Home Energy Nuclear Electricity Climate Change Lighting Control Contacts Links



By Charles Rhodes, P.Eng., Ph.D.

Electricity systems are inherently government approved monopolies that are ultimately responsible to elected politicians. However, these politicians often mistakingly think that because they understand the law they can manage the complexity of the electricity system. Frequently engineers cannot prevent politicians enacting legislation or approving regulations or setting electricity rates or imposing expenditure constraints that degrade the electricity system. Hence the electricity system design should inherently confine and limit the damage that any single politician or political body can do.

In this respect the electricity system in each political jurisdiction should be able to operate independent of its neighbors. Each political jurisdiction should have sufficient reliable generation in excess of its peak uncontrolled electricity load that it can function independent of other political jurisdictions. Thus the electricity system should consist of a collection of micro grids, one per political jurisdiction, each of which is coupled to its neighbors via interties. A micro grid can be as large as the Canadian province of Ontario or as small as a college campus provided that all electricity rate and governance matters are decided by a single body within that microgrid. There should be enough self excited reserve generation within each jurisdiction that its micro grid is stable and ideally can operate with any combination of its external interties disconnected.

Each micro grid needs its own regulatory authority which is responsible for affairs within that micro grid. The regulatory authority for each micro grid should set electricity rates within that micro grid, independent of legislated constraints. This regulatory authority must be responsible to jurisdiction politicians, but it is foolish for politicians to attempt to micro-manage the regulatory authority, especially with respect to electricity rate and technical matters. An example of the problems that politicians can cause is Ontario's Global Adjustment, which due to poorly thought legislation is applied to kWh instead of peak kW. The misapplication of the Global Adjustment discourages use of behind the meter energy storage, unnecessarily increases the blended retail electricity price per kWh and causes unnecessary use of fossil fuels.

Power transfers along interties must be mutually agreed to by all of the directly affected parties. Any party that experiences a power transfer along one of its interties that goes out of the specified tolerable range must have the right to disconnect that intertie without notice.

A micro grid is phase synchronized to its neighbours via only one intertie. Other interties require either DC isolation or dedicated generation that is used to suppress loop currents between interties.

This grid architecture forces every jurisdiction to pay attention to generation reliability and stability issues within that jurisdiction. If a particular jurisdiction becomes a problem to its neighbors, as indicated by excessive power flows along its interties, then its neighbors can isolate the offending jurisdiction until the offending jurisdiction gets its act together.

It is important to have a common national or international technical authority that can be used to resolve intertie related disputes because these matters are too technical for resolution via normal legal processes. To enable such resolution all intertie specifications should be expressed in a common technical language.

One of the consequences of this grid architecture is that all parties to an energy transfer along an intertie must be in agreement. This issue can potentially affect inter-state, inter-provincial and international commerce.

Politicians ignore the issues of sufficiency of reliable generation and grid stability at their peril. The issue of sufficiency of reliable generation is comparatively easy for non-technical persons to understand. However, the issue of grid stability is equally important but is far more complex to comprehend.

If the generation within a micro grid cannot meet that micro grid's own load and its neighbors do not agree to supply additional energy when required then due to the law of conservation of energy that micro grid will have a blackout. One of the issues that all parties must face is the practical necessity for at least 15% surplus power generation capacity within each microgrid.

An issue that the parties must also face is that there is no free lunch. The fad for allowing intermittent renewable generation to run unconstrained into the electricity grid must stop. There are very real costs related to providing reliable generation when needed. The more unconstrained renewable generation is connected the more the required grid balancing costs increase. Grid balancing via intertie power flows is not acceptable to non-consenting neighboring jurisdictions.

If wind and solar generation are contemplated the cost of the required balancing energy storage and related transmission must also be met. The value of non-fossil electricity lies primarily in reliable capacity, not energy. This issue must be recognized in the governance of each micro grid. Failure to recognize this issue in electicity rates leads to low load factors which increase blended electricity costs per kWh.

To be reliable an intermittent renewable generator needs a very large battery or a large hydroelectric reservoir with connecting transmission. The cost per kW of that energy storage facility in combination with its energy losses often far exceeds the cost per kW of a new nuclear power station.

A blunt reality is that the market value of intermittent renewable generation without sufficient balancing energy storage is only about $0.02 / kWh. There is a vast amount of misinformation relating to the costs of wind and solar energy as compared to the cost of reliable grid supplied energy. Often the grid supplied energy appears more expensive because part or all of the cost of capacity is included in the cost of grid energy per kWh. The only solution to this misinformation issue is to have separate charges to consumers for peak demand (kW) and energy (kWh) that reflect the actual or anticipated costs of capacity and energy. The sooner that regulatory bodies accept the necesswity for these two separate charges, the better.

In addition to sufficiency of generation it is important to have grid stability in the presence of step changes in load. With mechanical synchronous generation grid stability is provided by the moment of inertia of the rotating mass. A self excited mechanical generator is usually controlled to provide a power output versus frequency with a slight negative slope (droop). In a multi-generator system usually only one large generator (the master generator) has frequency error integrating feedback which causes the system to achieve a line frequency of exactly 60 Hz. The other generators act as slave generators. Fixed output dispatched generation is used to follow major load changes. When the line frequency is exactly 60 Hz the generator voltage is set to the desired level via rotor field adjustment.

Today it is common practice to couple wind and solar generation to the grid using an inverter. A voltage source inverter can be designed to emulate a mechanical generator. However, the power output capacity of a voltage source inverter that can emulate a mechanical synchronous generator is about double the continuous full load output rating, which makes this inverter expensive. A further issue with voltage source inverters is designing them so that power transients are proportionately shared over many such inverters.

To avoid these expense issues it has been common practice by renewable generators to use current source inverters. However, current source inverters are unable to emulate moment of inertia and require other generation to set the grid voltage and frequency and to provide black start capability. Thus use of current source inverters makes the grid less stable against step changes in load. A good rule of thumb is that at least 50% of the generation capacity in a micro grid must be self excited, must have stability equivalent to over damped synchronous mechanical generation and must be capable of stand alone black start.

Black starting is generally best done using an electronic frequency and phase reference to bring each generator close to the desired frequency and phase before attempting to synchronize to the micro grid.

After a micro grid is operating as an island that micro grid can be synchronized to its neighbors. A micro grid should not rely on an intertie to black start.

To understand grid response issues to step changes in load it is helpful to examine the power balance equation for a mechanical synchronous generator.

The kinetic energy Ek contained in the rotating moment of inertia I of the generator is:
Ek = (I W^2) / 2
I = moment of inertia
W = shaft angular frequency

Hence differentiating with respect to time T gives:
dEk / dT = I W (dW / dT)

Assume a slave prime mover feedback control function of the form:
Ps = Po - Ka(W - Wo)
Wo = shaft angular frequency setpoint
Ps = prime mover source power
Po = prime mover source power when W = Wo
Ka = feedback function constant

Then power balance on the generator shaft gives::
Ps = Pl + I W (dW / dT)
Pl = load power

Note that when:
Ps = Pl
dW / dT = 0

The generator shaft power balance equation gives:
Po - Ka(W - Wo) = Pl + I W (dW / dT)

At prior stable operating conditions:
Po = Plo
W = Wo
dW / dT = 0

Now assume that at T = To there is a step increase in load Pl from Plo to (Plo + D), where D is the amplitude of the power disturbance.

Po = Plo
the generator shaft power balance equation gives:
- Ka(W - Wo) = D + I W (dW / dT)
W (dW / dT) + (Ka / I) (W - Wo) + [(D / I)] = 0

At the instant when the step increase in load is applied:
W = Wo
Wo (dW / dT) + (D / I) = 0
(dW / dT) = [- D / (I Wo)]

For a stable solution at T = Tf where:
T >> To:
dW / dT = 0

Recall that:
W (dW / dT) + (Ka / I) (W - Wo) + [(D / I)] = 0
implying that at T = Tf:
(Ka / I) (W - Wo) + [(D / I)] = 0
(Wf - Wo) = - (D / Ka)
where Wf is the final value of W.

Wf = Wo - (D / Ka)

W (dW / dT) + [(Ka / I) (W - Wo)] + [(D / I)] = 0
that conforms to both the intial and final conditions.

Integrating the differential equation from T = To to T = T gives:
W^2 - Wo^2 + [Integral from T = To to T= T of:
{[(Ka / I) (W - Wo) dT] + [(D / I) dT}]} = 0
W^2 - Wo^2 + [Integral from T = To to T= T of:
{[(Ka / I) (W - Wo) dT] + [(Ka / I)(D / Ka) dT}]} = 0
W^2 - Wo^2 + [Integral from T = To to T= T of:
{(Ka / I) [W - Wo + (D / Ka)] dT}] = 0

Recall that:
Wf = Wo - (D / Ka)

W^2 - Wo^2 + [Integral from T = To to T= T of:
{(Ka / I) (W - Wf) dT}] = 0

The shaft angular frequency W is related to the AC line frequency F by the equation:
W = 2 Pi F / N
N = 1 for a 3600 RPM generator;
N = 2 for a 1800 RPM generator;
N = 4 for a 900 RPM generator

Wo = 2 Pi Fo / N
Wf = 2 Pi Ff / N

Hence the differential eequation becomes:
(2 Pi / N)^2 F^2 - (2 Pi / N)^2 Fo^2 + [Integral from T = To to T= T of {(Ka / I) ((2 Pi / N) F - (2 Pi / N) Ff) dT}] = 0
F^2 - Fo^2 + [Integral from T = To to T= T of {(Ka N / 2 Pi I) ( F - Ff) dT}] = 0

G = (Ka N / 2 Pi I)
which gives:
F^2 - Fo^2 + [Integral from T = To to T= T of {G dT ( F - Ff)}] = 0

In this equation:
T = time;
To = time at which a step change in load is applied;
Tf = value of T after disturbance has settled down;
F = AC line frequency as a function of time T;
Fo = (Wo N / 2 Pi) = AC line frequency setpoint;
Ff = Wf N / 2 Pi = line frequency after disturbance has settled down;
I = generator moment of inertia;
Po = prime mover power at T = To;
Pf = prime mover power at T = Tf;
Ka = feedback constant:
Ka = - (dPs / dW)
= - (dPs / dF) (dF / dW);

Recall that:
W = 2 Pi F / N
F = N W / 2 Pi
dF / dW = N / 2 Pi

Assume a 100 kW generator. To achieve reasonable frequency stability choose:
(dPs / dF) = - (100 kW / 5 Hz)

Ka = - (dPs / dF) (dF / dW)
= (100 kW / 5 Hz) (N / 2 Pi)
= 10 N kW / Pi Hz

dT = (1 s / 60)

G dT = (Ka N / 2 Pi I) dT
= [10 N kW/ Pi Hz] [N / 2 Pi I] [1 s / 60]
= [5 (N / Pi)^2 kW / Hz] [1 s / 60 I]

Express the differential equation in the form:
F^2 = Fo^2 - [Integral from T = To to T = T of {G dT ( F - Ff)}]

This differential equation can be numerically solved by iteration starting at F = Fo.

F^2 = Fo^2 - [Integral from T = To to T= T of {G dT (F - Ff)}]

F1^2 = Fo^2 - [G dT (Fo - Ff)]

F1 = {Fo^2 - [G dT (Fo - Ff)]}^0.5

(F2)^2 = Fo^2 - [G dT (Fo - Ff)] - [G dT (F1 - Ff)]

F2 = {Fo^2 - [G dT (Fo - Ff)] - [G dT (F1 - Ff)]}^0.5

In general:
(Fn)^2 = {Fo^2 - Sum i = 0 to i = n-1 of [G dT (Fi - Ff)]}
Fo = Fo
F(i+1) = {Fo^2 - Sum i = 0 to i = i of [G dT (Fi - Ff)]}^0.5

If the system is insufficiently damped this relationship leads to frequency oscillations about F = Ff and power oscillations about P = Pf.

Assume a moment of inertia of:
I = (N^2 / Pi) kg m^2

Assume a feedback control system that keeps the frequency in the range 57.5 Hz at full load to 62.5 Hz at no load. At T = To the load jumps from (1 / 2) load (50 kW at 60 Hz) to (3 / 4) load (75 kW at 58.75 Hz).
Fo = 60 Hz
Ff = 58.75 Hz
(2 Pi I) = 2 kg m^2
dT = (1 / 60) s

G dT = (Ka N / 2 Pi I) dT
= - (dPs / dF) (dF / dW) (N / 2 Pi I) dT
= (100 kW / 5 Hz) (N / 2 Pi) (N / 2 Pi I)(1 s/ 60)
= (20 kW / Hz) (1 Hz s) (N^2 / 4 Pi^2)(1 s / 60) (1 / I)
= [(1000 W / kW)(1 kW / 12 Hz)(1 Hz s)(N^2 / Pi^2)(1 s / I)]
= 8.443446234 N^2 W s^2 / I
= 26.5258462 / s

i Fi(Fi - Ff)[G dT (Fi - Ff)]Sum i = 0 to i = i of [G dT(Fi - Ff)]   {Fo^2 - Sum i = 0 to i = i of [G dT (Fi - Ff)]}^0.5
060 Hz1.25 Hz33.1573 s^-233.1573 s^-259.7230 Hz
159.7230 Hz0.97305 Hz25.81097 s^-258.9683 s^-259.5065 Hz
259.5065 Hz0.756568 Hz20.06862 s^-279.0369 s^-259.3377 Hz
359.3377 Hz0.5877 Hz15.5893 s^-294.6262 s^-259.2062 Hz
459.2062 Hz0.4562 Hz12.1010 s^-2106.7272 s^-259.1039 Hz
559.1039 Hz0.3539 Hz9.3879 s^-2116.1151 s^-259.0244 Hz
659.0244 Hz0.27444 Hz7.2798 s^-2123.3949 s^-258.9627 Hz
758.9627 Hz0.2127 Hz5.6432 s^-2129.0381 s^-258.9149 Hz
858.9149 Hz0.1649 Hz4.3733 s^-2133.4114 s^-258.8777 Hz
958.8777 Hz0.1277 Hz3.3884 s^-2136.7999 s^-258.84896 Hz
1058.84896 Hz0.09896 Hz2.6250 s^-2139.4249 s^-258.8266 Hz
1158.8266 Hz0.07665 Hz2.0333 s^-2141.4582 s^-258.8094 Hz
1258.8094 Hz0.05936 Hz1.5748 s^-2143.0330 s^-258.7960 Hz
The above table shows a system that is heavily over damped. This system takes:
12 X (1 s / 60) = (1 / 5) second
to give a
[(60 Hz - 58.796 Hz) / 1.25 Hz] X 100% = 96.32%
step response. However, if a power system has to accommodate a very high penetration of intermittent renewable generation using current source inverters this slow step response should be anticipated. It may be necessary to add additional spinning synchronous moment of inertia to an existing power system to stabilize it if there is a very high penetration of renewable generation. This amount of damping is also used in emergency power systems where ability to supply the inrush current related to on/off switching of elevators and fire pumps is more important than good frequency and power regulation.

Now try reducing I by a factor of 2 to:
I = (N^2 / 2 Pi) kg m^2
corresponding to:
G dT = 53.0516925 / s

i Fi(Fi - Ff)[G dT (Fi - Ff)]Sum i = 0 to i = i of [G dT (Fi - Ff)]   {Fo^2 - Sum i = 0 to i = i of [G dT (Fi - Ff)]}^0.5
060 Hz1.25 Hz66.3146 s^-266.3146 s^-259.4448 Hz
159.4448 Hz0.6948 Hz36.8603 s^-2103.1749 s^-259.1340 Hz
259.1340 Hz0.3840 Hz20.3697 s^-2123.5446 s^-258.9615 Hz
358.9615 Hz0.2115 Hz11.2191 s^-2134.7637 s^-258.8663 Hz
458.8663 Hz0.11626 Hz6.1677 s^-2140.9314 s^-258.8138 Hz
558.8138 Hz0.0638 Hz3.3872 s^-2144.3186 s^-258.7850 Hz
658.7850 Hz0.0350 Hz1.8591 s^-2146.1777 s^-258.7692 Hz
758.7692 Hz0.01923 Hz1.0201 s^-2147.1978 s^-258.7605 Hz
The above table shows the step response of an over damped system. This system takes more than twice as long to converge as does a critically damped system. However, this system provides margin to tolerate a substantial penetration of current source inverters. This step response is representative of real power systems that accommodate up to 50% renewable generation.

Now try reducing I by a further factor of 2 to:
I = (N^2 / 4 Pi) kg m^2
corresponding to:
G dT = 106.103385 / s
i Fi(Fi - Ff)[G dT (Fi - Ff)]Sum i = 0 to i = i of [G dT (Fi - Ff)]   {Fo^2 - Sum i = 0 to i = i of [G dT (Fi - Ff)]}^0.5
060 Hz1.25 Hz132.6292 s^-2132.6292 s^-258.8844 Hz
158.8844 Hz0.134384 Hz14.2586 s^-2146.8878 s^-258.7632 Hz
258.7632 Hz0.013187 Hz1.3992 s^-2148.287 s^-258.7513 Hz
358.7513 Hz0.00128 Hz0.1359 s^-2148.4229 s^-258.7501 Hz
The above table shows the rapid step response of a critically damped system. It converges more than twice as fast as the over damped system and more than four times as fast as the heavily over damped system. However, a critically damped system has little stability margin to accommodate renewable generation coupled using current source inverters.

Now try reducing I by a further factor of 2 to I = (N^2 / 8 Pi) kg m^2
corresponding to:
G dT = 212.20677 / s
i Fi(Fi - Ff)[G (Fi - Ff)]Sum i = 0 to i = i of [G (Fi - Ff)]   {Fo^2 - Sum i = 0 to i = i of [G (Fi - Ff)]}^0.5
060 Hz1.25 Hz265.2585 s^-2265.2585 s^-257.7472 Hz
157.7472 Hz- 1.00278 Hz-212.7966 s^-252.4619 s^-259.5612 Hz
259.5612 Hz0.81121 Hz172.1449 s^-2224.6068 s^-258.0981 Hz
358.0981 Hz- 0.6519 Hz- 138.3303 s^-286.2764 s^-259.2767 Hz
459.2767 Hz0.5266 Hz111.7629 s^-2198.0393 s^-258.3263 Hz
558.3263 Hz- 0.4237 Hz- 89.9058 s^-2108.1335 s^-259.0920 Hz
659.0920 Hz0.3420 Hz72.5784 s^-2180.7119 s^-258.47 Hz
758.47 Hz- 0.2753 Hz- 58.4250 s^-2122.2869 s^-258.9721 Hz
858.9721 Hz0.2221 Hz47.1392 s^-2169.4261 s^-258.5711 Hz
958.5711 Hz- 0.1789 Hz-37.9635 s^-2131.4626 s^-258.8943 Hz
1058.8943 Hz0.14429 Hz30.6194 s^-2162.0820 s^-258.6337 Hz
1158.6337 Hz-.11624 Hz-24.6666 s^-2137.4154 s^-258.8437 Hz
1258.8437 Hz0.0937 Hz19.8902 s^-2157.3056 s^-258.6745 Hz
The above table shows damped frequency oscillation about Ff and corresponding power oscillation caused by insufficient damping. In a practical power system the moment of inertia I should be sufficient to prevent system oscillation.

The amplitude of the frequency change is set by the power disturbance D. The change in line frequncy F resulting from a step change in load power Pl of size D is expressed in terms of generation parameters I, N, Fo and feedback parameter Ka.

The moment of inertia I of the generator(s) plays an important role in damping potential frequency and power oscillation on the AC power grid that results from step changes in load. Each micro grid should have sufficient damping to attenuate both its own load disturbances and power disturbances that are imported from other micro grids via interties.

The above examples indicate that a 60 Hz power system with critical damping requires a moment of inertia at least:
(10 N^2 / 4 Pi) kg m^2 / MW of synchronous generation. If the system is to accommodate up to a 50% penetration of intermittent generation coupled using current source inverters the mechanical synchronous generation moment of inertia should be doubled to:
(20 N^2 / 4 Pi) kg m^2 / MW of synchronous generation .

A major issue with both retail electricity rates and compensating owners of non-fossil generation is that the value of non-fossil electricity lies primarily in reliable capacity rather than in energy.

Assume that over a year the projected load is a time dependent function of the form L(T), where L is measured in kW. Then generator i can potentially supply fraction Fi of the load by providing a net power capacity function of the form:
Pi(T) = 1.15 Fi L(T)
and energy:
E = Integral from T = January 1 to T = December 31 of:
Fi L(T) dT

If for any reason a generator cannot meet its power supply commitment that generator must meet its commitment via spot market electricity purchases.

A group of generators can form a consortium to bid on meeting the power requirement, but the consortim members must all be proportionately liable for meeting the commitment.

The successful generators can earn extra revenue by selling additional firm energy and interruptible energy.

The maximum possible additional firm energy per annum is:
[Sum of (monthly power capacity bids) 730.5 hours] - E

Generators should be paid monthly for meeting their contracted capacity, for supplying their contracted energy, for supplying additonal firm energy and for supplying extra interruptible energy. A generator might earn extra revenue by providing spot market power and spot market energy to meet shortfalls by other generators.

A generator bids to supply a minimum specified capacity for each month for 12 successive months. A generator that fails to meet its capacity bid must pay for replacement capacity purchased at the spot market price. The generator's capacity bid implicitly includes 15% extra capacity. The capacity bid implies availability of at least:
1.15 (730.5 kWh) = 840.075 kWh
of firm energy per bid capacity kW. The generator might be able to sell unpurchased kWh into the interruptible kWh market. If another generator fails to meet its commitments the generator might also be able to sell spot market capacity and energy. Note that in the annual peak month the cost of spot market capacity heads toward infinity.

The capacity that can be bid by an intermittent generator is generally limited by that generator's reliable energy storage capacity.

Most wind and solar generators will have energy outputs that far exceed their firm capacity ratings. This extra energy must either be constrained or sold in a local distribution market that does not require use of the public transmission grid.

Typically the revenue of a reliable generator is about:
[($50 / kW) + ($0.01 / kWh)].
However, absent behind the meter energy storage and an interruptible energy market a generator may only be able to sell:
(350 kWh / month) / kW
of bid monthly power capacity.

If a generator fails to meet its capacity bid it must pay for replacement capacity. The cost of that replacement capacity may be as much as 20X the normal value of the missing capacity. Hence intermittent generators must have a extremely high certainty about their energy supply and energy storage before bidding into the capacity market. Another way to view this situation is that absent sufficient energy storage the value of intermittent generation is limited to the value of the fossil fuels that it can displace.

This web page last updated March 27, 2017.

Home Energy Nuclear Electricity Climate Change Lighting Control Contacts Links