**4. Grid dynamics and control with solar power penetration**

#### **4.1 Grid dynamics and control model**

Traditional grid dynamics models have been discussed in various books (Murty, 2008), (Machowski et al, 2008). The flow chart 11 below gives the Gauss-Seidel iterative method for load flow solutions for a n bus system with 1 slack bus. In this flow chart, P is the real power in kW while Q is reactive power in kVar. In real system, in order to secure the grid system, we need constrains in this dynamics model such as:


Fig. 9. Residential electricity consumption

Fig. 10. Residential power consumption of 10000 families

<sup>0</sup> <sup>20</sup> <sup>40</sup> <sup>60</sup> <sup>80</sup> <sup>100</sup> <sup>120</sup> <sup>140</sup> <sup>160</sup> <sup>180</sup> 3.16

Time (hour)

Fig. 9. Residential electricity consumption

3.17

Fig. 10. Residential power consumption of 10000 families

3.18

3.19

3.2

3.21

Power (kW)

3.22

3.23

3.24

3.25 x 104


Table 4. Control Time Scales (Abdallah, 2009)

• The state of change of the voltage must not exceed the slow state of the generator+tie line voltage control. Otherwise we have control saturation and the generator can no longer track the changes.

Table 4 gives the time scale of different disturbances and control signals in the grid. Once the disturbance introduced to the grid system exceeds the control limit, the cascading failure will happen (Ding et al, 2011). The test case (Dusko et al, 2006) in Fig. 12 and Fig. 13 represent a large European power system and has 1000 buses, 1800 transmission lines and transformers, and 150 generating units. The base case load has an active power demand of 33 GW and a reactive power demand of 2.5 GVar. It is seen from 12 that there is a sharp increase in blackout size at the critical loading of 1.94 times the base case loading, and (Dusko et al, 2006) also discussed that the expected energy not served (EENS) share similar distribution for critical loading and under critical loading cases, and for over critical loading cases, we have different patterns, with the exponent of the power law distributions ranging from -1.2 to -1.5 as shown in Table. 5.

#### **4.2 Grid fluctuation introduced by different levels of solar penetration**

A report published in 2009 by the North American Electric Reliability Corporation showed that the output power of a large PV systems, with ratings in the order of tens of megawatts, can change by ±70% in a five- to ten-min time frame (NAERC, 2009). And it should also be mentioned that if a number of small systems that are distributed over a large land area, the resulting combined fluctuations are much less due to the smoothing effect according to our previous analysis.

Fig. 11. Flow chart of Gauss-Seidel iterative method for load flow solutions for a n bus system with 1 slack bus

*<sup>i</sup>* , i=1,2...,n; i �= slack bus

Form Y-Bus

Let m=0

Let maximum voltage change <sup>Δ</sup>*V*(*m*) *max* <sup>=</sup> 0; i=1

Is i Slack bus?

> *YikV*(*m*+1) *<sup>k</sup>* <sup>−</sup> *<sup>n</sup>* ∑ *k*=*i*+1

No

*<sup>i</sup>* <sup>−</sup> <sup>Δ</sup>*V*(*m*) *max* <sup>=</sup>

Is <sup>&</sup>gt; 0? <sup>Δ</sup>*V*(*m*) *max* <sup>=</sup> �Δ*V*(*m*)

Yes

*YikV*(*m*) *k* 

*i* �

No

*m* = *m* + 1

<sup>−</sup> *<sup>i</sup>*−<sup>1</sup> ∑ *k*=1,*k*�=*i*

Let Δ*V*(*m*)

*V*(*m*)

*<sup>i</sup>* <sup>=</sup> *<sup>V</sup>*(*m*+1) *i*

No

*i* = *i* + 1

Is (*i* − *n*) ≤ 0?

Is <sup>Δ</sup>*V*(*m*) *max* <sup>−</sup> *V* > 0?

No

Yes

Calculate line flows and slack bus power

No

Detect new disturbance into grid from both supply and demand side.

Fig. 11. Flow chart of Gauss-Seidel iterative method for load flow solutions for a n bus

Initial voltage *V*(0)

*V*(*m*) *<sup>i</sup>* <sup>=</sup> <sup>1</sup> *Yij Pj*−*jQj V*(*m*) *i*

Yes

system with 1 slack bus


Table 5. Approximate power law exponents at criticality for several cascading failure models

Fig. 12. Expected energy not served (EENS) as a function of the loading factor with respect to the base case. (Dusko et al, 2006)

Fig. 14 shows an example of the PV output fluctuations in New York area, it is a normal distribution with a mean plot and a confidence interval of 68%(±1*σ*). Here we assume the PV output fluctuation have similar distribution with the solar radiation, which is reasonable according to the solar power output model (Dusabe et al, 2009). The X axis of the figure is time in hours and Y axis is the system output in MWh. We can conclude from this figure that the most severe fluctuation occurs around noon. In general, the change of solar power output is usually due to:


Fig. 13. Probability distribution of expected energy not served (EENS) at the critical loading of 1.94 times the base case loading (Dusko et al, 2006)


The negative effects introduced, especially to the stability of grid system as solar penetration level increases, is a major concern for the future grid. We can calculate the blackout probability of power system with different level of solar penetration as following:

For a grid system with *a*% of power from solar system, which follows a normal distribution:

$$\mathcal{W}\_{solar} \sim \mathcal{N}(a^{\circ}\!\!/ \_{\prime}\sigma\_{(a,i)}^{2}), i = 0, 1...23\tag{11}$$

where *a*% is the normalized expected power output from PV system at *i th* hour of the day with *a*% of penetration for the overall grid system, and *σ*(*a*,*i*) is the standard deviation of power output at the same time and same penetration level.

We can equate the solar system fluctuation to the inverse change of loading, for instance, a decrease of 1*MWh* of solar production is equivalent to an increase of the power load at the same time frame, therefore the equivalent load should also follow normal distribution.

$$L \sim \mathcal{N}(l, \sigma\_{(a,i)}^2) \tag{12}$$

<sup>100</sup> <sup>101</sup> <sup>102</sup> <sup>103</sup> <sup>104</sup> 10−6

Fig. 13. Probability distribution of expected energy not served (EENS) at the critical loading

The negative effects introduced, especially to the stability of grid system as solar penetration level increases, is a major concern for the future grid. We can calculate the blackout probability

For a grid system with *a*% of power from solar system, which follows a normal distribution:

with *a*% of penetration for the overall grid system, and *σ*(*a*,*i*) is the standard deviation of

We can equate the solar system fluctuation to the inverse change of loading, for instance, a decrease of 1*MWh* of solar production is equivalent to an increase of the power load at the same time frame, therefore the equivalent load should also follow normal distribution.

*<sup>L</sup>* <sup>∼</sup> *<sup>N</sup>*(*l*, *<sup>σ</sup>*<sup>2</sup>

(*a*,*i*)

(*a*,*i*)), *i* = 0, 1...23 (11)

) (12)

*th* hour of the day

EENS (MWh)

10−5

• Types of Clouds. • PV system topology.

of 1.94 times the base case loading (Dusko et al, 2006)

of power system with different level of solar penetration as following:

*Wsolar* <sup>∼</sup> *<sup>N</sup>*(*a*%, *<sup>σ</sup>*<sup>2</sup>

where *a*% is the normalized expected power output from PV system at *i*

power output at the same time and same penetration level.

10−4

10−3

Probability

10−2

10−1

100

Fig. 14. Fluctuations in the output power of a large PV system (1*σ* confidence interval) (NSRDB, 2005)

then following the same procedure as in Fig. 12 and Fig. 13, probability of failure due to different level of solar penetration is

$$P\_{(f,a)} = \int\_{-\infty}^{\infty} p\_{(L,a)} F\_{(EENS > 0 \, | \, (L,a) \,)} dL \tag{13}$$

where the probability *<sup>F</sup>*(*EENS*>0|(*L*,*a*) is the cumulative probability distribution of system failure

$$F\_{(EENS > 0 \mid (L, a))} = \int\_0^\infty (p\_{(EENS, L, a)}) dEENS \tag{14}$$

and *p*(*EENS*,*L*,*a*) is the probability density function in Fig. 13, in this example, *L* = 1.94 and *a* = 0.

For instance, if we choose *i* = 12, when solar radiation follows N(547,174), and assume solar system output follows the same distribution. The normalized solar penetration of *a* = 1, 10, 50, 100, and corresponding standard deviation of solar system output *σ*(*a*,*i*) = 0.0027, 0.0269, 0.1344, 0.2687. And the grid failure model with these levels of solar penetration is shown in Fig. 15, in which we show that as the level of solar penetration increases, the probability of system failure increases. This analysis does not take into account, additional solar backup. Thus as the proportion of solar power increases, the proportion of a controllable base load power source to meet demand fluctuations reduces, and hence, the system becomes more prone to failure. The availability of storage can ameliorate the problem. Fig. . 15 does not take into account real time matching of AC demand with solar supply, which and reduce failure probability.
