If turbine efficiency is above 95% in a modern turbine, what is the average overall efficiency including accelerated mass lost in the water as it leaves the bottom of the sluiceway?

The power generated during turbine operation is calculated as:

Where (H) is effective head is the head which works effectively for the turbine, and expressed as follows.

H = Hg - H_l

Hg is gross head and for reaction turbine (such as Franacis & kaplan) it is between Dam reservoir level and tailrace tunnel outfall tail water level and in case of impulse (Pelton wheel) it is head between reservoir level and nozzle level at turbine location in power house.

Now H_l is head loss in water conductor system which depend upon various factor such as friction loss in water conductor system, bend loss, entrance loss, exit loss, transition loss, gate loss etc.

η is the product of all of the component efficiencies, which are normally the turbine and generator.

As you pointed out nowadays Francis turbines  has a very high degree of efficiency of more than 96 %  and Kaplan turbines have 95 %.

But in addition to that generator efficiency also contribute in overall efficiency . Typical values for generator efficiency range from 91 to 98%. 

Say if turbine efficiency is 95 % and generator efficiency is 95 % than overall efficiency becomes = 0.95*0.95 = 0.9025 or 90.25 %

The losses in water conductor system varies w.r.t water conductor system and its project specific. For longer WCS the losses may be high.

The total head loss varies and is project specific but for clarification based on my project experience it might be in the range between (4-8) % ( note:- this might varies and mentioned only for clarification) .

So say if for any project head loss is 5% of the maximum gross head at maximum discharge than effective head will be 95 % of gross head.

The writer has used average efficiency in the name of capacity factor “ plant factor”. The capacity factor is a measure of the plant use. It is the ratio of the average load to plant capacity. It may be computed for a day , month year , or any other period of time.

Actually he has confused load factor with capacity factor as well. Load factor and Capacity factor are not same.

If the capacity factor of a hydroelectric plant approaches to 30- 35% then we can call it as a peaking plant.

He says that 14 to 15 was the average. That means the power plant was working 15/100 x 365 x 24 hours = 1314 hours in a year(8760hours) or 3.6 hours in a day all throughout a year. This is not right hydropower engineering. But hydropwer is actually by nature peaking. But there are solutions we can create like using pelton which works with less flows down to 10% of design flow without any problem.

I mean that that 3.6 hours must be minimum 8 hours a day (33%) for that hydro powerplant to be called as a real peaking plant. 4 hours in the morning + 4 hours in the evening.

By selecting a pelton turbine he could have compensated power because pelton turbine can generate same power with a discharge value of 10% -20% of the design flow he used. I gave a real time example below from Hoover Dam with its link. “five wide-head turbines, designed to work efficiently with less flow” are pelton turbines actually. They have not mentioned it there but it is so. You can read my text completely or the yellow coloured ones if you get bored of reading all.

The amount of electricity generated by Hoover Dam has been decreasing along with the falling water level in Lake Mead due to the prolonged drought in the 2010s and high demand for the Colorado River's water. Lake Mead fell to a new record low elevation of 1,071.61 feet (326.63 m) on July 1, 2016 before beginning to rebound slowly.^[97] Under its original design, the dam would no longer be able to generate power once the water level fell below 1,050 feet (320 m), which might have occurred in 2017 had water restrictions not been enforced. To lower the minimum power pool elevation from 1,050 to 950 feet (320 to 290 m), five wide-head turbines, designed to work efficiently with less flow, were installed.^[98] Due to the low water levels, by 2014 it was providing power only during periods of peak demand.^[99] Water levels were maintained at over 1,075 feet (328 m) in 2018 and 2019.^[100]

Source: https://en.wikipedia.org/wiki/Hoover_Dam

The load factor is an index of the load characteristics. It is the ratio of the average load over a designated period to the peak load occurring in that period. It may apply to a generating or a consuming station and is usually determined from recording powermeters.

Hydropower plant efficiency η depends on the speed of the turbine, on the head, on the flow processes.

Turbine Power Output

In general, the turbine converts the kinetic energy of the working fluid, in this case water, into rotational motion of the turbine shaft.

Swiss mathematician Leonhard Euler showed in 1754 that the torque on the shaft is equal to the change in angular momentum of the water flow as it is deflected by the turbine blades and the power generated is equal to the torque on the shaft multiplied by the rotational speed of the shaft. See following diagram.

Note that this result does not depend on the turbine configuration or what happens inside the turbine. All that matters is the change in angular momentum of the fluid between the turbine's input and output.

Hydroelectric power generation is by far the most efficient method of large scale electric power generation. The overall efficiency can never be 100% however since extracting 100% of the flowing water's kinetic energy means the flow would have to stop.

The conversion efficiency of a hydroelectric power plant depends mainly on the type of water turbine employed and can be as high as 96% for large installations. Smaller plants with output powers less than 5 MW may have efficiencies between 80 and 85 %.

Overall efficiency of a HPP  is 85% maximum 86%.

The capacity factor for hydroelectric power in the world has been fairly consistent at 40–44% from 1980 to 2008. The capacity factor for hydroelectric power in the United States was 37% in 2008.

Capacity factor has a major influence on the cost of power

 While hydroelectric projects which take their flow directly or indirectly from large reservoirs are suitable for development as peak-load plants , provided that the conduit (headrace tunnel + penstock) required is not too long or too expensive , they are not necessarily peak load plants. (Conduit (headrace tunnel + penstock) is the main artery of a HPP) This is because market conditions may not require all the peak load hydro power available at such a site. Consequently many plants of this type operate on a daily capacity factor of 30 to 100%. Preferably many plants of this type must operate 15 to 20-22 hours a day.

     Although at the time a reservoir plant is installed market conditions may not make it practicable to install capacity on a peak-load basis , the possibility that such conditions may soon change should be considered. If the incremental cost of installation is low , provision should usually be made in the design for the later installation of a large amount of additional capacity.

     In many systems purely peak-load plants prove economical , particularly in connection with storage projects. Assume that , as the load increases in future years , storage is provided on the headwaters of the river system to such an extent that during the minimum December flow and maximum demand week , the plant has available , say , 1,200,000 kw-hr instead of the 438,000 kw-hr which is available without storage. The net effect of the increase in load and this additional energy is to leave at all times the sharp peaks of the load curve projecting above the band that can be served by this plant.

Consequently , these peaks might be served by installing a peak-load hydro plant at the reservoir. In many cases , the additional expense of such a peak load hydro plant need not exceed $70 per kw of installed capacity. This low additional amount results from the fact that the costs of dams and reservoir for storage is already incurred and the additional sum is for intake , conduits (hedrace tunnel + penstock) powerhouse and equipment.

Each hydropower site has unique characteristics. Thus, each hydroelectric project and powerhouse design is different. Each solution must be tailored to the unique characteristics of the site, the transmission and distribution system, and economic and financial resources.

Annual capacity factor at which hydro plants operate is usually limited by the variation in water supply (except in the case of hydro plants like those at Niagara, where installation is less than minimum stream flow and where annual capacity factors may exceed 95%). That means the installed power has been arranged according to be working under less than the minimum stream flow. So in most of the time in a year this plant shall find available water feeding its turbines. But shall not be able to take benefit of a lot of amount of water in most of the time in a year because water will be wasted by overflowing from the spillway. This is not right hydropower engineering.

When the peak load just equals the plant capacity , the capacity factor and load factor are obviously the same. If the maximum demand is less than the plant capacity , the capacity factor may be either greater or less than the load factor depending largely on the load factor itself.

The load for the peak day of the year determines the required generating capacity , while the requirements of the peak week or month (or seasonal or annual) dictate the amount of energy storage required in the form of water. If the scheme is only for power development, then the best use of the water will be by releasing according to the power demand

The average capacity factors are in the typical range for hydropower (≈ 35 to 55%). Capacity factor can be indicative of how hydropower is employed in the energy mix (e.g., peaking vs base-load generation), water availability, or an opportunity for increased generation through equipment upgrades and operation optimization. Potential generation increases achievable by equipment upgrades and operation optimization have generally not been assessed.

The capacity of a power plant is not easily defined. Nameplate capacity or rated capacity of a turbine is usually given in kilowatts or horse power for a given head , discharge and speed at which the best efficiency is obtained. Obviously each of these quantities may vary within definite limits. The rated capacity of a-c generators is usually stated in terms of definite speed, power factor and temperature rise and is usually given in kilovolt-amperes. Each of these quantities may also vary within definite limits.

The IEEE definition of generating station capacity is “ the maximum net power output that a generating station can produce without exceeding the operating limit of its component parts.”

The station or plant capacity can therefore be determined for a given station. It may be stated for a peak load over a given period as 15 min. or 1 hr or for a continuous load. It would be higher for short periods than for continuous service if storage regulation exists but is limited by the temperature riğse of the generators. Until the station capacity has been fixed the various factors having to do with capacity can not acquire definite meanings. Where the capacity of a plant has not been fixed , it is customary to take nameplate capacity of the generators as the plant capacity , which is often called installed capacity.

The average load of a plant or system during a given period of time is a hypothetical constant load over the same period that would produce the same energy output as the actual loading produced (IEEE).

The peak load is a maximum load consumed or produced by a unit or a group of units in a stated period of time. It may be the maximum instantenous load or a maxiumum average load over a designated interval of time.

The maximum average load is generally used. In commercial transactions involving peak load , it is taken as the average load during a time interval of specified duration occurring within a given period of time , that time interval being selected during which average power is greatest (IEEE).

The load factor is an index of the load characteristics. It is the ratio of the average load over a designated period to the peak load occurring in that period. It may apply to a generating or a consuming station and is usually determined from recording powermeters. We may thus have a daily , weekly , monthly  or yearly load factor ; it may apply to a single plan tor to a system. Some plants of a system may be run continuously at a high load factor , whereas variations in load are taken by other plants of the system , either hydro or steam. Hydro plants designed to take such variations must have sufficient regulating storage to enable them to operate on a low load factor. They are often called peak load plants. Operating on a 50 percent load factor , there must be sufficient storage to enable such a plant , in effect , to utilize twice the inflow for half the time : on a 25 percent load factor , the plant should be able to utilize four times the inflow for a quarter of the time , etc. – the lower the load factor , the greater the storage required.

The capacity factor is a measure of the plant use. It is the ratio of the average load to plant capacity. It may be computed for a day , month year , or any other period of time. When the peak load just equals the plant capacity , the capacity factor and load factor are obviously the same. If the maximum demand is less than the plant capacity , the capacity factor may be either greater or less than the load factor depending largely on the load factor itself.

The period usually considered is a month for purposes of billing , although the sale rates of power are often based on the yearly load factor of the consumer.

Firm Power : Power intended to have assured availability to the customer to meet his load requirements.

Primary Energy : Hydroelectric energy which is available from continuous power.

Secondary Energy : All hydroelectric energy other than primary energy.

Surplus System Capacity : The difference between assured capacity and the system peak load for a specified period.

A typical pumped-storage plant (round-trip efficiency now 80%) is a net consumer of energy: it returns approximately 3 kilowatt-hours (kWh) of electricity for each 4 kWh required for pumping. However, it offers the following important benefits:

• The energy generated during peak periods has a higher monetary value than the energy required for pumping during off-peak periods;

• It permits continuous operation of the highest efficiency plants in the utility's system;

• It provides rapid and flexible response to system load changes. Typically, very large load swings can be accommodated; and

• The utility's overall fuel consumption is reduced because the pumped-storage plant's on-peak generation  avoids or displaces generation at the least efficient thermal plants in the system.

Thanks for your question.  Both NYPA's Robert Moses and OPG's Sir Adam Beck are flagship power stations.  They've been studied and optimized for decades, so it is not surprising that their turbines can operate near 97% efficiency. The economics behind the scheduling and marketing of energy are complex, and I am sure NYPA strives to dispatch the plants to produce at peak efficiency.  But there are doubtless heavy load hours every year when it's beneficial to run turbines below their peak.  And, perhaps, one or more of their turbine runners are optimized to produce additional power, at a sacrifice of some efficiency, whereas others are optimized for peak efficiency, at a sacrifice of some power.

The overall plant efficiency includes unavoidable losses which come about from moving water through pipes toward the turbine (friction and momentum changes), conveying water away from the turbine downstream, electrical resistance losses in the generator, and stepping up electricity as needed to connect it to the grid.  I suspect the overall efficiency of the Robert Moses plant is more like 85% when all these factors are included.  But the figures 15% and lower must be referring to something other than overall plant efficiency.

And well designed pumped storage systems like these usually have a round-trip efficiency above 75%.  (My information is a little dated and I'd welcome a correction!)  By round-trip, think of a blob of water, then consider the electric energy consumed to pump it uphill then subtract the electric energy produced when it spins the turbine going downhill.

Anyway, in summary 85% efficiency is not uncommon from a hydro power station. The low-percentage figures you remember hearing must be referring to some other aspects.