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**GRID TOPOLOGY:**

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 an 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 entity can do.

In this respect the electricity system in each political jurisdiction should be able to dependably function independent of its neighbors. Each political jurisdiction should have sufficient dependable generation in excess of its peak uncontrolled electricity load and sufficient moment of inertia or its electronic equivalent that the jurisdiction can electrically function independent of other political jurisdictions. Thus the total electricity system should consist of a collection of micro grids, one per political jurisdiction, each of which is coupled to its neighbors via power 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 technically informed body within that microgrid. There should be enough self excited reserve generation within each jurisdiction that its micro grids are stable and can reliably operate with any combination of its external power interties disconnected.

**LEGAL REGULATION:**

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 and stability requirements 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 sophisticated 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. The practical political problems entailed in correcting such faulty legislation are immense.

Politicians must grasp that grid black start generally requires a significant fraction of large stable central generation. Electricity rate structures that make large central generation plants financially insolvent are a risk to all electricity consumers that rely on reliable grid supplied electricity.

**FREQUENCY AND POWER STABILITY:**

The issue of electricity system stability is deeply rooted in math and physics and is seldom understood by the general public. Electricity system stability entails more than just ability to meet the steady state peak power load.

In an electricity system a step change in total load is sensed via a change in electricity angular frequency W. This change in W causes a change in the system stored energy (I W^2 / 2), where I is the system equivalent rotating moment of inertia. This change in system stored energy manifests itself as an energy transient superimposed on the load step. Limiting the system frequency and voltage excursion resulting from a load step requires a sufficient value of equivalent rotating moment of inertia I. Historically the main source of I was rotating generator mass. Some loads contribute to I but at any particular time those loads may or may not be present. Suitably designed voltage source static inverters can also contribute to I. However, such voltage source inverters require switching and energy storage components with very high transient power ratings.

Politicians have reduced apparent cost of renewable generation by allowing wind and solar generators to use less expensive current cource inverters that do not contribute to system equivalent moment of inertia I. Hence the larger the fraction of system power supplied by such renewable generation the less is the ratio of I to peak power and the more unstable a grid becomes. If a grid has to rely on other grids for stability, then it cannot dependably function on its own. Moreover, an unstable grid degrades other grids to which it is connected. The necessity for providing additional equivalent rotating moment of inertia in grids with large fractions of connected wind and solar generation is becoming increasingly apparent.

A physical analogy is to think of trying to run a railway train using uniform radius wheels rather than wheels with flanges. If the track was perfectly straight and if the wheel radius was perfectly uniform the train could theoretically run along the tracks without derailing. However, in the real world neither the tracks nor the wheels are perfect. Wheel flanges are necessary to keep railway cars on railroad tracks. The more imperfect the railroad tracks the larger the required wheel flanges.

The role of equivalent moment of inertia in an electricity system is similar to the role of the wheel flanges on railway cars. Sufficient equivalent moment of inertia is required to realize essential electricity system stability. Absent that stability on the occurrence of a sufficient load step the electricity grid will have a frequency or voltage excursion sufficient to trip a safety device and the grid will black out.

**INTERTIES:**

Power transfers along interties must be mutually agreed to by both of the directly affected parties. A 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. This issue constrains wheeling of power between non-adjacent political jurisdictions.

A micro grid should be phase synchronized to its neighbours via only one power 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, stability and black start issues within that jurisdiction. If a particular jurisdiction becomes a problem to its neighbors, as indicated by excessive power flows or transients 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 power wheeling, inter-state, inter-provincial and international commerce.

**COMPLICATIONS:**

**Politicians ignore the issues of sufficiency of dependable generation, grid stability and grid black start at their peril.** The issue of sufficiency of dependable generation is comparatively easy for non-technical persons to understand. However, the issues of grid stability and grid black start are equally important but are far more difficult for non-specialists to comprehend. In jurisdictions where there is a high penetration of wind and solar generation with AC line connected current source inverters electricity grid instability is an increasingly serious problem. It manifests itself by safety trips that can potentially lead to cascade blackouts. Recovery from cascade blackouts becomes increasingly difficult as the fraction of intermittent generation increases and as the ratio of I to peak power decreases.

**SUFFICIENCY OF DEPENDABLE GENERATION:**

If the generation within a micro grid cannot meet that micro grid's own load and its neighboring microgrids 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. In large microgrids, such as the Province of Ontario, a full blackout restoration sequence can take days to implement. One of the issues that all parties must face is the practical necessity of maintaining at least 15% reserve power generation capacity above and beyond the historical peak load within each microgrid. A related issue is maintaining dependable spinning reserve equal to the largest single dependable generation unit. This issue is particularly important at times when there is little available intermittent electricity generation.

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 without matching dependable balancing generation and without sufficient rotating moment of inertia must stop. There are very real costs related to providing dependable generation when needed to balance intermittent renewable generation. There are very real costs related to providing extra moment of inertia to compensate for use of current source inverters. The more unconstrained renewable generation that is connected the more the required grid balancing generation costs increase. Grid balancing via intertie power flows is not acceptable to non-consenting neighboring jurisdictions.

If wind and solar generation are contemplated the costs of the required balancing generation, energy storage and related transmission must also be met. The monetary value of non-fossil electricity lies primarily in dependable power capacity and rotating moment of inertia, 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 energy storage system and connecting transmission. The cost per kW of that energy storage facility in combination with its related energy losses often far exceeds the cost per kW of a new nuclear power plant.

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 dependable grid supplied energy. Often the grid supplied energy appears more expensive because part or all of the cost of dependable and stable capacity is included in the cost per kWh of grid energy. The only solution to this misinformation issue is to have separate charges to consumers for peak demand (kW or kVA) and energy (kWh) that reflect the actual costs of marginal capacity and marginal energy. The sooner that regulatory bodies accept the necessity for these two separate charges, the better.

**GRID STABILITY:**

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 I of the system rotating mass. A self excited synchronous 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 output voltage is set to the desired level via rotor field adjustment.

Today it has become common practice to couple wind and solar generators to the grid using current source static power inverters. A voltage source static power inverter can be designed to emulate a mechanical generator. However, the power rating of a voltage source static power inverter that can emulate a mechanical synchronous generator in production of equivalent rotating moment of inertia is about double the inverter's continuous full load output rating, which makes voltage source power inverters expensive. Another issue with voltage source static power inverters is designing them so that power transients are proportionately shared over many such power inverters.

Reference: Big Wheels

**BLACK START:**

To minimize capital cost renewable generators usually use current source power inverters. However, current source power inverters are unable to emulate mechanical moment of inertia and require other generation to set the grid voltage and grid frequency and to provide black start capability. Thus use of current source inverters makes the electricity grid less stable against step changes in load and make black start much more difficult. A good rule of thumb is that at least 50% of the generation capacity in a micro grid should be self excited, should have stability equivalent to critically damped synchronous mechanical generation and should be capable of stand alone black start.

A major potential problem with natural gas fuelled generation is gas line pressure. Most natural gas fuelled combustion turbines require a natural gas injection pressure of about 120 psig (9 bar absolute) to operate. If the natural gas transmission line pressure is derived from electrically driven compressors and there is a total electricity blackout the natural gas combustion turbines cannot start. Black start of the electricity grid then relies on hydro-electric, coal, oil and nuclear generation. As coal and oil generation are phased out and nuclear generation capacity is not replaced grid black start capability is an issue of increasing concern.

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.

During grid black start there are complications relating to reflected power on unloaded transmission lines. In an unterminated transmission line due to resonance the voltage can head towards infinity. Generally the transmission line loading must be increased as the connected generation is increased. Otherwise the transient voltages caused by changes in reflected power will trip safety devices. The first generator connected to a transmission line needs to be sufficiently large to both energize the transmission line and absorb the subsequent step changes in reflected power that occur as the transmission line is loaded.

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

**TRANSIENT STEP RESPONSE:**

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

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 during stable operation 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 amplitude of the load power disturbance step.

Since:

Po = Plo

the generator shaft power balance equation gives:

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

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 = Tf where:

Tf >> To:

**dW / dT = 0**

Recall that:

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

implying that at T = Tf:

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

or

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

where Wf is the final value of W.

Hence:

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

**FIND SOLUTION TO 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**

Similarly:

**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:

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

or

F = N W / 2 Pi

or

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

**Assume a 100 kW generator**. To achieve reasonable frequency stability choose:

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

Thus:

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

= (100 kW / 5 Hz) (N / 2 Pi)

= 10 N kW / Pi Hz

Choose:

**dT = (1 s / 60)**

Then:

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)]}**

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 leads to frequency oscillations about F = Ff and power oscillations about P = Pf.

EXAMPLE - HEAVY OVER DAMPING:

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

= - (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

=

Then:

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 |

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 the system if there is a high penetration of renewable generation. Additional moment of inertia is also required 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.

EXAMPLE - OVER DAMPING:

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

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 to:

**I = (N^2 / 4 Pi) kg m^2**

corresponding to:

**G dT = 106.103385 / s**

giving:

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 to
**I = (N^2 / 8 Pi) kg m^2**

corresponding to:

**G dT = 212.20677 / s**

giving:

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 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 system plays an important role in damping potential frequency and power oscillations that result 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 grids via interties.

The above examples indicate that a 60 Hz power system with critical damping requires a rotating moment of inertia or its electronic equivalent of at least:

**(10 N^2 / 4 Pi) kg m^2 / MW of 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 or its electronic equivalent should be doubled to:

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

An issue that is seldom adequately appreciated is that in practical two bearing generation equipment the flywheel weight and hence the moment of inertia is limited by the sustained load bearing capacity of the bearings. If voltage source inverters are used in place of synchronous generators the switching component and battery ratings must be much larger than in comparable current source inverters.

**CONSEQUENCE OF INSUFFICIENT MOMENT OF INERTIA:**

If a grid has insufficient moment of inertia per unit of power generation capacity then sooner or later a load transient will cause a voltage and/or frequency excursion which is sufficiently large to trip a major safety device. That safety device trip will dump load which will increase the size of the load transient seen by the remaining grid connected safety devices.

If these remaining grid connected safety devices start a cascade of tripping there will be a propagating blackout. There is no reliable solution to this problem other than adding sufficient mechanical or electrical inertia to the grid, both of which are expensive. An alternative technical solution is to require that all renewable generation be fitted with voltage source inverters and appropriate controllers, regardless of the extra cost. However, communicating this complex technical requirement to regulators, lawyers and the public and then enforcing this requirement via the courts is not simple.

**REVENUE 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 dependable 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 consortium members must all be proportionately liable for meeting the commitment. If individual consortium nmembers are not able to provide performance bonds then the issue of joint and several liability arises.

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 and moment of inertia, for supplying their contracted energy, for supplying additonal firm energy, for suppling additional firm moment of inetia and for supplying extra interruptible energy. A generator might earn extra revenue by providing spot market power, spot market moment of inertia and spot market energy to meet shortfalls by other generators. A generator may also be paid a premium if it has capacity to support grid black start.

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 replacement 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. There may also be a moment of inertia limitation.

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

Typically the revenue of a dependable 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.

**HARMONICS**

An increasing fraction of electrical power is being sourced from power inverters that generate significant harmonics. A typical power inverter approximates a sine wave by six succesive voltage levels. In so doing it introduces significant harmonic power content. When this power passes through transformers the harmonics are attenuated more than the fundamental. Much of the harmonic energy is converted into heat. From an end user's perspective, if the purpose of the electricity is to smoothly drive a line synchronous AC electric motor, the harmonic energy has no value. Hence if a generator is paid in accordance with total energy delivered to the grid, and that energy has significant harmonic power content, then from the end user's perspective the generator has been over paid. Further, dissipation of harmonic energy in transformers requires derating the transformer, so from a transmission/distribution cost perspective the harmonic energy adds to the system cost. Hence the compensation paid to the generator should be further reduced to reflect the increased transmission cost as well as the reduced energy value delivered to the end user.

A fundamental question that needs resolution is "should an electricity customer pay for total energy absorbed from the grid as if that customer was a purely resistive load or should the customer only pay for energy provided at the fundamental frequency as if that customer had only an induction motor load?" Similarly, "should generators be paid for total energy delivered to the grid or just for energy delivered at the fundamental frequency?" A related issue is "Who is responsible for the extra costs of managing the harmonic energy?"

Unused harmonic energy contributes to reflected energy. Reflected energy is discussed on the web page titled: ELECTRICITY METERING.

This web page last updated October 30, 2021.

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