Grid-connected power stations form a diverse interconnection of fossil, nuclear and renewable units whose objective is to meet the area’s power demands as safely, economically and securely as feasi­ble. On a continuous basis centralized management selects genera­tion from those available units best able to meet these objectives. Sinusoidal ac power at nominal frequencies of 50 or 60 Hz has many advantages from the viewpoints of generation, transmission and utilization [35]. Specifically, two-pole cylindrical alternators with water-cooled conductors provide the highest commercially available power generation per unit volume, but even these machines are limited to around 3000 or 3600 rpm which corresponds to 50 or 60 Hz respectively. With present units of between 100 and 660 MW this rotational speed is near optimum for steam turbine efficiency and blade reliability. Consequently a turbine and an alternator can be directly coupled together with a bolted flange to avoid the complexi­ties and inefficiencies of high-power gear trains. Electrical transmis­sion losses countrywide are reduced by the use of high voltage (e. g., 400 kV) to current ratios, but use in industrial and domestic situations requires relatively lower voltages (440 V or 230 V). For frequencies of 50 or 60 Hz, transformers provide a highly efficient (>96%) and reliable execution of this task.[38] However, as explained in the context of equation (1.62), high voltage dc is more cost-effective for cable transmission. A similar argument for dc transmission also applies to very long (~650km) overhead lines for which corona losses [147] increase with line voltage[39] and length.

Figure 1.1 illustrates the partially predictable seasonal and daily changes in the power demands on a Grid network. In addition, there are unforeseen material fluctuations induced for example by the start-up or shutdown of large industrial plant, or the substantial loss of a 400 kV Supergrid transmission line. Consequently, instantaneous electricity generation and demand cannot be identically matched by pure prediction, and thermal constraints also restrict each station’s rate of change of power.[40] As electricity cannot be stored in the required quantities, a mismatch between instantaneous Grid generation and demand must be accommodated in the short term by thermal energy stored in the coupled generating units and in the rotational energies of all Grid-connected generators and motors. Because the synchronizing torque per degree electrical of an ac machine is so large compared to its inertia [35], all directly connected ac machines can be considered for present purposes to be “locked” together at a synchronous speed V rad/s given by

Synchronous speed (V) (3.9)

= 2p x Grid frequency fG)/pole-pairs of a machine

where fG — Grid frequency (Hz).

Thus mismatches in Grid power appear throughout as a common frequency fluctuation about the nominal, and three principal reasons for its tight constraint now follow.

Firstly, each turbo-alternator is a multi-machine system of inertias linked by resilient shafts and therefore exhibit mechanical resonances at certain critical speeds [148] above and below the nominal 3000 or 3600 rpm. Unless the Grid frequency is controlled within narrow limits, these resonances could persist long enough to inflict serious damage. In practice during the Grid synchronization of a turbo-alternator these resonant speeds are accelerated through as quickly as possible.

Secondly, a large number of industrial and domestic consumers are still metered by electro-mechanical units which were installed by virtue of their good stability of calibration, wide measurement range and low cost of mass production [149]. Measurement accuracy with these single — and three-phase induction instruments depends on maintaining a 90° phase relationship between line voltage and the voltage coil’s magnetic flux. Though a degree of compensation is provided by “shaded poles,” measurement errors still occur when the frequency deviates from the calibration frequency of 50 or 60 Hz. Because the United Kingdom’s national electricity consumption is currently of the order of 350 TWh/year, even very small errors are fiscally significant.

Accordingly, UK consumers are protected by a Parliamentary Statute that requires the 24-h average deviation to be within ± 0.5 Hz, though National Grid plc self-imposes stricter limits of ± 0.2 Hz. Measure­ments [151] in 1972 characterized UK Grid fluctuations by a Normal distribution having a standard deviation of 0.05 Hz, which is consistent with the now continuously updated data on the Internet [150]. Safety trip limits for UK steam turbines impose operation between 48 and 52 Hz.

Finally, synchronous[41] or induction motors [35] are generally used to drive power station boiler feedpumps, whose pressure rise is approxi­mately proportional to the square of their rotational speed. When power demand exceeds generation, the Grid frequency and feedpump speed fall, so boiler pressures are reduced contrary to the required increased steam flow and alternator output. On the other hand, when power demand is less than generation, feedpump speed rises so boiler pressures increase, and the life expectancy of turbine blading in its low-pressure cylinder could thereby be prejudiced by a potential over­expansion of the steam [117]. A suitably controlled boiler inlet-pressure is therefore necessary, so an excess feedpump pressure must be devel­oped and the flow throttled to provide the required operating condi — tions.[42] Thermodynamics show that the steady-state power required for an incremental pump pressure change dP is

Pumping power = (W/hp)SP

where

W — mass flow rate (kg/s) h—pump efficiency p — water density (kg/m3)

The above conditions imply that a +1% frequency deviation corre­sponds to an extra pumping-power loss of about 1.6MW(e) for a 1200MW(e) station having an ac motor-driven feedpump [80].

The required control of Grid frequency is achieved by closely balancing instantaneous generated power with demand. For this pur­pose previous statistics as well as meteorological forecasts and mass entertainment data are involved to continuously predict demand so as to accommodate intrinsic plant start-up delays and thermal rate con­straints. In the United Kingdom, regional quotas are allocated on the basis of these predictions with consideration for plant outages and the overall security of supply. Accordingly, a Grid control region has more nominal capacity than historic demands. Individual stations are initially selected for operation by regional controllers in terms of a Merit Order based on fuel costs (£ per kWh) and reliability. This selection clearly favors relatively high capital but low fuel-cost nuclear stations, though these can no longer meet the minimum UK consumption. Thus contri­butions to the daily predicted load are required from fossil and renew­able plants. When wind turbines are available, they too appear as an economic option for this purpose. However, in a UK winter, extensive areas of meteorological high pressure would render a large number of turbines impotent, so that fossil-fired units are required for balancing during this season of greatest demand. As described in Section 1.7 combined cycle gas turbine (CCGT) plants have progressively replaced less thermally efficient and more polluting end-of-life coal-fired stations since 1991. Though these factors in part favor the selected operation of CCGT units, the principally coal-fired 3960 MW(e) Drax plant is usually operational by virtue of its high capacity factor (‘75%) and relatively high thermal efficiency11 (‘40%). Due to the absence of ponderous coal pulverizers and a more favorable fuel- combustion chemistry,[43] [44] CCGT generation has the additional advantage of faster dynamics for meeting major predicted and unscheduled load changes. Consequently a number of CCGT stations are operated at around 75% of full-load (a spinning reserve) or at almost zero output (hot starts) to meet these load changes. Currently the United Kingdom has access to some 650 MW(e) of auxiliary diesel or gas turbine units along with 1800 MW(e) from the Dinorwic pumped storage scheme having a 10 s access time [40], and 2000 MW(e) of rapidly disconnect — able load. With many individual stations maneuvering to effect an instantaneous Grid power balance there is clearly the problem of overall network stability to be addressed.

In this context, the energy stored in an inertia (I) rotating at V rad/s is

E = 1/2.i V2

so the rate of change of energy as it is delivered or withdrawn is

Power = E = IV (3.10)

dt

As described above, the speeds of all Grid-connected units can be considered locked together at the existing synchronous Grid frequency fG. Because Grid frequency is necessarily maintained within ±0.5 Hz about the nominal 50 or 60 Hz, equation (3.10) can be approximated by

Grid power perturbation d Power = KRTfG (3.11)

where RT is the sum of name-plate ratings for all synchronized

13

generators and motors. An allowance for some multi-pole pair units is accommodated by

0.2<K<0.4per VA of RT (3.12)

Coupled (or boiler-follows-turbine) stations buffer frequency fluctua­tions, but decoupled (or turbine-follows-boiler) stations contribute only to RT. Figure 3.6 illustrates stability considerations for a hypothetically isolated coupled controlled station. As a result of meeting physical rate constraints and a circumspect engineering design,[45] [46] its open loop frequency deviation to output-power transfer function can be approxi­mated by a SISO function H(s) which relates just the measured frequency deviation through the turbine control-valve dynamics to a release rate of stored energy in the plant. By means of comprehensive non-linear simulations, a set{H(iw) g can be derived for a representative number of load factors including synchronization. Simulations then confirm that the isolated station can deliver demanded power up to its
name plant rating, but the impact of its Grid connection on the stability of the overall network needs to be addressed as in Figure 3.7.

Intuitively it might seem that a parallel combination of individually stable isolated units would always be stable in the coupled-control mode, but theoretically this is untrue. Consider just two such stations with identical nameplate ratings R, but with different transfer functions H1(s) and H2(s). Equation (2.10) and Figure 3.7 yield the open loop Real Frequency Response of this hypothetical arrangement as

[H1(ia)+H2(ia)]/iaK2R = l/2[H1(im)/imKR + H2(im)/imKR]

If at some frequency V the individual station responses were complex conjugates of each other, then the open loop response function of their parallel combination would be

Re [H1(iv/iVKR] = Re [H2(iv)/iVKR]

I National Power Demand |

t— Hi do))

f set point

Power output change from Station К

Figure 3.7 Grid Network Stability Model

Figure 3.8 Two Stable Individual Stations; but Unstable in Parallel

Though each station is stable in isolation under coupled control, Figure 3.8 shows they could be unstable in a parallel combination. However, the depicted situation patently cannot arise if their open loop frequency responses do not cross the negative real axis. That is, the stations are each unconditionally stable.[47] Accordingly, the sufficient Grid stability criterion [80,117] devised by Butterfield et al. [150], is

“The Nyquist Diagrams for each station in conceptual isolation must imply unconditional and adequate stability at all output powers.”

Given a stable multi-station Grid network with totalized nameplate ratings RT, there is the academic question—what if a conditionally stable station with coupled control and a nameplate rating R were then to be synchronized? After connection the modified open loop response is derived from Equation 2.10 and Figure 3.7 as

— [RT (HT (iv) / itvKRj)+ R(H(iv)/ivKR)] (3.13)

Rt + R

where:

K

Ht (iv) = Hk (iv);

k=1

Hk(iv) — transfer function for kth coupled controlled station

and

H(iv) — transfer function for the additional coupled station

Because for the United Kingdom

20GW <RT < 60GW and R < 1GW then by defining

r = R/Rt < 1

equation (3.13) approximates to

(1 — r)[HT (iv) / ivKRT ] + r[H (iv) / ivKR]

Thus the connection of an additional coupled-controlled station affects stability margins by between 12/3 to 5%, which is negligible. If a decoupled station is synchronized, the open loop response of the Grid network becomes modified to

(1 — r’)[HT (iv)/ivKRT] with Г = R/(Rt + R)

which is again negligible.

These examples so far demonstrate the value of

i. a comprehensive non-linear simulation,

ii. engineering insight and experience,

iii. the utility of Real Frequency Response functions (Nyquist diagrams) in solving complex practical problems.

Though these “working functions” evolve as more tractable SISO systems, their proper formulation is rooted in MIMO system theory.

Finally, popular UK media often comment that some particular Grid-connected wind farm can supply a certain number of homes. Such

statistics often assume that all turbines are providing their rated maximum outputs and a daily average energy consumption of about 1V2 kW per household. Table 1.4 shows that the capacity factors of wind turbines is around 20%, so the predicted number of homes should at least be reduced by a factor of 5. During intervals in globally popular events like the World Cup, a large number of homes simultaneous brew tea or coffee and a typical electric kettle alone consumes 2 kW. In fact at the end of a 1970s Miss World competition UK power demand surged at 2 GW/min. As just described Grid operation necessitates a close instantaneous match between generated and demanded powers, and not daily averaged values. Such media statistics are therefore fallacious and suggest quite unrealistic contributions from wind energy. Adopting their same argument would suggest that 2 kW electric kettles could be properly fused on the basis of a daily averaged milliampere current.