r (24)

We obtained new formulas for the Thoma criterion, which are more substantiated than the known ones; particularly (2), since they take into account the real properties of turbines.

Of practical interest are the values of the required parameters of stabilization of the regulators T i and 60 when there are no diversion system and surge tank. The answer is given by formulas (20) if we set T d - 0, r - 0, # - 0. In this case, the system is described by a second-order equation and the conditions of stability have the form

~C ~>~(CA--o'SqleB); } (25)

T kcT~ i > ~ (CA-- 0,5q, eB).

It is noteworthy that the first of these conditions conforms also to a system with a surge tank and corresponds to the requirement a 0 > 0.

Let us examine a particular example. The scheme of the pressure conduits is shown in Fig. 3a. The parameters of the system are as follows. Heads: design 88 m, minimum 60 m. Mixed-flow turbines RO 115/810, diameter 5.67 m, rotational speed 125 min -I, maximum discharge 301o5 ma/sec. The geometric parameter of the pressure conduit from the water intake to the surge tank (including the turbine) Z~/F - 6.9 m -I.

According to the turbine characteristic (Fig. 2), n'in - 75.5 rpm, Q'Inn - 1.03 mS/sec, a n - 37 mm. Chezy coefficient of the diversion system C - 86, coefficient of local losses r--l.4.

The stability calculations are carried out for the most severe conditions, with a minimum head and maximum opening. In this case, n'i. = 92 rpm and from the characteristic Q'I. = 0.98 m3/sec. The discharge of the two turbines Qe = 2"0.98"5.672~ = 488 m3/sec, v e 488/129 - 3.78 m/sec.

I. Determine F~ from (2). The coefficient according to (3) ~m= 2.9.8/(862"3) + (1,4--0,5)/420 = 0,031.~h = 1291(0,0031-60)=693 m 2 .

2. Determine F~ from (23). From the universal turbine characteristic for the given regime ,n~,=838/1176==0,715, q,e=l, by (6) A=I, B=0,73, by (8) b=1,3, by (7) C=1,37. By (12)Td=420.488/(9,8.129• 60i=2,7 sec. The hydraulic losses in the diversion system ha~420.3.782/(862.3)+1,4.3,78~119,6=1.3 m, v:/ 2g=0,73 m, ~=0.022, r By (21) r - 0.044 - 0.012 = 0.032.

By (23) TtTh = 2,7(1,37.1--0,5.1.0,73)/(0,73.0,032)=116,2 and in conformity with (13) PT~=T,ThQ~/Ho=-

]16,2"488/60==945 m 2 . It was 35% greater than by (2). But another difference can occur in other cases.

3. Determine the required values of the stabilization parameters for conditions when F t - 65 m z (almost 15 times less than according to the Thoma criterion).

We will consider two values of the gain kc - 0.15 and kc = 0.25. We plot the boundaries of the stability region by (22) in coordinates T i and 6 (the calculations are performed by the iterative method with linear interpolation). The calculated curves are shown in Fig. 3b (solid lines). Here also are given the limiting values of 6 and T i calculated by (25) (dashed lines). The difference represents the require~ increase of 6 and T i caused by the presence of the surge tank.

407

r

%

�9 I

a

0 2 g 6 0 ~ ~ lq ~ see b

Fig. 3. Pressure system of the hydrostation and boundaries of the stability regions: a) scheme, of pressure conduits: b) calculated curves.

The results obtained show that the stability of the system when F~ - 65 m 2 can be com- pletely provided. Obviously, the most favorable values are in the region of a sharp change in curvature, for example, 6 - 0.55, T i - 3.5 sec for kc - 0.15 and 6 - 0.7, T i - 3.3 sec for kc - 0.25.

To provide a good quality of regulation, T i and 6 should be greater than the values corresponding to the boundary of the stability region. A search for optimal adjustment can be carried out by the method proPOsed in [4]. The need to take extremely high values of the stabilization parameters leading to slowing of the reaction of the system to the control signals of a planned and unplanned change in power can be a constraint for using the described principle of reducing the required cross-sectional area of a surge tank. It is recommended preliminarily not to exceed the following limitations: T i < 15 sec and 6 < 1.2. However, this problem requires additional investigations with consideration of the possibility of creating a structure of the control signals such that the action of the stabilizing elements is excluded at the control stage.

CONCLUSIONS

The analysis showed that appropriate selection of the parameters of adjusting the system automatically regulating the power of a hydrostation makes it possible to provide stability of its oper~ation and with a smaller cross-sectional area of the surge tank than is required according to the Thoma criterion. The necessary values of the parameters of adjustment are determined by conditions (22) with consideration of the indicated constraints.

LITERATURE CITED

I. V. Ya. Karelin, and G. I. Krivchenko (eds.), Hydroelectric Stations [in Russian], Energo- atomizdat, Moscow (1987).

408

2. G.I. Krivchenko, E. V. Kvyatkovskaya, A. B. Vasil'ev, and V. B. Vladimirov, "New designs of outlets of hydropower plants," Gidrotekh. Stroit., No. 7 (1985).

3. Go V. Aronovich, N. A. Kartvelishvili, and Ya. K. Lyubimtsev, Water H;mmer and Surge Tanks [in Russian], Nauka, Moscow (1968).

4. G.I. Krivchenko, Optimization of automatic control systems of turbine-generator units," Energomashinostroenie, No. 4 (1986).

EVALUATION OF THE STABILITY OF A CHANNEL WITH PARTIAL REVETMENT

M. Ya. Krupnik and Yu. A. Ibad-Zade UDC 556.535.6

Partial revetment of the channel of rivers and canals, changing the velocity field of the flow, can cause deformation of the unrevetted part of the channel~ In this case, the stability both of the structures on the bank and of the revetment itself is disturbed.

It was shown in [I, 2] that the condition of stability of particles on the wetted surface of a prismatic channel with an arbitrary cross section in cohesive soils has the form

, I t ] \~ ~ tg ? ~ ~-'~-'s ~J ~ tg'- ~ + a~ cos ~ ~ ' ( I )

where ~ is the angle of internal friction of the water-saturated soil; [C] is the cohesive force of the soil; ~ is the angle between the tangent to a point on the wetted surface of the channel and horizon; ~ is the shear stress at this point

o=([xJ--[Cl)/tg ~, ( 2 )

[r] is the allowable shear stress.

The quantity T, as is known, can be estimated by the expression

~=~,v~(x, y)/C=, (3)

where 7 is the unit weight of water; v is the average velocity on a vertical with abscissa x and depth y.

On a bend of the channel the-average vertical velocity of the water flow is determined with an accuracy sufficient for practical purposes by the formula [2]

ct/', v(x,, y)= -~-o R( :~) I , (4)

where C is the Chezy coefficient; R(x) is the local hydraulic radius R(x) =y/L/I +y'-~; I is the hydraulic gradient; r, r 0 are respectively the radius of curvature of the flow on the vertical with abscissa x and on the channel line (Fig. I),

Keeping in mind that y' = tan ~ and denoting

[fl=vI#~, (5)

where hma x is the maximum allowable depth for the given channel.

With consideration of (3), Eq. (i) will take the form

[C] / ~ " : : (~. y) tg ~ + �9 cos = ] ~" tg~ = + C ~ ~ cos ~

Replacing a according to (2) and v according to (4), we obtain the condition of stability of particles on the wetted surface of a curved channel in the form

i i i i

Translated from Gidrotekhnicheskoe Stroitel'stvo, No. 7, pp. 31-33, July, 1988.

0 0 1 8 - 8 2 2 0 / 8 8 / 2 2 0 7 - 0 4 0 9 5 1 2 . 5 0 �9 1 9 8 9 P l e n u m P u b l i s h i n g C o r p o r a t i o n 409