Dynamic analysis of the impact of grid connection of "La Higuera" hydropower plant to the transmission grid

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Dynamic analysis of the impact of grid connection of "La Higuera" hydropower plant to the transmission grid

Isbi Felix

Master of Science Thesis

Stockholm, Sweden 2006

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Abstract

Studies regarding the development of a new hydropower plant exploiting the water resources offered by the upper Tinguiririca river, located about 150 km south-east of the capital Santiago in Chile, has been done by Pacific Hydro Ltd from Australia and Lahmeyer International from Germany. These studies have resulted in proposals to construct two Hydropower Stations, “La Higuera” and “Confluencia”. Both hydropower stations will have a total installed capacity of 300 MW.

When setting up a new hydropower plant, it is important to foresee how the hydropower plant would affect the existing transmission grid in different situations during operation as well as how events in the grid may affect the La Higuera and/or Confluencia hydropower stations. In this report three kind of analysis are highlighted, which are static analysis, large signal stability and rotor angle stability. To perform these analyses a simulation tool named DigSilent is used. DigSilent is used to perform these analyses in a simulated network of the studied transmission system.

These two hydropower stations as shown in the results will improve the existing transmission system by enhancing the stability margins in the presence of a fault. When performing the simulation of the existing transmission system with the newly installed hydropower plant we could see that it had a poor damping after a disturbance; this might be due to the large distance between production plants and the existing loads. This phenomenon can be alleviated if a power system stabilizer (PSS) is integrated in the hydropower plant.

The final conclusion is that the integration of the two hydropower plants will improve the existing transmission system in Chile.

Keywords

Electric Power System, Load flow, Voltage Stability, Hydropower, DigSilent

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Acknowledgements

First of all I would like to express my deepest gratitude for being given this opportunity by Gunnar Frostberg at SwedPower International AB to perform my Master Thesis abroad.

In the beginning it was difficult to perform a consulting job without contacts in a foreign country like Chile, but it turned out well thanks to the people that I learned to know. Specially thanks to, Ing. Anibal Ramos Romero who helped me through the design of the existing transmission system in the DigSilent environment, Ing.Hugo Tapia Muñoz for the given data and the people at the newly formed company Hidroeléctrica “La Higuera” for their hospitality and kindness.

Also I would like to express my gratitude to my supervisor Valerij Knyazkin and examiner Professor Mehrdad Ghandhari for the help and skilled guidance through the project.

Finally, thanks to my mother who has given me the strength and inspiration to become an engineer.

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1. Introduction

South America is facing important challenges in electricity supply to allow for future economic development. Current electricity market designs are being reviewed to avoid supply difficulties and couple the existing outlook of primary energy resources and the investment interest by the private sector. Examples of these developments are the giant Brazil, the economically troubled Argentina, and the pioneer of electricity reform, Chile. While Brazil and Chile progress into a second stage of reforms with public power purchase agreement auctions in a private environment, Argentina makes a backward these diverging approaches, their primary challenge is to ensure sufficient capacity and investment to reliably serve their growing economies.

1.1 Background

Chile has limited indigenous energy resources and relies on imports for most of its hydrocarbon needs. Crude oil comes primarily from Argentina, Ecuador, Nigeria, and Venezuela, and is processed by one of the country's three state-owned refineries.

Natural gas is imported via pipelines from Argentina. Coal is imported mainly from Australia. Chile imported 95% of its oil consumption, 80% gas and 92% coal. A significant amount of the country's electricity is supplied by domestic hydropower.

With respect to electricity supply, Chile can be divided in four areas, as follows. The map can be seen in the Appendix A.[1]

• The Northern Interconnected System (SING) which supplies the northern zone of the country, from Arica in the north to Antofagasta in the south. The distance between both locations is around 700 km. The installed capacity is 99% thermo-electric and reached 3,634 MW in December 2003. The energy generation is based on natural gas and coal. Some fuel-oil and diesel oil units exist in the system, but are not dispatched in normal conditions.

• The Central Interconnected System (SIC) which supplies the central zone of the country, from Taltal in the north to Quellón (in the Chiloé island) in the south. The distance between both locations is around 2100 km. In December 2003 the installed capacity reached 6,996 MW of which 58% was hydroelectric.

• The Aysén Electric System consisting of 5 isolated electric systems powered by small hydroelectric plants and diesel units. The total installed capacity reached 37 MW in December 2003.

• The Magallanes Electric System supplying the cities located in the Magallanes zone (Punta Arenas, Puerto Natales and Puerto Porvenir) powered by thermal generation based on natural gas. The total installed capacity reached 65 MW in December 2003.

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Table 1.1.1 Characteristics of the four mentioned systems.[1]

Installed Capacity System

MW % hydro

Generation GWh

SING 3634 0% 11424

SIC 6996 58% 33708

Aysén 37 42% 103

Magallanes 65 0% 187

Self Producers 684 10% 2836

Total 11416 38% 48244

As a result of Argentina’s energy policies following devaluation of its currency in December 2001, no investments in gas exploration and exploitation were made while domestic gas consumption increased sharply. In March 2004 when gas shortages occurred, the Argentine government breached the Gas Treaty with Chile and unilaterally curtailed exports to Chile.

On average the reduction has been about 30% since March 2004. This event combined with low inflow to the hydroelectric plants, spot prices in SIC increased sharply as can be seen from the next two graphs. The average energy spot price in 2003 was 16.8 USD/MWh, while the 2004 first semester average has been 42.0 USD/MWh.

Fig. 1.1.1 Daily Spot Prize for 2003 and 2004 in one node of the SIC[1]

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This new situation in Chile resulted in chaos during winter 2005, since the prices where still increasing, and the households where then aware of the power supply situation in Chile. The government realized that the Chilean Power System was in crisis and not as robust as they thought it was. To solve the problem the government has been forced to be more open to building new hydropower plants since the water flow is more constant than the government relation to Argentina. Regardless this situation the implementation of “La Higuera” was planned long before, but now it is on the spotlight since the hydro power plant will increase the power generation in the SIC with 155 MW at the beginning.

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2 Stability in the Power System

Stability of a power system refers to its capability of returning to normal operation following a disturbance, much as a sailing vessel rights itself after being heeled over by wind or wave.

System stability is largely determined by the behaviour of generators and their controls.

Oscillatory instability is typically reflected in uncontrollable oscillations of frequency and voltage leading to breakdown.

When such an occurrence creates an unbalance between the system generation and load, it results in the establishment of a new steady-state operating condition, with the subsequent adjustment of the voltages angles. The perturbation could be a major disturbance such as the loss of a generator previously mentioned, a fault or the loss of a line, or a combination of such events. Adjustment to the new operating condition is called the transient period. The system behaviour during this time is called the dynamic system performance, which is of concern in defining system stability. The dynamic behaviour of the generators within a power system is of fundamental importance to the overall quality of the power supply. The synchronous generator converts mechanical power to electrical power at a specific voltage and frequency (50 Hz). A dynamic phenomenon is initiated either by a disturbance of the system or by the variation of some system variable. The disturbance can originate either in the occurrence of some fault or from a switching action. Disturbances originating in some fault, like phase to ground faults or short circuits can cause a large change in the system’s state, which in the worst case that can render the system or parts of the system unstable.[5]

The main criterion for stability is that the synchronous machines maintain synchronism at the end of the transient period.

Definition: “If the oscillatory response of a power system during the transient period following a disturbance is damped and the system settles in a finite time to a new steady state operating condition, we say the system is stable. If the system is not stable, it is considered unstable.”[4]

According to the definition of stability, it requires that the system oscillations be damped.

This condition is sometimes called asymptotic stability and means that the system contains inherent forces that tend to reduce oscillations. Note that in some cases the system can go unstable with virtually no oscillations. For example “voltage collapse”.

When looking at a large power system with its numerous machines, lines, and loads and consider the complexity of the consequences of any impact, we may take the first step in a stability study, which is to make a mathematical model of the system during the transient. The elements included in the model are those affecting the acceleration of the machine rotors. The complexity of the model depends upon the type of transient and the components of the power system that influence the electrical and mechanical torques of the machines.

To define the power system stability technically it is necessary to find its equilibrium point which is the initial conditions. These are defined from the differential algebraic equations governing the dynamics of the power system, which defines the system in numerous variables which becomes the representation of the power system in a mathematical manner. This is more specifically explained in the next chapter.

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2.1 Definition of Power System Rotor Stability

The definition of power system rotor stability is explained briefly below, a more extended explanation is found in the ref [5].

2.1.1 Small-Signal Stability

Small-signal instability is a problem of insufficient damping of oscillations in a large power system, when it is subject to small disturbances. Such disturbances occur continually on the system because of small variations in loads and generation. The instability that may result can be of two forms, one steady increase in rotor angle due to lack of sufficient synchronizing torque, or rotor oscillations of increasing amplitude due to insufficient damping torque.

The nature of system response to small disturbances depends on a number of factors including the initial operating condition; the transmission system robustness, and the type of generator excitation controls used. For a radially connected generator to a large power system, in the absence of automatic voltage regulators the instability might occur due to lack of sufficient synchronizing torque.

This results in instability through a non-oscillatory mode. With continuously acting voltage regulators, the small-disturbance stability problem is one of ensuring sufficient damping of system oscillations, since instability is normally trough oscillations of increasing amplitude.

This type of stability can be shown in local modes, which are associated with the swinging of units at a generating station with respect to the rest of the power system or in inter-area modes which is associated with the swinging of many machines in one part of the system against machines in other parts, they are caused by two or more groups of closely coupled machines being interconnected by weak ties.

The behaviour of the studied dynamic system is described by the swing equation, which is a non-linear ordinary differential equation of the form described in the next chapter. The equilibrium points are those points where all the derivatives are simultaneously zero. In a nonlinear system there may be more than one equilibrium point.

The equilibrium points are in fact characteristic of the behaviour of the dynamic system, and therefore we can draw conclusion about stability from their nature.

Since we are dealing with a non-linear system, we have to linearize the swing equation around the equilibrium points of the studied power system in order to analyze the small-signal stability in the system. As the perturbation is assumed to be small, the nonlinear functions of the swing equation can be expressed in terms of Taylor’s series expansion. These can be studied in the next chapter.

2.1.2 Definition of Power System Transient Stability

Transient stability is defined as the ability of the power system to maintain in synchronism during a severe transient disturbance, such as fault on transmission facilities, loss of generation or loss of a large load. This means that the stability depends on the initial operating state and the severity of the disturbance.

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The disturbances that are considered in transient stability studies are either symmetric or asymmetric short circuits; these are occurred relatively often on the transmission lines.

The fault is often cleared by opening the appropriate circuit breaker to isolate the faulted element. To study this kind of stability the so called equal area criterion is used, this method helps in understanding basic factors that influence the transient stability of any system, but is not applicable to multi-machine systems with detailed representation of synchronous machines.

2.2 Definition of Power System Voltage Stability

The concept of voltage stability can be divided in two subcategories, namely Small and Large Disturbance Voltage Stability. This is done since the instability phenomena can occur in different time frames, from fractions of a second to tens of minutes.

A power system is said to be small-disturbance voltage stable if it is able to maintain voltages identical or close to the steady values when subjected to small disturbances. Consequently a power system that is large-disturbance voltage stable is able to maintain voltages identical or close to the steady state values when subjected to large perturbations.

A power system enters in a state of voltage instability as a disturbance-increase in load demand, or changes in system condition-causes a progressive and uncontrollable drop in voltage.

There are some factors that may cause voltage instability.

- One of the main factors causing instability is the inability of the power system to meet the demand of reactive power, the voltage drop occurs when active power and reactive power flow through inductive reactance associated with the transmission network.

- Overload on the transmission lines

- Voltage sources are too far from the loads centers - The source voltages are too low

- There is insufficient load reactive compensation.

A criterion for voltage stability is that, at a given operating condition for every bus in the system, the bus voltage magnitude increases as the reactive power injection at the same bus is increased. A system is voltage unstable if, for at least one bus in the system, the bus voltage magnitude decreases as the reactive power injection at the same bus is increased. [5]

A voltage stable power system is capable of maintaining the post-fault voltages near the pre- fault values. If a power system is unable to maintain the voltage within acceptable limits, the system undergoes voltage collapse, which may be total (blackout) or partial.

To analyze power system voltage stability we may consider the relationships between the transmitted power, receiving end voltage, and the reactive power injection. System dynamics influencing voltage stability are usually slow. Therefore, many aspects of the problem can be

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effectively analyzed by using static methods, which examine the viability of the equilibrium point represented by a specified operating condition of the power system.

The static analysis techniques allow examination of a wide range of system conditions, and can provide much insight into the nature of the problem and identify the key contributing factors.

Dynamic analysis is useful for detailed study of specific voltage collapse situations, coordination of protections and controls and testing of remedial measures. Dynamic simulations also examine whether and how the steady-state equilibrium point will be reached.

As mentioned active power and voltage curves (PV-curves) are useful for conceptual analysis of voltage stability and for study of radial systems. The method is also used for large meshed networks where P is the total load in an area and V is the voltage at a critical or representative bus. We can analyze at the same time the reactive power (Q) versus the voltage in a V-Q curve, V-Q curves plot voltage at a test or critical bus versus reactive power on the same bus.

The advantage of V-Q curves is closely related to reactive power, and a V-Q curve gives reactive power margin at the test bus. The reactive power margin is the MVAr distance from the operating point or either the bottom of the curve, to a point where the voltage squared characteristic of an applied capacitor is tangent to V-Q curve.

In the chapter 5 we will analyze the existing power system and some critical buses, and further on with their representative P-V-curves.

In the next chapter we will emphasize importance of the balance between generation and loads, and the most important components of a transmission system in the power system, such as: the synchronous machine, the two/tree winding transformer, the reactive power compensation by the capacitors, transmission lines, and finally the loads of the system. A brief review of these components as the applied theory is presented below.

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3 Applied Theory

3.1 Balance between generation and loads

If just looking at the power, there is power generation in one end and in the other end there are the consumers (loads). In between there are losses that also are mathematically treated as loads. The equation of generation equals load must be balanced at all times and the system needs constant power regulation to remain stable and within its safety margins.

3.2 Synchronous machine

The synchronous machine is the main component responsible for the generation of electric power. The synchronous machine converts mechanical energy to electricity. The mechanical to electric energy conversion takes place in an electric generator, usually of the three-phase synchronous type, and is based on Faraday’s induction law.[5]

A feature of the synchronous machine is that the power factor of the machine can be controlled; which means that the field current could control the amount of produced active and reactive power. The machine behaves as a variable inductor or capacitor as the field current changes.

When two or more synchronous machines are interconnected, the stator voltages and currents of all the machines must have the same frequency and the rotor mechanical speed of each is synchronized to this frequency. Therefore, the rotors of all interconnected synchronous machines must be in synchronism.

An unloaded synchronous machine is called a synchronous condenser and may be used to regulate the voltage in a part of the grid; there are two with such of description in the designed power system. This is just a brief explanation of a synchronous machine, more information can be found in [5].

The generating units that are modelled in DigSilent for the dynamic studies are shown in Appendix C and Appendix D.

3.3 Transmission System

Transmitting power over significant distances to consumers spread over a wide area requires a transmission system comprising subsystems operating at different voltages levels. Electric power is produced at generating units, and transmitted to the consumers trough a complex network of individual components, including transmission lines, transformers, and switching devices. The transmission system interconnects all major generating stations and main load centres in the system, the model used is explained more specifically in the next chapter.

The Chilean transmission system that is modelled in DigSilent for the dynamic studies is shown in the Appendix B.

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3.4 Transformer

A transformer is an electrical device that by electromagnetic induction transforms electric energy from one or more circuits to one or more other circuits at the same frequency. By varying magnetic relationships or values of the input versus the output, a transformer produces changed values of voltage and current.

One particular example is that of using transformers to step up voltages at a power station to a very high level before transmission, due to reduce power losses in transmission lines resistances. To regulate the voltage at the secondary side it is a common way to use tap changers in the winding, which gives a voltage change.

3.4.1 The two-winding Transformer

The definition above concludes in that the main and most common task of the transformer is to change the voltage from one level to another. Transformers used in most power systems are three-phase transformers. These transformers works in a similar way as the one-phase transformers, they are like three parallel single-phase transformers one in each phase.

In the primary winding the electric energy is converted to magnetic flux which is leaded to the secondary winding by the iron core. The secondary winding converts the magnetic flux to electric energy once again. In an ideal transformer, the amount of energy taken out form the secondary side matches the amount of energy injected in to the primary winding.

The law of induction also gives that the voltage at the secondary side terminals is proportional to the number of turns of the secondary winding. This fact can be seen in the equation below, which represents the connection between the number of turns at each winding and the voltages terminals. [8]

3.4.2 The three-winding Transformer

The 3-winding transformer is similar to a 2-winding transformer except that the secondary side has two sections and is quite often suitable for series or parallel connection. In our studied system a typical example would be a 500 kV primary with a secondary of 220/154 kV. "Windings" can be affected by putting taps at various places on one large winding or by making separate coils (either primary or secondary). This kind of transformer can be seen in the Appendix F specifically the Part2, Part4, and Part5.

3.5 Load Forecast

The load varies every single instant, so there are no exact loads forecast. What we can do is to assume a load quantity according to the season of the year e.g. summer or winter. In the performed simulation the loads are represented as impedances according to the given data from the CDEC-SIC in Chile.

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3.6 Load-flow calculation

When analyzing the steady state in a power system a load flow calculation is performed by the simulation tool DigSilent with the appropriate input data for the power system.

This mentioned tool solves the system equations using iterative procedures since the power system is a nonlinear problem. This procedure is known as Newton Raphson´s method.

The load-flow is preformed to see if the system voltages converge and also to find out if the transmission system remains within acceptable operating ranges or if they are exceeded. If the iterative procedure converges, then we have fulfilled a necessary condition for stability in the power system.

The purpose of the load-flow calculation is to find the voltage magnitude and phase angle at all buses. The power balance in each bus in the power system has to be satisfied according to Kirchhoff´s first law, which means that the generated power must be equal to the demanded power in each bus added to the power losses.

When these are known, the power flow on the lines is straight forward to calculate in accordance with the following equation:

Where:

To compute the power injected in the bus “i”, the separate powers Pin should be summed up as follows:

The real part of the equation gives the active power flow and the imaginary part gives the reactive power flow. This is shown in the equations above, where “i” and “n” are two connected nodes in the power system, “X” the total reactance of the line between nodes “m”

and “n”, “U” the voltage in each node and finally the angle “θ “at each node.

) sin (cos _,

2

in n

i n

i

in i j j

X U j U X

S =U ⋅ − ⋅ ⋅ θ + ⋅ θ

{ }

i n in in in

X U S U

P =Re = ⋅sinθ

{ }

i i n in in in

X U U X S U

Q Im cosθ

2 − ⋅

=

∑

=

= ^N

n in

i P

P

1

∑

=

= ^N

n in

i Q

Q

1

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3.7 State-Space Representation

The behaviour of a dynamic system, such as this analyzed power system, may be described by a set of “n” first order nonlinear ordinary differential equations of the following form [5] [7].

Assume that there are m synchronous generators in the system. The rotor dynamics of each generator can be described by the following equations:

s i i

dt

dδ ω ω

−

=

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ − −

= 1 sin( )

' '

i i di

i i mi i

X U P E

M dt

dω δ θ

Depending on the model used the state vector will be augmented by several other state variables that describe the dynamics of the extra pieces of equipment such as the exciter, the governer, PSS, etc

x=(δ,ω)T

The power system dynamics can be described as follows:

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ − −

−

=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

...

) 1 sin(

..

' ' 1 2

1

i i di

i i mi

s

k

X U P E

M

x

x x x

θ δ ω

Where:

- δ and ω stand for the rotor angle [rad] and angular frequency [rad/s].

- stand for the transient EMF(Electromotive force)[p.u]. reactance of the synchronous machine. M

'

Ei X_di^'

i moment of inertia [s²]. Pmi mechanical power [p.u].

- Ui andθ_istand for voltage magnitude and angle of the generation node.

The algebraic equations describing the power balance in the generator stator are as follows:

i i i di

i

i P

X U

E − +

−

= sin( )

0 _'

' δ θ

i i i di

i i di

i Q

X U E X

U − − +

−

= cos( )

0 _'

' '

2 δ θ

jx’di '

Ei U_i

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The following equations reflect the power balance in the network.

∑

=

−

− ⁿ

k i k ik i k ik

i

Li U UU Y

P

1

0 ) cos(

)

( θ θ α

∑

=

−

− ⁿ

k

ik k i ik k i i

Li U UU Y

Q

1

0 ) sin(

)

( θ θ α

Where the Yik and αik are defined as Y_ik =Y_ike^j^α^ik, with Y_ik being the (i,k)- element of the network admittance matrix. Since typically the resistances in transmission system can be neglected, we assume that αik = л/2 for all “i” and “k”.

The column vector ”x” is referred to as the state vector and its entries xi as the state variables.

In our case the dimension of this vector is m, where “m” is the number of generators in the system. To simplify notation, the derivative of the state variable “x” with respect to time is denoted by . x

Now let us define another new vector variable “y” defined as showed below:

) , (U_i _i

y= θ

with U_iand θ_idefined as above.

Finally the differential algebraic equations describing the system can be put in vector form:

x= f( yx, ) {1}

0=g(x,y) {2}

3.8 Linearization of the Power System

Linear analysis has proven to be an invaluable tool for the investigation of the dynamics of power systems. Linearization of power system equations must be done around an operating equilibrium point, which defines the steady state of the system. To find the equilibrium points it is necessary to equate equation{1}{2} simultaneously to zero

x = f(x_o,y₀)=0 and solve the nonlinear equations for the unknowns x_o y_o.

It should be noted that a linear system has only a unique equilibrium state, while a nonlinear system may have more than one equilibrium point.

To linearize the system add a small disturbance to the system to obtain:

x x

x= ₀ +∆ ; y= y₀ +∆y ; where the prefix ∆ denotes a small deviation.

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Taylor’s formula gives:

Ö ( , ) ...

0 0

. ∆ +

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

∂ + ∂

⎥⎦ ∆

⎢⎣ ⎤

⎡

∂ + ∂

=

∆

= =

y y x f x

y f x f x

y x y

x

...

) , ( 0

0 0

0

0 ∆ +

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

∂ + ∂

⎥⎦ ∆

⎢⎣ ⎤

⎡

∂ + ∂

=

= =

y y x g x

y g x g

y x y

x

The higher order terms of the Taylor series expansion are neglected.

Let us denote the various Jacobians as follows:

0 y y0

x

x y

D g x

C g

= ⎥⎥ =

⎦

⎤

⎢⎢

⎣

⎡

∂

= ∂

⎥⎦⎤

⎢⎣⎡

∂

= ∂

Therefore, the linearized system can be written in matrix form as follows.

y D x C

y B x A x

∆ +

∆

=

∆ +

∆

=

∆ 0

.

where A, B, C, and D are matrices of appropriate dimensions.

Since the equations above are now linear, the unknown algebraic equations can be eliminated as follows:

x A x C BD A

x= − ∆ = _new∆

∆ ^. ( ⁻¹ )

By taking the Laplace´s transform of the above equations, we obtain the state equations in the frequency domain.

0 ) det(sI −A_new =

The polynomial equation in the variable “s” and the values of “s” which satisfy the above are known as eigenvalues (λ) of matrix Anew, and it is referred as the characteristic equation of matrix Anew. These eigenvalues give us the characterization of the stability of the equilibrium points.

when:

0 y y0

x

x y

B f x

A f

= ⎥⎥ =

⎦

⎤

⎢⎢

⎣

⎡

∂

= ∂

⎥⎦⎤

⎢⎣⎡

∂

= ∂

- All the eigenvalues have negative real parts; the original system is asymptotically stable.

- At least one of the eigenvalues has a positive real part, the original system is unstable.

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- The eigenvalues have real parts equal to zero, it is not possible to define if it is stable or unstable

A real eigenvalue corresponds to a non-oscillatory mode. A negative real eigenvalue represents a decaying mode. The larger its magnitude is the faster the decay. A positive real eigenvalue represents aperiodic instability. Complex eigenvalues occur in conjugate pairs, and each pair corresponds to an oscillatory mode (complex eigenvalue).

The real part of the eigenvalues gives the damping, and imaginary component gives the frequency oscillation. Thus,

i i

i σ jω

λ = ±

The frequency of oscillation can be expressed in Hz as follows:

π ω 2

i

fi =

To define the rate of decay of the amplitude of the oscillation, we define the damping ratio, which is:

2 2

i i

i

i σ ω

ξ σ

+

= −

The eigenvalues of a matrix are given by the values of the scalar parameter λ, for which there exist a non-trivial solution (i.e., other than) φ_i to the equation.

i i i

Anewφ =λφ

Where Anew is the system state matrix andφ_i is a vector of appropriate dimensions. For any eigenvalue λi, the vector φ_iwhich satisfies the equation (A_new −λI)φ_i =0 is called the right eigenvector of Anew associated with the eigenvalue λi.

The right eigenvectors can be combined in the right eigenvector matrix as shown below:

[

φ₁ φ₂ ... φ_n

]

= Φ

Since the equation (A_new −λI)φ_i =0 is homogeneous, kφ_i is also a solution. Thus, the eigenvectors are determined only to within a scalar multiplier. Similarly, the vector ψi which satisfies:

i i new

iA λψ

ψ =

The equation above is called the left eigenvector associated with the eigenvalue λi. The left eigenvectors can be combined in the left eigenvector matrix as shown below:

[

ψ₁T ψT₂ ... ψ^Tn

]

^T

= Ψ

The left and right eigenvector will be used to explain the participation factor of the power system.

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3.9 Participation factor

To identify how each dynamic variable affects a given eigenvalue an aid called participation factor analysis is used. Due to the large size of the Chilean Power system, it was necessary to only retain a few modes (complex frequencies) to perform the stability studies; these can be seen in chapter 6.

The participation factor is a tool for identifying the state variables that have significant participation in a selected mode. The result is given by a graph were we can see the different generating units oscillate against each other (see chapter 6)

The participation matrix “P” combines the right and left eigenvectors as follows.

] ...

[p₁ p₂ p_n P =

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

Ψ Φ

=

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

=

in ni

i i

ni i i

i

p p p

p ² ²

1 1 2

1

...

The element p_ki =Φ_kiΨ_ik is termed the participation factor. It is a measure of the relative participation of the kth state variable in the ith mode, and vice versa. Since measures the activity of x

Φki

k in the ith mode and weighs the contribution of this activity to the mode, the product measures the total participation. [5]

Ψik

pki

Determining the dynamic behaviour of the analyzed power system with more than one generator is a difficult task, than describing the dynamic behaviour of a system with only one generator, since the electric power system is a nonlinear system. Not only is the model large- scale, it is also nonlinear, which makes analytical stability studies infeasible. Therefore, one normally resorts to numerical studies, which are greatly facilitated by specialized software, such as DigSilent

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4 Inspection and description of the existing power system

The participating power generation companies of the SIC must be able to supply their power demand every year. For this purpose, they have to consider potential dry-hydrology conditions in their hydroelectric Power Plants and the average availability in their thermal units. (see Appendix C) Every year the SIC checks that its integrating companies are indeed able to assure the demand supply of their customers.

In order to determine the generating capacity of a self-generating company, and the supplies of other generating entities, that operate synchronized with the system and whose partial or total output has been assured through a signed agreement at a freely agreed price, and also to determine the power demand to be taken into consideration.

As an input of the firm capacity of the hydroelectric Power Plants for dry hydrological conditions (see Appendix D), generation is defined according to hydrology of the year with hydrologic statistic potential surplus close to 90%. For thermoelectric Power Plants, the maximum annual power that they can generate as an average is considered, taking into account failures and maintenance periods. To inspect the existing power system we use a reduce network model in the environment of the tool DigSilent, which is described in the next chapter.

4.1 Reduced Network Model

The reason why using a reduced network model in these analyzes is in order to highlight the most important involved parts in the Chilean transmission network. A model of the Chilean transmission network that was suggested from the CDEC-SIC is implemented in DigSilent (see next chapter which describes briefly the software DigSilent).

The implemented network includes the highest operating voltage levels in the country, 500 kV, 220 kV and 154 kV. The initial conditions for dynamic simulations are taken from the results of the power flow in the system, with a given typical scenario of generation and demand. The parts of the power system with low voltages levels are modelled as loads.

The calculation of the equivalent system is done from the results of a power flow that includes all voltage levels and the typical scenario of generation. The estimation of the loads at the nodes (substations) is found by simply considering that the generated power should be equal to the demanded power; if this relationship is not fulfilled, then either a generation or consumption of power exists in that substation. Following this procedure lower voltage levels are eliminated from the network.

Of course, the reduction of the network to high and extra high voltage exclusively does not include the generators; however there is a reduction in the number of them. Synchronous generators that are connected directly to either 220 kV or 154 kV through a step up transformer make part of the equivalent network, but if they are connected to lower voltage levels, they are not considered and make part of the calculation of the load. The behaviour of the network with those three voltage levels should give a good resemblance of the behaviour of the interconnected transmission system.

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5 Introduction to DigSilent Power Factory

The development of DigSilent (DIgital SImuLator for Electrical NeTwork) software began in 1976, since the inception of DIgSILENT, the program has grown to incorporate a vast array of analysis features that are required to plan, operate and maintain any aspect of the power system.

The voltage stability analysis performance of a power system in DigSilent is represented in P- V curves of the selected buses and loads. To compensate reactive power, shunt capacitors are used. These can be effectively used up to a certain point to extend the voltage stability limits by correcting the receiving and power factor.

To perform the different calculations it is first necessary to build the power system in DigSilent environment. The reduced network model of the Chilean transmission system is build according to the following structure process.

1. Design of the buses in the power system that will be analyzed.

2. Definition of the designed buses as the type and locate the data at the indicated fields.

3. Design of the lines/cables of the transmission system,

4. Define the lines/cables type and locate the data at the indicated places.

5. Design of the three winding transformators

6. Define the data of the three winding transformators 7. Design of the generators and loads at the buses

8. Define the designed generators type and locate the data at the indicated places.

9. Define the designed loads type and locate the data at the indicated places 10. Run a loadflow to verify that the power system actually converges.

The designed power system in the DigSilent environment is shown in Appendix F.

The whole process of performing a transient simulation in DigSilent typically takes the following steps:

1. Calculation of Initial Values, which includes a load-flow calculation.

2. Defining result variables and/or simulation events.

3. Optionally defining result graphs.

4. Run Simulation

5. Creating additional result graphs

6. Changing settings, repeating calculations.

7. Printing results.

Designing one power plant, and specifically “La Higuera” the models mentioned below where used. The graph of the model is followed. This design must be performed for each one of the generating units, these to se the dynamical behaviour of the power system.

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g

curex

u

uerrs

speed

pt

PMU

* PCO

*

Syn.Generator ElmSym*

0

1 0

1

2 VCO

upss * 0 1 2

Complete Frame:

DIgSILENT

In DigSilent the model of the generating unit is used, which has been described in the chapter 3.6.1. The specific data for each generator is placed in the fields that are required when modelling each generator in the environment of DigSilent. To model each power plant, the followed models are used for each part of the power plant.

Fig 5.1 Complete model frame of a power plant

- The voltage controller is of the type vcoBBSEX1 which is a typical IEEE model of type AC1. This can be seen in the Appendix E fig 9.5.1.

- The model of the hydro governor of a hydropower plant according to the IEEE model (pcu_HYGOV). This can be seen in the Appendix E fig 9.5.2.

- The model of the turbine of a hydropower plant is modelled according to the build-in model in DigSilent. This can be seen in the Appendix E fig 9.5.3.

The angular speed (w) is regulated by the PCO which is the hydro governor, trough the hydro governor the opening of the gate is regulated (g). The water though the gate generates mechanical power, which is the output signal from the hydro turbine. The mechanical power is now the input signal for the synchronous generator, which generates electrical power and has as output signals the current, voltage and voltage angle/speed.

Now when the power plants, transmission lines, and loads are generated (shown in the Appendix F) in the DigSilent environment we can proceed to analyze the transmission system, which is done in the next chapter.

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6 Results & Analysis

The main purpose in this chapter is to analyze the results of the simulation in the DigSilent environment of the existing power system versus the power system in the presence of the newly installed generator.

The analyses are performed for five buses of the power system. The buses were chosen based on the load flow analysis which revealed that those buses were more critical in stability terms than the others and are geographically near the newly installed hydropower plant.

These are:

1. Rancagua 154 kV 2. Punta Cortez 154 kV 3. Tilcoco 154 kV

4. San Fernando/1 154 kV 5. Teno/1 154 kV

These buses can be found in the Appendix F figure 8.5.3, which is the figure before the integration of “La Higuera” to the Chilean power system.

As described in the previous chapters we can see that there are wide variety of stability phenomena to be considered in the system.. In this chapter we constrain ourselves to:

- Static analysis contained by voltage levels and voltage stability. This is done in order to get a picture of how the hydro power station of “La Higuera” improves the Chilean transmission system. The results for the static analysis are shown as voltage levels and PV-curves for the five selected buses. This can be seen in the next chapter.

- Rotor Angle stability analysis contained by small-signal analysis. Since the analysis is done before and after the integration of “La Higuera” in the transmission system, we are able see the improvement of the transmission system. This can be seen when the eigenvalues moves further away from the imaginary axis in the left hand side of the complex “s-plane” half plane.

- Large disturbance analysis contained by the simulation of a disturbance such as the disconnection of one transmission line between two significant buses. This kind of analysis gives us knowledge of how rapidly the transmission system recovers from the disturbance.

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6.1 Static Analysis

6.1.1 Voltage levels before and after the integration of “La Higuera”

The table below illustrates the voltage levels before and after “La Higuera” in the above mentioned buses.

Table 5.1.1.1 Voltages levels

N° Station Before After

1. Punta Cortez 154kV 0.965 p.u 0.970 p.u 2. Tilcoco/1 154 kV 0.955 p.u 0.970 p.u 3. San Fernando/1 154kV 0.950 pu 0.975 p.u 4. Teno/1 154 kV 0.965 p.u 0.980 p.u 5. Rancagua 154 kV 0.970 p.u 0.970 p.u

As we can see in the table above, the voltage levels increases with the incorporation of

“La Higuera” to values closer to 1. p.u in this case 154 kV. This new situation gives us a more stable power system, since the electrical components are operated closer to their designed conditions.

6.1.2 Voltage stability

In this chapter we will analyze PV-curves of the 5 mentioned buses with (straight line) and without (broken line) the integration of “La Higuera”. The PV-curve in each bus is performed with the maximal power factor that is allowed for preserving the stability.

Voltage (p.u)

0.9

0.8

0.7 PmaxWithout = 263 MW

PmaxWith = 443 MW 0.6

Fig. 6.1.2.1 Bus 1: Rancagua 154 kV with and without “La Higuera”

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As it is shown above in the figure 6.1.2.1 we can see that the system has improved or enlarged the limits of convergence in Bus1 from 263 MW to 443 MW. These two mentioned points of convergence are also the so-called sadde nodes (Bifurcations- nodes). [5]

Fig. 6.1.2.2 Bus 2: Punta Cortez 154 kV with and without “La Higuera

As it is shown above in the figure 6.1.2.2 we can see that the system has improved or enlarged the limits of convergence in Bus1 from 180 MW to 421 MW.

Voltage (p.u)

PmaxWith = 421 MW

0.8

0.7

0.6

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Fig. 6.1.2.3 Bus 3: Tilcoco 154 kV with and without “La Higuera”

As it is shown above in the figure 6.1.2.3 we can see that the system has improved or enlarged the limits of convergence in Bus1 from 180 MW to 421 MW.

Voltage (p.u)

0.9

PmaxWith = 425 MW

0.7

0.6

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432.35 332.35

232.35 132.35

32.35 1.00

0.90

0.80

0.70

0.60

0.50

x-Axis: P-V curves: Total Load of selected loads in MW TAP OFF SAN FERNANDO 1\S.FERNA.154/1: Voltage, Magnitude in p.u.

^{P-V curves}

Date: 12/26/2005 Annex: /1

DIgSILENT

Fig. 6.1.2.4 Bus 4: San Fernando 154 kV with and without“La Higuera”

As it is shown above in the figure 5.1.2.7 and figure 5.1.2.8 we can see that the system has improved or enlarged the limits of convergence in Bus1 from 400 MW to 650 MW Voltage (p.u)

0.9

0.8

PmaxWithout = 400 MW PmaxWith = 650 MW

0.7

0.6

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Voltage (p.u)

Fig. 6.1.2.5 Bus 5 Teno 154 kV with and without “La Higuera”

As it is shown above in the figure 6.1.2.5 we can see that the system has improved or enlarged the limits of convergence in Bus1 from 400 MW to 650 MW

0.9

0.8

PmaxWithout = 635 MW PmaxWith = 810 MW 0.7

0.6

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6.2 Rotor Angle stability analysis 6.2.1 Small signal stability analysis

This analysis is performed using the eigenvalues and participation matrixes. It is shown below the eigenvalues for which the power system is stable before “La Higuera”

Then a graph is shown for the eigenvalues that are nearest the imaginary axis. These eigenvalues will be our measure instrument. This since we can se the improvement, when integrating the new generating unit, by seeing that the eigenvalues has been moved a distance from the imaginary axis.

Fig.6.2.1.1 Eigenvalues near the imaginary and real axis.Z =σ ± jϖ

The critical eigenvalues of the system are shown below. These are those that are near the imaginary axis.

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Here we can observe that the integration of the new power plant “La Higuera” places the eigenvalue further away in the left halfplane from the imaginary axis, which is an indication of improvement of small-signal stability.

Fig.6.2.1.3 Critical Eigenvalues with “La Higuera”

To summarize the placement of the eigenvalues, we can observe in the followed table the damping ratio and the frequency for each mode, with and without “La Higuera”.

Table 6.2.1.1 Eigenvalues with and without “La Higuera”

Without ”La Higuera” With ”La Higuera”

Modes ξ (damping ratio) f(frecuency) ξ (damping ratio) f(frecuency)

181 0.943 0.556 0.866 0.998

202 0.075 0.999 0.565 0.837

257 0.223 0.629 0.551 0.633

The participation factor illustrates how the different generating units oscillate against each other. This can be seen as a sensitivity measure of an eigenvalue to a diagonal entry of the system A-matrix.

In the next page we can see the participation factor for the generating units in the Chilean transmission system. These for one of the critical eigenvalues, the participation factor for the rest of the critical eigenvalues are shown in the Appendix H.

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Table 6.2.1.1 Participation factor for the system mode 181 before “La Higuera”

DIgSILENTPowerFactory

Damping ratio: 0.943 Date: 12/26/2005

Element System Study:

Modal Analysis Busbar Participation Factors

Mode 181

G ABANICO ABANICO/ABANICO.154 <<<<<<<<<<<|

G ACONCAGUA POLPAICO/POLPAICO.220 <|

G ALFALFAL ALFALFAL/ALFALFAL.220 |>>>>>

G ANTUCO ANTUCO/ANTUCO.220 |>>>>>>>>>>>>>>>

G ARAUCO CORONEL/CORONEL.154 <<<<<<<<|

G BOCAMINA BOCAMINA/BOCAMINA.154 <<<|

G CANUTILLAR CANUTILLAR/CANUTIL.220 <<<<<<<<<<<<<<<<<|

G CAPULLO BLANCO/B.BLANCO.220 |

G CHACABUQUITO NAVIA/C.NAVIA.220 |

G CIPRESES CIPRESES/CIPRESES.154 |>>

G COLBUN COLBUN/COLBUN.220 <<|

G CONSTITUCION MAULE/MAULE.154 <|

G CORONEL CORONEL/CORONEL.154 <|

G CURILLINQUE CURILLINQUE/CURILLI.154 |>

G FLORIDA JAHUEL/A.JAHUEL.220 <|

G GUACOLDA GUACOLDA/GUACOLDA.220 |

G HUASCO TV MAITENCILLO/MAITENC.220 |

G HUASCO TG MAITENCILLO/MAITENC.220 |

G ISLA ISLA/ISLA.154 <<<<<<<<<<<<<|

G ITATA CHILLAN/T.O.CHILL.4 |>>>>>

G L.ALTA ALTA/L.ALTA.220 <<<<|

G LAJA VERDE HUALPEN/HUALPEN.154 |

G LOS MOLLES NAVIA/C.NAVIA.220 <|

G MACHICURA MACHICURA/MACHICURA <<<<|

G MAITENES JAHUEL/A.JAHUEL.220 |

G MAMPIL MAMPIL/MAMPIL.220 |>>>>>>

G NEHUENCO TG SAN LUIS/S.LUIS.220 <<<<<<<<|

G NEHUENCO TV SAN LUIS/S.LUIS.220 <<<<<<<<<<<<<<|

G NUEVA RENCA NAVIA/C.NAVIA.220 |

G PANGUE PANGUE/PANGUE.220 |>>>>>

G PEHUENCHE PEHUENCHE/PEHUEN.220 |>>

G PETROOPOWER HUALPEN/HUALPEN.154 <<<<<<<<<<<<<<|

G PEUCHEN PEUCHEN/PEUCHEN.220 |>>>>>>>>>>>>>>>>

G PILMAIQUEN BLANCO/B.BLANCO.220 |>>>>>>

G PULLINQUE TEMUCO/TEMUCO.220 |>

G PUNTILLA JAHUEL/A.JAHUEL.220 <|

G RALCO RALCO/RALCO.220 |>>>>>>>>

G RAPEL RAPEL/RAPEL.220 <<|

G RUCUE RUCUE/RUCUE.220 |>>>>>>>>>>>>>>>>>>

G SAN FCO MOST. PAINE/A.PAINE.154 <|

G SAN IGNACIO ITAHUE/ITAHUE.154 <<|

G SAN ISIDRO TG SAN LUIS/S.LUIS.220 <<<<<<<<|

G SAN ISIDRO TV SAN LUIS/S.LUIS.220 <<<<<<<<<|

G SAUZAL JAHUEL/A.JAHUEL.220 |>>>>>>>

G TALTAL PAPOSO/PAPOSO.220 |>>>

G VALDIVIA CIRUELOS/L.CIRUEL.220 <|

G VENTANAS QUILLOTA/QUILLOTA.220 |>>>>>>>

G VENTANAS QUILLOTA/QUILLOTA.220 |>>>

G VOLCAN JAHUEL/A.JAHUEL.220 |

G LOS MOLLES AZUCAR/P.AZUCAR.220 |

G DIEGO DE

ALMAGRO ALMAGRO/D.ALMAG.220 |

Dynamic analysis of the impact of grid connection of "La Higuera" hydropower plant to the transmission grid