2025 Tutorial 0

At the edge of the Mojave Desert in California, millions of solar panels occupy about 4084 acres of land. The Sun shines at least 300 days of the year on the Cadmium telluride (CdTe) photovoltaic (PV) panels at the Desert Sunlight Solar Farm. The solar farm can generate a maximum of about 550 megawatts of power. The plant’s solar panels power about 160,000 California homes. The site’s millions of 2-by-4-ft. (0.6 m by 1.2 m) panels are each covered by a thin film of glass that absorbs sunlight and captures electrons, creating an electrical current that flows into wires in the back of each module. Each panel generates approximately 90 to 100 watts. From another website, we know that the typical solar panel efficiency is estimated to be about 20%. $[1 {km}^{2} = 247.1 acres]$ .

a. What is the total land area used in km $^{2}$ ? How does it compare with the area of NTU?
b. An important piece of information related to the placement of solar panels is the solar irradiance $I$ , which has units of energy per time per unit area. Write down the base SI units for $I$ .
c. Assuming at noon time, the sun’s rays come vertically down and the intensity is $I_{o}$ . If the area of a solar panel is $A$ and it is tilted at an angle $θ$ from the horizontal, what is the power (energy per unit time) incident on the panel? If needed, we will assume that $θ = 0^{\circ}$ for subsequent calculations. [See if you can use the sensor app installed on your phone to do a simple experiment.]
d. Using the efficiency information provided, estimate the Sun’s power incident on each 2-by-4-ft panel.
e. The average Sun-Earth distance is about 150 million kilometres. (You may assume that the Earth moves in a circle round the Sun.) Using your answer in (d), determine the energy per second generated by the Sun.
f. If a system can capture 1 s of the energy generated by the Sun as calculated in (e), how many California homes can the system support for a year with this energy?
From 2015 – 2018, Desert Sunlight generated an average 1,325,000 megawatt-hours (MWh) of clean energy per year while preventing about 300,000 metric tons of carbon dioxide emissions annually. The average energy generated (in MWh) in the year 2018 is given in the table below:

Jan, 62367
Feb, 87862
Mar, 100464
Apr, 125593
May, 151388
Jun, 163110
Jul, 140988
Aug, 141969
Sep, 134393
Oct, 105885
Nov, 72832
Des, 57941

g. Provide 3 reasons for the variation of energy production across the months.
h. Determine the average daily energy production in the year 2018.
i. Assuming that the solar farm captures 550 megawatts of power uniformly in daytime, estimate the number of daylight hours in Jun 2018. Do you think your answer is an overestimate or underestimate? Why?

ANSWER

a. What is the total land area used in km²? How does it compare with the area of NTU?

Excellent first step. This part is all about unit conversion. We are given the area in acres and a conversion factor to square kilometers.

Identify the given information:
- Total land area = 4084 acres
- Conversion factor = $1 t e x t {k m}^{2} = 247.1 t e x t a c r e s$
Set up the calculation:
To convert from acres to km², we can divide the total acres by the number of acres in one km².

3. * * C a l c u l a t e t h e r e s u l t : * *

\text{Area} \approx 16.53 \text{ km}^2

S o, t h e t o t a l l a n d a r e a u s e d b y t h e s o l a r f a r m i s a b o u t * * $ 16.5 t e x t {k m}^{2} $ * * . * * C o m p a r i s o n w i t h N T U : * * T h e m a i n c a m p u s o f N a n y a n g T e c h n o l o g i c a l U n i v e r s i t y (N T U) i n S i n g a p o r e i s a p p r o x i m a t e l y 2 k m ² . T o c o m p a r e t h e m, w e c a n f i n d t h e r a t i o :

\frac{\text{Area of Solar Farm}}{\text{Area of NTU}} = \frac{16.53 \text{ km}^2}{2 \text{ km}^2} \approx 8.3

T h i s m e a n s t h e s o l a r f a r m i s m o r e t h a n * * 8 t i m e s * * t h e s i z e o f t h e N T U c a m p u s I t h e l p s t o p u t i n t o p e r s p e c t i v e j u s t h o w l a r g e t h e s e r e n e w a b l e e n e r g y p r o j e c t s a r e . * * b . A n i m p o r t a n t p i e c e o f i n f o r m a t i o n r e l a t e d t o t h e p l a c e m e n t o f s o l a r p a n e l s i s t h e s o l a r i r r a d i a n c e $ I $, w h i c h h a s u n i t s o f e n e r g y p e r t i m e p e r u n i t a r e a . W r i t e d o w n t h e b a s e S I u n i t s f o r $ I $ . * * 1. * * S t a r t w i t h t h e d e f i n i t i o n o f I r r a d i a n c e ($ I $) : * *

I = \frac{\text{Energy}}{\text{Time} \times \text{Area}}

2. * * B r e a k d o w n e a c h c o m p o n e n t i n t o i t s b a s e S I u n i t s : * * * * * E n e r g y : * * T h e S I u n i t f o r e n e r g y i s t h e J o u l e ($ J $) . A J o u l e i s d e f i n e d a s t h e w o r k d o n e w h e n a f o r c e o f o n e N e w t o n i s a p p l i e d o v e r o n e m e t e r ($ J = N c d o t m $) . A N e w t o n i s t h e f o r c e r e q u i r e d t o a c c e l e r a t e a 1 k g m a s s a t $ 1 t e x t {m / s}^{2} $ ($ N = k g c d o t m / s^{2} $) . P u t t i n g i t a l l t o g e t h e r : $ t e x t E n e r g y r i g h t a r r o w J r i g h t a r r o w (k g c d o t m / s^{2}) c d o t m = k g c d o t m^{2} c d o t s^{- 2} $ * * * T i m e : * * T h e b a s e S I u n i t i s t h e s e c o n d ($ s $) . * * * A r e a : * * T h e b a s e S I u n i t f o r l e n g t h i s t h e m e t e r ($ m $), s o t h e u n i t f o r a r e a i s s q u a r e m e t e r s ($ m^{2} $) . 3. * * C o m b i n e t h e b a s e u n i t s : * *

I = \frac{kg \cdot m^2 \cdot s^{-2}}{s \cdot m^2}

2. * * S i m p l i f y t h e e x p r e s s i o n : * * N o t i c e t h a t t h e $ m^{2} $ i n t h e n u m e r a t o r a n d d e n o m i n a t o r c a n c e l o u t .

I = \frac{kg}{s \cdot s^2} = \frac{kg}{s^3} = kg \cdot s^

S o, t h e b a s e S I u n i t s f o r s o l a r i r r a d i a n c e ($ I $) a r e * * $ k g c d o t s^{- 3} $ * * . A s a q u i c k c h e c k, y o u m i g h t k n o w t h a t p o w e r i s e n e r g y p e r t i m e (W a t t s, $ W = J / s $), s o i r r a d i a n c e i s a l s o p o w e r p e r a r e a ($ W / m^{2} $) . L e t^{'} s s e e i f t h a t g i v e s t h e s a m e r e s u l t : $ W / m^{2} = (J / s) / m^{2} = (k g c d o t m^{2} c d o t s^{- 2} / s) / m^{2} = (k g c d o t m^{2} c d o t s^{- 3}) / m^{2} = k g c d o t s^{- 3} $ . I t w o r k s p e r f e c t l y D o e s b r e a k i n g i t d o w n t o t h e b a s e u n i t s m a k e s e n s e ? * * c . I f t h e a r e a o f a s o l a r p a n e l i s $ A $ a n d i t i s t i l t e d a t a n a n g l e $ t h e t a $ f r o m t h e h o r i z o n t a l, w h a t i s t h e p o w e r i n c i d e n t o n t h e p a n e l ? * * T h i s i s a c r u c i a l c o n c e p t i n s o l a r p a n e l i n s t a l l a t i o n T h e p o w e r a p a n e l r e c e i v e s i s n^{'} t j u s t a b o u t i t s o w n a r e a; i t^{'} s a b o u t t h e * e f f e c t i v e a r e a * i t p r e s e n t s t o t h e s u n l i g h t . I m a g i n e h o l d i n g y o u r h a n d o u t i n t h e r a i n . I f y o u r p a l m i s f a c i n g u p (h o r i z o n t a l), i t c a t c h e s t h e m o s t r a i n . I f y o u t i l t i t, i t c a t c h e s l e s s . T h e p a n e l a n d t h e s u n^{'} s r a y s w o r k t h e s a m e w a y . 1. * * D e f i n e I n c i d e n t P o w e r : * * T h e p o w e r i n c i d e n t o n a s u r f a c e i s t h e i n t e n s i t y o f t h e r a d i a t i o n m u l t i p l i e d b y t h e a r e a t h a t i s p e r p e n d i c u l a r t o t h a t r a d i a t i o n .

P_{\text{incident}} = I \times A_{\text{effective}}

3. * * D e t e r m i n e t h e E f f e c t i v e A r e a ($ A_t e x t e f f e c t i v e $) : * * T h e s u n^{'} s r a y s a r e c o m i n g v e r t i c a l l y d o w n . T h e p a n e l i s t i l t e d a t a n a n g l e $ t h e t a $ f r o m t h e h o r i z o n t a l . T h e " e f f e c t i v e a r e a " i s t h e p r o j e c t i o n o f t h e p a n e l^{'} s a r e a o n t o t h e h o r i z o n t a l p l a n e, w h i c h i s t h e p l a n e p e r p e n d i c u l a r t o t h e i n c o m i n g s u n l i g h t . L e t^{'} s v i s u a l i z e t h i s : A s y o u c a n s e e f r o m t h e d i a g r a m, t h e e f f e c t i v e a r e a t h a t i n t e r c e p t s t h e v e r t i c a l s u n r a y s i s t h e " s h a d o w " t h e p a n e l w o u l d c a s t o n t h e g r o u n d . T h i s i s g i v e n b y :

A_{\text{effective}} = A \cos(\theta)

3. * * W r i t e t h e f i n a l e x p r e s s i o n f o r p o w e r : * * S u b s t i t u t i n g t h i s b a c k i n t o o u r p o w e r e q u a t i o n, w e g e t :

P_{\text{incident}} = I_o \cdot A \cos(\theta)

This formula tells us that the maximum power is received when $\\theta = 0^\\circ$ (the panel is horizontal), because $\\cos(0^\\circ) = 1$. As the panel tilts more, $\\theta$ increases and $\\cos(\\theta)$ decreases, reducing the incident power. For subsequent calculations, we will follow the instruction and assume $\\theta = 0^\\circ$, so we can use $P\_{\\text{incident}} = I\_o \\cdot A$. **d. Using the efficiency information provided, estimate the Sun’s power incident on each 2-by-4-ft panel.** Now we'll use the concept of efficiency to work backward from the panel's electrical output to find out how much solar power was hitting it in the first place. 1. **Define Efficiency ($\\eta$):** Efficiency is the ratio of the useful power output to the total power input.

\eta = \frac{P_{\text{output}}}{P_{\text{input}}}

2. * * I d e n t i f y t h e g i v e n i n f o r m a t i o n : * * * T h e e l e c t r i c a l p o w e r g e n e r a t e d ($ P_t e x t o u t p u t $) i s 90 t o 100 w a t t s . L e t^{'} s u s e t h e a v e r a g e v a l u e f o r o u r c a l c u l a t i o n : $ P_t e x t o u t p u t = 95 t e x t W $ . * T h e e f f i c i e n c y ($ e t a $) i s a b o u t 20 3. * * R e a r r a n g e t h e f o r m u l a a n d s o l v e f o r t h e i n c i d e n t p o w e r ($ P_t e x t i n p u t $) : * *

P_{\text{input}} = \frac{P_{\text{output}}}{\eta}

P_{\text{input}} = \frac{95 \text{ W}}

P_{\text{input}} = 475 \text

S o, t h e e s t i m a t e d p o w e r f r o m t h e S u n i n c i d e n t o n e a c h p a n e l i s a p p r o x i m a t e l y * * 475 W a t t s * * . I t^{'} s f a s c i n a t i n g t o s e e t h a t 80 * * e . U s i n g y o u r a n s w e r i n (d), d e t e r m i n e t h e e n e r g y p e r s e c o n d g e n e r a t e d b y t h e S u n . * * T h e k e y i d e a i s t h a t t h e S u n r a d i a t e s e n e r g y u n i f o r m l y i n a l l d i r e c t i o n s (i s o t r o p i c a l l y) . T h i s e n e r g y s p r e a d s o u t o v e r t h e s u r f a c e o f a n e v e r - e x p a n d i n g i m a g i n a r y s p h e r e . 1. * * F i r s t, c a l c u l a t e t h e s o l a r i r r a d i a n c e ($ I $) a t E a r t h^{'} s d i s t a n c e . * * W e k n o w t h e i n c i d e n t p o w e r ($ P_t e x t i n p u t = 475 t e x t W $) o n a p a n e l o f a c e r t a i n a r e a ($ A $) . T h e p a n e l a r e a i s g i v e n a s $ 0.6 t e x t m t i m e s 1.2 t e x t m $ .

A = 0.6 \text{ m} \times 1.2 \text{ m} = 0.72 \text{ m}^2

Now, we can find the intensity (assuming the panel is horizontal, so $\\theta=0^\\circ$):

I = \frac{P_{\text{input}}}{A} = \frac{475 \text{ W}}{0.72 \text{ m}^2} \approx 659.7 \text{ W/m}^2

T h i s v a l u e i s t h e s o l a r p o w e r t h a t a r r i v e s p e r s q u a r e m e t e r a t E a r t h^{'} s l o c a t i o n . 1. * * C a l c u l a t e t h e t o t a l p o w e r o f t h e S u n ($ P_t e x t S u n $) . * * I m a g i n e a g i a n t s p h e r e c e n t e r e d o n t h e S u n, w i t h a r a d i u s e q u a l t o t h e S u n - E a r t h d i s t a n c e ($ R $) . A l l o f t h e S u n^{'} s e n e r g y m u s t p a s s t h r o u g h t h e s u r f a c e o f t h i s s p h e r e . * S u n - E a r t h d i s t a n c e, $ R = 150 t i m e s 10^{6} t e x t k m = 150 t i m e s 10^{9} t e x t m $ . * T h e s u r f a c e a r e a o f t h i s s p h e r e i s $ A_t e x t s p h e r e = 4 p i R^{2} $ . T h e S u n^{'} s t o t a l p o w e r i s t h i s i n t e n s i t y m u l t i p l i e d b y t h e e n t i r e s u r f a c e a r e a o f t h e s p h e r e .

P_{\text{Sun}} = I \times A_{\text{sphere}} = I \times (4\pi R^2)

3. * * P l u g i n t h e n u m b e r s a n d c a l c u l a t e : * *

P_{\text{Sun}} = (659.7 \text{ W/m}^2) \times 4\pi (150 \times 10^9 \text{ m})^2

P_{\text{Sun}} \approx (659.7) \times 4\pi (2.25 \times 10^{22}) \text

P_{\text{Sun}} \approx 1.86 \times 10^{26} \text

T h e a c c e p t e d v a l u e f o r t h e S u n^{'} s t o t a l p o w e r o u t p u t (i t s l u m i n o s i t y) i s a b o u t $ 3.8 t i m e s 10^{26} $ W a t t s . T h e d i f f e r e n c e c o m e s f r o m f a c t o r s l i k e t h e s u n^{'} s i n t e n s i t y v a r y i n g, a t m o s p h e r i c a b s o r p t i o n (o u r c a l c u l a t e d $ I $ i s a t t h e g r o u n d, n o t a t t h e t o p o f t h e a t m o s p h e r e), a n d t h e i n i t i a l n u m b e r s b e i n g e s t i m a t e s . S o, t h e S u n g e n e r a t e s a p p r o x i m a t e l y * * $ 1.9 t i m e s 10^{26} $ J o u l e s o f e n e r g y e v e r y s i n g l e s e c o n d * * * * f . I f a s y s t e m c a n c a p t u r e 1 s o f t h e e n e r g y g e n e r a t e d b y t h e S u n a s c a l c u l a t e d i n (e), h o w m a n y C a l i f o r n i a h o m e s c a n t h e s y s t e m s u p p o r t f o r a y e a r w i t h t h i s e n e r g y ? * * W e^{'} l l c o m p a r e t h e S u n^{'} s i m m e n s e t o t a l o u t p u t t o o u r e a r t h l y e n e r g y n e e d s . 1. * * R e c a l l t h e S u n^{'} s 1 - s e c o n d e n e r g y o u t p u t : * * F r o m o u r c a l c u l a t i o n i n p a r t (e), t h e t o t a l e n e r g y t h e S u n g e n e r a t e s i n o n e s e c o n d i s :

E_{\text{Sun}, 1s} \approx 1.9 \times 10^{26} \text

2. * * C a l c u l a t e t h e a v e r a g e a n n u a l e n e r g y u s e p e r h o m e : * * W e a r e t o l d t h e f a r m g e n e r a t e s a n a v e r a g e o f $ 1, 325, 000 $ m e g a w a t t - h o u r s (M W h) p e r y e a r a n d s u p p o r t s $ 160, 000 $ h o m e s . W e c a n u s e t h i s t o f i n d t h e a v e r a g e c o n s u m p t i o n p e r h o m e .

E_{\text{per home}} = \frac{\text{Total Farm Energy}}{\text{Number of Homes}} = \frac{1,325,000 \text{ MWh/year}}{160,000 \text{ homes}}

E_{\text{per home}} \approx 8.28 \text

2. * * C o n v e r t e n e r g y u n i t s t o b e c o n s i s t e n t : * * T o c o m p a r e t h e t w o e n e r g y f i g u r e s, w e m u s t h a v e t h e m i n t h e s a m e u n i t s . L e t^{'} s c o n v e r t t h e h o m e^{'} s a n n u a l e n e r g y u s e f r o m M W h t o J o u l e s . * 1 h o u r = 3600 s e c o n d s * 1 W a t t = 1 J o u l e / s e c o n d = 1 W h = 3600 J * 1 M W h = $ 10^{6} $ W h = $ 10^{6} t i m e s 3600 t e x t J = 3.6 t i m e s 10^{9} t e x t J $ N o w, c o n v e r t t h e a n n u a l e n e r g y p e r h o m e t o J o u l e s :

E_{\text{per home}} = 8.28 \text{ MWh} \times (3.6 \times 10^9 \text{ J/MWh}) \approx 2.98 \times 10^{10} \text

4. * * C a l c u l a t e t h e n u m b e r o f h o m e s t h a t c a n b e s u p p o r t e d : * * W e c a n n o w f i n d o u t h o w m a n y o f t h e s e " a n n u a l h o m e e n e r g y p a c k e t s " f i t i n t o t h e S u n^{'} s 1 - s e c o n d o u t p u t .

\text{Number of Homes} = \frac{E_{\text{Sun}, 1s}}{E_{\text{per home}}}

\text{Number of Homes} = \frac{1.9 \times 10^{26} \text{ J}}{2.98 \times 10^{10} \text{ J/home}} \approx 6.38 \times 10^{15} \text

J u s t o n e s e c o n d o f t h e S u n^{'} s t o t a l e n e r g y o u t p u t c o u l d p o w e r o v e r * * 6 q u a d r i l l i o n h o m e s * * f o r a n e n t i r e y e a r . * * g . P r o v i d e 3 r e a s o n s f o r t h e v a r i a t i o n o f e n e r g y p r o d u c t i o n a c r o s s t h e m o n t h s . * * L o o k i n g a t t h e d a t a, w e c a n c l e a r l y s e e a p e a k i n t h e s u m m e r m o n t h s (J u n e) a n d a t r o u g h i n t h e w i n t e r m o n t h s (D e c e m b e r / J a n u a r y) . T h i s v a r i a t i o n i s d u e t o v e r y p r e d i c t a b l e a s t r o n o m i c a l a n d a t m o s p h e r i c f a c t o r s . 1. * * L e n g t h o f D a y l i g h t H o u r s : * * T h e m o s t d i r e c t c a u s e i s t h e c h a n g e i n t h e n u m b e r o f h o u r s t h e s u n i s s h i n i n g . D u e t o t h e E a r t h^{'} s a x i a l t i l t, d a y s a r e m u c h l o n g e r i n t h e s u m m e r t h a n i n t h e w i n t e r . M o r e d a y l i g h t h o u r s m e a n t h e p a n e l s a r e g e n e r a t i n g e l e c t r i c i t y f o r a l o n g e r p e r i o d e a c h d a y . 2. * * A n g l e o f t h e S u n (S o l a r A l t i t u d e) : * * T h e s u n i s m u c h h i g h e r i n t h e s k y d u r i n g t h e s u m m e r . T h i s h a s t w o i m p o r t a n t e f f e c t s : * * * H i g h e r I n t e n s i t y : * * W h e n t h e s u n i s d i r e c t l y o v e r h e a d, i t s l i g h t t r a v e l s t h r o u g h t h e l e a s t a m o u n t o f a t m o s p h e r e t o r e a c h t h e g r o u n d . W h e n i t^{'} s l o w e r o n t h e h o r i z o n (a s i n w i n t e r), i t s r a y s m u s t p a s s t h r o u g h m o r e o f t h e a t m o s p h e r e, w h i c h s c a t t e r s a n d a b s o r b s s o m e o f t h e e n e r g y, r e d u c i n g t h e i n t e n s i t y ($ I $) a t t h e s u r f a c e . * * * M o r e D i r e c t A n g l e o f I n c i d e n c e : * * S o l a r p a n e l s a r e m o s t e f f i c i e n t w h e n s u n l i g h t h i t s t h e m p e r p e n d i c u l a r l y . E v e n w i t h f i x e d p a n e l s, t h e s u n^{'} s p a t h b e i n g h i g h e r i n t h e s k y d u r i n g s u m m e r m e a n s t h e a n g l e o f i n c i d e n c e i s m o r e f a v o u r a b l e f o r a g r e a t e r p o r t i o n o f t h e d a y . 3. * * W e a t h e r a n d C l i m a t e : * * W h i l e t h e M o j a v e D e s e r t i s v e r y s u n n y, i t^{'} s n o t i m m u n e t o w e a t h e r . S e a s o n a l w e a t h e r p a t t e r n s c a n p l a y a r o l e . F o r e x a m p l e, t h e r e m i g h t b e a h i g h e r c h a n c e o f c l o u d c o v e r, h a z e, o r d u s t i n t h e a i r d u r i n g c e r t a i n m o n t h s, w h i c h w o u l d b l o c k s u n l i g h t a n d r e d u c e t h e e n e r g y o u t p u t . * * h . D e t e r m i n e t h e a v e r a g e d a i l y e n e r g y p r o d u c t i o n i n t h e y e a r 2018. * * W e n e e d t o s u m u p a l l t h e m o n t h l y p r o d u c t i o n a n d t h e n a v e r a g e i t o v e r t h e n u m b e r o f d a y s i n t h e y e a r . 1. * * S u m t h e t o t a l e n e r g y f o r 2018 : * * F i r s t, l e t^{'} s a d d u p a l l t h e m o n t h l y v a l u e s f r o m y o u r t a b l e (i n M W h) : $ E_t e x t t o t a l = 62367 + 87862 + 100464 + 125593 + 151388 + 163110 + 140988 + 141969 + 134393 + 105885 + 72832 + 57941 $

E_{\text{total}} = 1,344,792 \text

(T h i s i s v e r y c l o s e t o t h e 1, 325, 000 M W h a v e r a g e g i v e n e a r l i e r, w h i c h i s a n i c e c o n s i s t e n c y c h e c k) 2. * * D i v i d e b y t h e n u m b e r o f d a y s i n t h e y e a r : * * T h e y e a r 2018 i s n o t a l e a p y e a r, s o i t h a s 365 d a y s .

\text{Average Daily Energy} = \frac{1,344,792 \text{ MWh}}{365 \text{ days}}

\text{Average Daily Energy} \approx 3684.4 \text

T h e a v e r a g e d a i l y e n e r g y p r o d u c t i o n i n 2018 w a s a b o u t * * 3, 684 M W h * * . * * i . E s t i m a t e t h e n u m b e r o f d a y l i g h t h o u r s i n J u n 2018. D o y o u t h i n k y o u r a n s w e r i s a n o v e r e s t i m a t e o r u n d e r e s t i m a t e ? W h y ? * * T h i s i s a g r e a t q u e s t i o n t h a t t e s t s o u r u n d e r s t a n d i n g o f t h e a s s u m p t i o n s w e m a k e i n p h y s i c s p r o b l e m s . 1. * * R e l a t e E n e r g y, P o w e r, a n d T i m e : * * T h e f u n d a m e n t a l r e l a t i o n s h i p i s :

\text{Energy} = \text{Power} \times \text

W e c a n r e a r r a n g e t h i s t o s o l v e f o r t h e t o t a l t i m e t h e p l a n t w a s g e n e r a t i n g p o w e r :

\text{Time} = \frac{\text{Energy}}{\text{Power}}

2. * * U s e t h e v a l u e s f o r J u n e : * * * T o t a l E n e r g y i n J u n e = $ 163, 110 $ M W h (f r o m t h e t a b l e) . * A s s u m e d P o w e r = $ 550 $ M W (t h e f a r m^{'} s m a x i m u m c a p a c i t y) . 3. * * C a l c u l a t e t h e t o t a l o p e r a t i n g h o u r s i n J u n e : * *

\text{Total Hours} = \frac{163,110 \text{ MWh}}{550 \text{ MW}} = 296.56 \text

T h i s i s t h e t o t a l n u m b e r o f h o u r s f o r t h e e n t i r e m o n t h o f J u n e . 4. * * F i n d t h e a v e r a g e h o u r s p e r d a y : * * J u n e h a s 30 d a y s .

\text{Average Daily Hours} = \frac{296.56 \text{ hours}}{30 \text{ days}} \approx 9.89 \text

S o, o u r e s t i m a t e i s t h a t t h e s u n w a s s h i n i n g f o r a b o u t * * 9.9 h o u r s p e r d a y * * i n J u n e . * * O v e r e s t i m a t e o r U n d e r e s t i m a t e ? * * M y a n s w e r i s a n * * u n d e r e s t i m a t e * * o f t h e a c t u a l n u m b e r o f d a y l i g h t h o u r s . * * R e a s o n i n g : * * T h e c a l c u l a t i o n r e l i e s o n t h e c r u c i a l a s s u m p t i o n t h a t t h e s o l a r f a r m p r o d u c e s p o w e r a t i t s * m a x i m u m * l e v e l o f 550 M W u n i f o r m l y t h r o u g h o u t t h e d a y . I n r e a l i t y, a s o l a r f a r m^{'} s p o w e r o u t p u t l o o k s m o r e l i k e a b e l l c u r v e . I t s t a r t s a t z e r o a t s u n r i s e, g r a d u a l l y i n c r e a s e s t o a p e a k (w h i c h m i g h t b e 550 M W) a r o u n d n o o n w h e n t h e s u n i s h i g h e s t a n d b r i g h t e s t, a n d t h e n d e c r e a s e s b a c k t o z e r o a t s u n s e t . S i n c e t h e a c t u a l p o w e r o u t p u t i s * l e s s t h a n * 550 M W f o r m o s t o f t h e d a y (i n t h e m o r n i n g a n d a f t e r n o o n), t h e p l a n t m u s t o p e r a t e f o r a * l o n g e r t i m e * t o g e n e r a t e t h e s a m e t o t a l a m o u n t o f e n e r g y . O u r m o d e l a s s u m e s t h e " p o w e r r e c t a n g l e " s h o w n i n t h e d i a g r a m, w h i l e r e a l i t y i s t h e " p o w e r b e l l c u r v e . " T o g e t t h e s a m e a r e a (e n e r g y), t h e b a s e o f t h e b e l l c u r v e (a c t u a l d a y l i g h t h o u r s) m u s t b e w i d e r t h a n t h e b a s e o f t h e r e c t a n g l e (o u r c a l c u l a t e d h o u r s) . T h e r e f o r e, t h e a c t u a l n u m b e r o f d a y l i g h t h o u r s m u s t b e l o n g e r t h a n o u r 9.9 - h o u r e s t i m a t e .