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**GRID ORGANIZATION:**

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 electricty 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 microgrid 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 is applied to kWh rather than kW, which distorts electricity rates so much as to affect the entire electricity system implementation.

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 neighbours 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.

**SUFFICIENCY OF GENERATION:**

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 that micro grid will have a blackout. One of the issues that all parties must face is the 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 energy storage, whatever it is, 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 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 exceeds the cost per kW of a new nuclear power station.

A blunt reality is that the market value of intermittent renewable generation without 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. The grid supplied energy appears more expensive because typically the cost of capacity is included in the grid energy cost per kWh. The only solution to this misinformation issue is to have separate charges to consumers for peak demand (kW) and energy (kWh). The sooner that regulatory bodies accept this reality, the better.

**GRID STABILITY:**

In addition to sufficiency of generation it is important to have grid stability in the face of step changes in load. With mechanical synchronous generation grid stability is provided by the moment of inertial 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 voltage is set to the desired level via rotor field adjustment.

Today it is commonm 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 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 issues expense it has been common practice by renewable generators to use current source inverters. An alternative to a synchronous generator is a voltage source inverter. 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 grid.

**TRANSIENT STEP RESPONSE:**

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

The kinetic energy Ek contained in the rotating moment of inertia I of the generator is:

Ek = (I W^2) / 2

where:

I = moment of inertia

and

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)

where:

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)

where:

Pl = load power

Note that when:

Ps = Pl

then:

dW / dT = 0

The generator shaft power balance equation gives:

Po - Ka(W - Wo) = Pl + I W (dW / dT)

**PRIOR STABLE CONDITION:**

At prior stable operating conditions:

Po = Plo

and

W = Wo

and

dW / dT = 0

**INITIAL CONDITION:**

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

Since:

Po = Plo

the generator shaft power balance equation gives:

- Ka(W - Wo) = D + I W (dW / dT)

or

**D + Ka(W - Wo) + I W (dW / dT) = 0**

or

W (dW / dT) + (Ka / I) (W - Wo) + [(D / I)] = 0

At the instant when the step increase in load is applied:

**W = Wo**

giving:

Wo (dW / dT) + (D / I) = 0

or

**(dW / dT) = [- D / (I Wo)]**

**FINAL CONDITION:**

For a stable solution at T >> To:

**dW / dT = 0**

Recall that:

W (dW / dT) + (Ka / I) (W - Wo) + [(D / I)] = 0
implying that:

(Ka / I) (W - Wo) + [(D / I)] = 0

or

**(Wf - Wo) = - (D / Ka)**

where Wf is the final value of W.

Hence:

**Wf = Wo - (D / Ka)**

FIND SOLUTION OF DIFFERENTIAL EQUATION

**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

or

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

or

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)

Hence:

**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

where:

**N = 1 for a 3600 RPM generator;
N = 2 for a 1800 RPM generator;
N = 4 for a 900 RPM generator**

Thus:

W = 2 Pi F / N

and

Wo = 2 Pi Fo / N

and

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

or

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

Define:

**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:

F = AC line frequency as a function of time T;

T = time;

To = initial time at which a step change ion load is applied;

Fo = (Wo N / 2 Pi) = AC line frequency setpoint;BR>
I = generator moment of inertia;

Ff = Wf N / 2 Pi = line frequency after disturbance has settled down;

Tf = value of T after disturbance has settled down;

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

or

F = N W / 2 Pi

or

**dF / dW = N / 2 Pi**

To achieve reasonable frequency stability choose:

(dPs / dF) = - (100 kW / 5 Hz)

Thus:

**Ka** = - (dPs / dF) (dF / dW)

= 100 kW N / 2 Pi (5 Hz)

= 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)}]**

The differential equation can be 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)]}**

where:

Fo = Fo

and

**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 can cause frequency oscillations about F = Ff and power oscillations about P = Pf.

EXAMPLE - HEAVY OVER DAMPING:

**G dT = 26.5258462 / s**

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) to (3 / 4) load (75 kW).

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

=

Hence:

**I** = 8.443446234 N^2 W s^2 / (26.5258462 / s)

= [8.443446234 N^2 W s^2 / (26.5258462 / s)] X (1 J / s W) X (1 kg m^2 / s^2 J)

= **0.31831 N^2 kg m^2**

= **(1 / Pi) N^2 kg m^2**

Then for ** (G dT) = 26.5258462 / s** and **I = (N^2 / Pi) kg m^2**

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 |
---|---|---|---|---|---|

0 | 60 Hz | 1.25 Hz | 33.1573 s^-2 | 33.1573 s^-2 | 59.7230 Hz |

1 | 59.7230 Hz | 0.97305 Hz | 25.81097 s^-2 | 58.9683 s^-2 | 59.5065 Hz |

2 | 59.5065 Hz | 0.756568 Hz | 20.06862 s^-2 | 79.0369 s^-2 | 59.3377 Hz |

3 | 59.3377 Hz | 0.5877 Hz | 15.5893 s^-2 | 94.6262 s^-2 | 59.2062 Hz |

4 | 59.2062 Hz | 0.4562 Hz | 12.1010 s^-2 | 106.7272 s^-2 | 59.1039 Hz |

5 | 59.1039 Hz | 0.3539 Hz | 9.3879 s^-2 | 116.1151 s^-2 | 59.0244 Hz |

6 | 59.0244 Hz | 0.27444 Hz | 7.2798 s^-2 | 123.3949 s^-2 | 58.9627 Hz |

7 | 58.9627 Hz | 0.2127 Hz | 5.6432 s^-2 | 129.0381 s^-2 | 58.9149 Hz |

8 | 58.9149 Hz | 0.1649 Hz | 4.3733 s^-2 | 133.4114 s^-2 | 58.8777 Hz |

9 | 58.8777 Hz | 0.1277 Hz | 3.3884 s^-2 | 136.7999 s^-2 | 58.84896 Hz |

10 | 58.84896 Hz | 0.09896 Hz | 2.6250 s^-2 | 139.4249 s^-2 | 58.8266 Hz |

11 | 58.8266 Hz | 0.07665 Hz | 2.0333 s^-2 | 141.4582 s^-2 | 58.8094 Hz |

12 | 58.8094 Hz | 0.05936 Hz | 1.5748 s^-2 | 143.0330 s^-2 | 58.7960 Hz |

EXAMPLE - OVER DAMPING:

Now try reducing I by a factor of 2.

Then for **G dT = 53.0516925 / s** and **I = (1 / 2 Pi) N^2 kg m^2**

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 |
---|---|---|---|---|---|

0 | 60 Hz | 1.25 Hz | 66.3146 s^-2 | 66.3146 s^-2 | 59.4448 Hz |

1 | 59.4448 Hz | 0.6948 Hz | 36.8603 s^-2 | 103.1749 s^-2 | 59.1340 Hz |

2 | 59.1340 Hz | 0.3840 Hz | 20.3697 s^-2 | 123.5446 s^-2 | 58.9615 Hz |

3 | 58.9615 Hz | 0.2115 Hz | 11.2191 s^-2 | 134.7637 s^-2 | 58.8663 Hz |

4 | 58.8663 Hz | 0.11626 Hz | 6.1677 s^-2 | 140.9314 s^-2 | 58.8138 Hz |

5 | 58.8138 Hz | 0.0638 Hz | 3.3872 s^-2 | 144.3186 s^-2 | 58.7850 Hz |

6 | 58.7850 Hz | 0.0350 Hz | 1.8591 s^-2 | 146.1777 s^-2 | 58.7692 Hz |

7 | 58.7692 Hz | 0.01923 Hz | 1.0201 s^-2 | 147.1978 s^-2 | 58.7605 Hz |

EXAMPLE - CRITICAL DAMPING:

Now try reducing I by a further factor of 2.

Then for **G dT = 106.103385 / s** and **I = (N^2 / 4 Pi) kg m^2**:

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 |
---|---|---|---|---|---|

0 | 60 Hz | 1.25 Hz | 132.6292 s^-2 | 132.6292 s^-2 | 58.8844 Hz |

1 | 58.8844 Hz | 0.134384 Hz | 14.2586 s^-2 | 146.8878 s^-2 | 58.7632 Hz |

2 | 58.7632 Hz | 0.013187 Hz | 1.3992 s^-2 | 148.287 s^-2 | 58.7513 Hz |

3 | 58.7513 Hz | 0.00128 Hz | 0.1359 s^-2 | 148.4229 s^-2 | 58.7501 Hz |

EXAMPLE - UNDER DAMPING:

Now try reducing I by a further factor of 2.

For **G dT = 212.20677 / s** and **I = (N^2 / 8 Pi) kg m^2**

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 |
---|---|---|---|---|---|

0 | 60 Hz | 1.25 Hz | 265.2585 s^-2 | 265.2585 s^-2 | 57.7472 Hz |

1 | 57.7472 Hz | - 1.00278 Hz | -212.7966 s^-2 | 52.4619 s^-2 | 59.5612 Hz |

2 | 59.5612 Hz | 0.81121 Hz | 172.1449 s^-2 | 224.6068 s^-2 | 58.0981 Hz |

3 | 58.0981 Hz | - 0.6519 Hz | - 138.3303 s^-2 | 86.2764 s^-2 | 59.2767 Hz |

4 | 59.2767 Hz | 0.5266 Hz | 111.7629 s^-2 | 198.0393 s^-2 | 58.3263 Hz |

5 | 58.3263 Hz | - 0.4237 Hz | - 89.9058 s^-2 | 108.1335 s^-2 | 59.0920 Hz |

6 | 59.0920 Hz | 0.3420 Hz | 72.5784 s^-2 | 180.7119 s^-2 | 58.47 Hz |

7 | 58.47 Hz | - 0.2753 Hz | - 58.4250 s^-2 | 122.2869 s^-2 | 58.9721 Hz |

8 | 58.9721 Hz | 0.2221 Hz | 47.1392 s^-2 | 169.4261 s^-2 | 58.5711 Hz |

9 | 58.5711 Hz | - 0.1789 Hz | -37.9635 s^-2 | 131.4626 s^-2 | 58.8943 Hz |

10 | 58.8943 Hz | 0.14429 Hz | 30.6194 s^-2 | 162.0820 s^-2 | 58.6337 Hz |

11 | 58.6337 Hz | -.11624 Hz | -24.6666 s^-2 | 137.4154 s^-2 | 58.8437 Hz |

12 | 58.8437 Hz | 0.0937 Hz | 19.8902 s^-2 | 157.3056 s^-2 | 58.6745 Hz |

**SUMMARY:**

The amplitude of the frequency changes 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 are expressed in terms of generation parameters I, Fo and feedback parameter Ka.

The moment of inertia I of the generator(s) plays an inportant role in dampening potential frequency and power oscillation on the AC power grid that results from step changes in load. Each microgrid 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 ought to have 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 50% intermittent generation coupled with current source inverters this moment of inertia should be doubled to:

(20 N^2 / 4 Pi) kg m^2 / MW of synchronous generation .

**VALUE OF NON-FOSSIL GENERATION:**

A major issue with both consumer metering 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 memebers must all be jointly and severally liable for meeting the commitment.

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

The 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 money by providing spot market power and spot market energy to meet shortfalls by other generators.

A generator bids to supply minimum specified capacity 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 includes 1.15 (730.5 kWh) = 840.075 kWh of firm energy per bid capacity kW. The generator might be able to sell addiitional 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 may be extremely high.

The capacity that can be bid by an intermittent generator is generally limited by that generator's 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 value of capacity to a generator is about [($50 / kW) + ($0.01 / kWh)]. However, absent behind the meter 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 very high degree of certainty about their energy supply and energy storage before bidding into the capacity market. Another way to view this situation is that absent energy storage the value of intermittent generation is limited to the value of fossil fuels that it can displace.

This web page last updated March 26, 2017.

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