Efficiency Optimization of Power Stations with Different Generating Units

Abstract

There are many power stations in the world with different generators. These include thermal power stations, hydropower stations, etc. They convert the energy of raw materials into electrical energy. In these power stations, with the same input of raw materials, there is maximum power generation. To achieve the maximum, we should decide how many generators to use and how many tasks should be assigned to each generator.

There are many power stations in the world with different generators. These include thermal power stations, hydropower stations, etc (Fig. 16.1). They convert the energy of raw materials into electrical energy. In these power stations, with the same input of raw materials, there is maximum power generation. To achieve the maximum, we should decide how many generators to use and how many tasks should be assigned to each generator.

Fig. 16.1

Power stations

16.1 Power Generation in ESPS

A generalized generator must reach the rated speed n₀ before it can be connected to the grid. Taking the input amount of raw materials as a variable, such as the amount of coal input in a thermal power station or the water flow in a hydropower station, the efficiency of a generalized generator is shown in Fig. 16.2.

Fig. 16.2

The efficiency of a generalized power generator

In Fig. 16.2, L is the adjustable input of the generator, η is the efficiency function of the generator, L₀ is L at zero load of the generator, η_e is the maximum efficiency corresponding to L_e, L_e is the optimal input, and L_m is the maximum L.

We define the load factor γ as

$$ \gamma = \frac{L}{{L_{e} }} $$

(16.1)

We define η_N(γ) to be the normalized efficiency function. The normalized efficiency function η_N(γ) has a shape shown in Fig. 16.3.

Fig. 16.3

The normalized efficiency function η_N(γ)

In Fig. 16.3, γ is the load rate and a variable, η_N(γ) is the normalized efficiency. η_N(γ) and η(L) have the following relationship

$$ \eta_{N} \left( \gamma \right) = \eta \left( {\gamma L_{e} } \right) $$

(16.2)

If the normalized efficiency functions of two generators are identical, then

$$ \eta_{N1} \left( \gamma \right) = \eta_{N2} \left( \gamma \right) $$

(16.3)

We define them as efficiency similarity generators. The power station which includes the efficiency similarity generators is called the efficiency similarity power station (ESPS).

The electricity energy generated by an ESPS is expressed as

$$\begin{aligned} W_{t} &= \mathop \sum \limits_{i = 1}^{n} W_{i} = \frac{{W_{0} }}{{L_{t} }}\mathop \sum \limits_{i = 1}^{n} L_{i} \eta_{i} \left( {L_{i} } \right) = \frac{{W_{0} }}{{L_{t} }}\mathop \sum \limits_{i = 1}^{n} \gamma_{i} L_{ie} \eta_{i} \left( {\gamma_{i} L_{ie} } \right) = \frac{{W_{0} }}{{L_{t} }}\mathop \sum \limits_{i = 1}^{n} \gamma_{i} L_{ie} \eta_{N} \left( {\gamma_{i} } \right) \\ & = W_{0} \eta_{0} \end{aligned}$$

(16.4)

$$ \gamma_{i} = \frac{{L_{i} }}{{L_{ie} }} $$

$$ \mathop \sum \limits_{i = 1}^{n} \gamma_{i} L_{ie} = L_{t} $$

$$ \eta_{0} = \frac{1}{{L_{t} }}\mathop \sum \limits_{i = 1}^{n} \gamma_{i} L_{ie} \eta \left( {\gamma_{i} L_{ie} } \right) = \frac{1}{{L_{t} }}\mathop \sum \limits_{i = 1}^{n} \gamma_{i} L_{ie} \eta_{N} \left( {\gamma_{i} } \right) $$

where n is the total number of generators, W₀ is the ideal energy input from raw materials, W_t is the total power generation of the power station, W_i is the power generation of the i-th generator, L_t is the total input of the power station, η₀ represents the total power generation efficiency, η_i(L_i) represents the operating efficiency of the i-th generator, η_N($\gamma_{i}$) represents the normalized efficiency of the i-th generator, L_i represents the adjustable input of the i-th generator, and L_ie represents the maximum input of the i-th generator. Optimal input, L_i0 represents L when the i-th generator is zero load, γ_i represents the load rate of the i-th generator, there is

$$ L_{t} = \mathop \sum \limits_{i = 1}^{n} L_{i} > \mathop \sum \limits_{i = 1}^{n} L_{i0} $$

(16.5)

16.2 Optimal Control in an ESPS

Load distribution theorem: For the optimization problem of W_t at the fixed L_t, the maximization of the total electricity energy output W_t of the ESPS

$$ \mathop {\max }\limits_{\begin{subarray}{l} s.t. L_{m} \ge L_{i} > L_{0} \\ \quad \quad \mathop \sum \limits_{i = 1}^{n} L_{i} = L_{t} \end{subarray} } W_{t} $$

(16.6)

is given by

$$ \max W_{t} = W_{0} \eta_{N} \left( {\frac{{L_{t} }}{{\mathop \sum \nolimits_{i = 1}^{n} L_{ie} }}} \right) $$

(16.7)

That is the maximization of the overall efficiency η₀ of the power station

$$ \mathop {\max }\limits_{\begin{subarray}{l} s.t. L_{m} \ge L_{i} > L_{0} \\ \quad \quad \mathop \sum \limits_{i = 1}^{n} L_{i} = L_{t} \end{subarray} } \eta_{0} $$

(16.8)

is given by

$$ \max \eta_{0} = \eta_{N} \left( {\frac{{L_{t} }}{{\mathop \sum \nolimits_{i = 1}^{n} L_{ie} }}} \right) $$

(16.9)

Proof

We begin our inductive proof by considering the case where n = 2.

The constraint condition then becomes

$$ L_{1} + L_{2} = L_{t} $$

(16.10)

where

$$ L_{m} \ge L_{1} > L_{0} $$

(16.11)

$$ L_{m} \ge L_{2} > L_{0} $$

That is

$$ \gamma_{1} L_{1e} + \gamma_{2} L_{2e} = L_{t} $$

(16.12)

The objective function W_t is expressed as

$$ W_{t} = \frac{{W_{0} }}{{L_{t} }}\left( {\gamma_{1} L_{1e} \eta_{N} \left( {\gamma_{1} } \right) + \gamma_{2} L_{2e} \eta_{N} \left( {\gamma_{2} } \right)} \right) $$

(16.13)

The optimal condition is given for

$$ W_{t}^{\prime } \left( {\gamma_{1} } \right) = 0 $$

(16.14)

We have

$$ L_{1e} \eta_{N} \left( {\gamma_{1} } \right) + \gamma_{1} L_{1e} \eta_{N}^{\prime } \left( {\gamma_{1} } \right) - L_{1e} \eta_{N} \left( {\gamma_{2} } \right) - \left( {\gamma_{2} L_{1e} } \right)n_{N}^{\prime } \left( {\gamma_{2} } \right) = 0 $$

(16.15)

It is then easily verified that

$$ \gamma_{1} = \gamma_{2} = \frac{{L_{t} }}{{\mathop \sum \nolimits_{i = 1}^{2} L_{ie} }} $$

(16.16)

is an optimal point.

We then check the second derivative,

$$ W_{t}^{\prime \prime } < 0 $$

(16.17)

So, the optimal point is only maximum.

The maximal value of the total electricity energy output W_t of the power station is

$$ \max W_{t} = W_{0} \eta_{N} \left( {\frac{{L_{t} }}{{\mathop \sum \nolimits_{i = 1}^{2} L_{ie} }}} \right) $$

(16.18)

The maximal value of the overall efficiency η₀ of the power station is

$$ \max \eta_{0} = \eta_{N} \left( {\frac{{L_{t} }}{{\mathop \sum \nolimits_{i = 1}^{2} L_{ie} }}} \right) $$

(16.19)

We then assume that this holds for n = k. The above conclusion is readily extended to the case of n = k, and the optimal point is then

$$ \gamma_{1} = \gamma_{2} = \ldots = \gamma_{k} = \frac{{L_{t} }}{{\mathop \sum \nolimits_{i = 1}^{k} L_{ie} }} $$

(16.20)

The maximal value of W_t is

$$ \max W_{t} = W_{0} \eta_{N} \left( {\frac{{L_{t} }}{{\mathop \sum \nolimits_{i = 1}^{k} L_{ie} }}} \right) $$

(16.21)

The maximal value of the overall efficiency η₀ of the power station is

$$ \max \eta_{0} = \eta_{N} \left( {\frac{{L_{t} }}{{\mathop \sum \nolimits_{i = 1}^{2} L_{ie} }}} \right) $$

(16.22)

Our inductive case is then given by n = k + 1. For the total electricity energy output W_t we have

$$ W_{t} = \frac{{W_{0} }}{{L_{t} }}\left( {\mathop \sum \limits_{i = 1}^{k} \gamma_{i} L_{ie} \eta_{N} \left( {\gamma_{i} } \right) + \gamma_{k + 1} L_{{\left( {k + 1} \right)e}} \eta_{N} \left( {\gamma_{k + 1} } \right)} \right) $$

(16.23)

and the maximum of the first item is

$$ \max \frac{{W_{0} }}{{L_{t} }}\mathop \sum \limits_{i = 1}^{k} \gamma_{i} L_{ie} \eta_{N} \left( {\gamma_{i} } \right) = \frac{{W_{0} }}{{L_{t} }}\left( {L_{t} - \gamma_{k + 1} L_{{\left( {k + 1} \right)e}} } \right)\eta_{N} \left( {\frac{{L_{t} - \gamma_{k + 1} L_{{\left( {k + 1} \right)e}} }}{{\mathop \sum \nolimits_{i = 1}^{k} L_{ie} }}} \right) $$

(16.24)

where

$$ \gamma_{1} = \gamma_{2} = \ldots = \gamma_{k} $$

(16.25)

The expression for W_t becomes

$$ W_{t} = \frac{{W_{0} }}{{L_{t} }}\left( {\left( {L_{t} - \gamma_{k + 1} L_{{\left( {k + 1} \right)e}} } \right)\eta_{N} \left( {\frac{{L_{t} - \gamma_{k + 1} L_{{\left( {k + 1} \right)e}} }}{{\mathop \sum \nolimits_{i = 1}^{k} L_{ie} }}} \right) + \gamma_{k + 1} L_{{\left( {k + 1} \right)e}} \eta_{N} \left( {\gamma_{k + 1} } \right)} \right) $$

(16.26)

The optimal condition is given for

$$ W_{t}^{\prime } \left( {\gamma_{k + 1} } \right) = 0 $$

(16.27)

We have

$$ \begin{gathered} - L_{{\left( {k + 1} \right)e}} \eta_{N} \left( {\frac{{L_{t} - \gamma_{k + 1} L_{{\left( {k + 1} \right)e}} }}{{\mathop \sum \nolimits_{i = 1}^{k} L_{ie} }}} \right) - \left( {L_{t} - \gamma_{k + 1} L_{{\left( {k + 1} \right)e}} } \right)\eta_{N}^{\prime } \left( {\frac{{L_{t} - \gamma_{k + 1} L_{{\left( {k + 1} \right)e}} }}{{\mathop \sum \nolimits_{i = 1}^{k} L_{ie} }}} \right)\frac{{L_{{\left( {k + 1} \right)e}} }}{{\mathop \sum \nolimits_{i = 1}^{k} L_{ie} }} \hfill \\ + L_{{\left( {k + 1} \right)e}} \left. {\eta_{N} (\gamma_{k + 1} } \right)+ \gamma_{k + 1} L_{{\left( {k + 1} \right)e}} \eta_{N}^{\prime } \left( {\gamma_{k + 1} } \right) = 0 \hfill \\ \end{gathered} $$

(16.28)

It is then easily verified that

$$ \gamma_{k + 1} = \frac{{L_{t} - \gamma_{k + 1} L_{{\left( {k + 1} \right)e}} }}{{\mathop \sum \nolimits_{i = 1}^{k} L_{ie} }} $$

(16.29)

is an optimal point.

That is

$$ \gamma_{k + 1} = \frac{{L_{t} }}{{\mathop \sum \nolimits_{i = 1}^{k + 1} L_{ie} }} $$

(16.30)

and

$$ \gamma_{1} \mathop \sum \limits_{i = 1}^{k} L_{ie} = L_{t} - \gamma_{k + 1} L_{{\left( {k + 1} \right)e}} $$

(16.31)

$$ \gamma_{1} = \frac{{L_{t} - \gamma_{k + 1} L_{{\left( {k + 1} \right)e}} }}{{\mathop \sum \nolimits_{i = 1}^{k} L_{ie} }} = \frac{{L_{t} }}{{\mathop \sum \nolimits_{i = 1}^{k + 1} L_{ie} }} $$

Therefore, the optimal point is then

$$ \gamma_{1} = \gamma_{2} = \ldots = \gamma_{k} = \gamma_{k + 1} = \frac{{L_{t} }}{{\mathop \sum \nolimits_{i = 1}^{k + 1} L_{ie} }} $$

(16.32)

and the maximal value of the total electricity energy output W_t of the power station is

$$ \max W_{t} = W_{0} \eta_{N} \left( {\frac{{L_{t} }}{{\mathop \sum \nolimits_{i = 1}^{k + 1} L_{ie} }}} \right) $$

(16.33)

The maximal value of the overall efficiency η₀ of the power station is

$$ \max \eta_{0} = \eta_{N} \left( {\frac{{L_{t} }}{{\mathop \sum \nolimits_{i = 1}^{k + 1} L_{ie} }}} \right) $$

(16.34)

Load distribution theorem: In an ESPS which consists of n generators that do not have all the identical model, the optimal control method is to keep each generator to have the same load rate.

16.3 Optimization Discriminant of the ESPS

We define $\gamma \left( {L_{t} ,n} \right)$ as

$$ \gamma \left( {L_{t} , n} \right) = \frac{{L_{t} }}{{\mathop \sum \nolimits_{i = 1}^{n} L_{ie} }} $$

(16.35)

In an ESPS, suppose that it has M-unit generators in total, if n is the optimal, as shown in Fig. 16.4, there must be

$$\begin{aligned} W_{0} \eta _{N} \left( {\frac{{L_{t} }}{{\mathop \sum \nolimits_{{i = 1}}^{n} L_{{ie}} }}} \right) & = \max \left( {W_{0} \eta _{N} \left( {\frac{{L_{t} }}{{\mathop \sum \nolimits_{{i = 1}}^{1} L_{{ie}} }}} \right),W_{0} \eta _{N} \left( {\frac{{L_{t} }}{{\mathop \sum \nolimits_{{i = 1}}^{2} L_{{ie}} }}} \right), \ldots ,} \right. \\ & \quad \left. {W_{0} \eta _{N} \left( {\frac{{L_{t} }}{{\mathop \sum \nolimits_{{i = 1}}^{M} L_{{ie}} }}} \right)} \right) \\ \end{aligned} $$

(16.36)

Fig. 16.4

The n is the optimal

Namely

$$ \eta_{N} \left( {\frac{{L_{t} }}{{\mathop \sum \nolimits_{i = 1}^{n} L_{ie} }}} \right) = {\text{max}}\left( {\eta_{N} \left( {\frac{{L_{t} }}{{\mathop \sum \nolimits_{i = 1}^{1} L_{ie} }}} \right),\eta_{N} \left( {\frac{{L_{t} }}{{\mathop \sum \nolimits_{i = 1}^{2} L_{ie} }}} \right), \ldots ,\eta_{N} \left( {\frac{{L_{t} }}{{\mathop \sum \nolimits_{i = 1}^{M} L_{ie} }}} \right)} \right) $$

(16.37)

In Fig. 16.4, n1 and n2 are arbitrary number which corresponds to different device combinations. n1 is less than or equal to M, and n2 is less than or equal to M also.

Note: Even though the same n₁, there are many different generator combinations.

16.4 Optimal Switch in the ESPS

Now we consider the optimal switch point for an ESPS. Then we take n to be less than or equal to M and the optimum, the total electricity energy output W_t of the ESPS is the maximum.

$$ \mathop {\max }\limits_{\begin{subarray}{l} s.t. L_{m} \ge L_{i} > L_{0} \\ \quad \quad \quad \mathop \sum \limits_{i = 1}^{n} L_{i} = L_{t} \\ \quad \quad \quad n \le M \end{subarray} } W_{t} = W_{0} \eta_{N} \left( {\frac{{L_{t} }}{{\mathop \sum \nolimits_{i = 1}^{n} L_{ie} }}} \right) $$

(16.38)

The optimal overall efficiency is then

$$ \max \eta_{0} = \eta_{N} \left( {\frac{{L_{t} }}{{\mathop \sum \nolimits_{i = 1}^{n} L_{ie} }}} \right) $$

(16.39)

If the L_t varies, the optimal n may change as well. The optimal switching point depends on whether L_t is increasing or decreasing.

Theorem

The optimal switch point is at η_N(L_t, n) = η_N(L_t, n₂), if L_t is increasing.

Proof

From Figs. 16.5 and 16.6, the η_N(L_t, n₁) is less than η_N(L_t, n) and greater than other efficiency values on the right side of γ = 1,the η_N(L_t, n₂) is less than η_N(L_t, n) and greater than other efficiency values on the left side of γ = 1, we see that

$$ \eta_{N} \left( {L_{t} ,n} \right) = \max \left( {\eta_{N} \left( {L_{t} ,n1} \right),\eta_{N} \left( {L_{t} ,n} \right),\eta_{N} \left( {L_{t} ,n2} \right)} \right) $$

(16.40)

Fig. 16.5

η_N(L_t, n) curve when γ(L_t, n) is less than 1

Fig. 16.6

η_N(L_t, n) curve when γ(L_t, n) is greater than 1

As seen in Fig. 16.5, if γ (L_t, n) < 1 and L_t increases, then η_N(L_t, n) will continue to increase until γ(L_t, n) reaches 1, immediately after which point it will begin to decrease. At the same time, η_N(L_t, n₂) will increase until it will eventually become greater than η_N(L_t, n) and will be the new maximum. This change of the maximum efficiency defines the optimal switch point at η_N(L_t, n) = η_N(L_t, n₂).

If γ (L_t, n) > 1, then η_N(L_t, n) will decrease with the increase of L_t while η_N(L_t, n₂) will increase as can be inferred from Fig. 16.6. Ultimately, η_N(L_t, n₂) will increase to such a point that it will become the new maximum and η_N(L_t, n) = η_N(L_t, n₂) is the optimal switch point.

If γ (L_t, n) > 1, then η_N(L_t, n) will decrease as L_t increases, while η_N(L_t, n₂) will increase, as shown in Fig. 16.6. Eventually, η_N(L_t, n₂) will increase to the point where it becomes the new maximum, and η_N(L_t, n) = η_N(L_t, n₂) is the optimal switching point.

Theorem

The optimal switch point for an M-unit system is at η_N(L_t, n) = η_N(L_t, n₁) if L_t is decreasing.

Proof

Similarly, if L_t decreases and γ(L_t, n) > 1, Fig. 16.6 shows that η_N(L_t, n) will increase until γ(L_t, n) reaches the value 1, at which point it will immediately start decreasing. At the same time, η_N(L_t, n₁) will increase. As before, there is a point where η_N(L_t, n₁) exceeds η_N(L_t, n) as a new maximum. This is our optimal switching point, given by: η_N(L_t, n) = η_N(L_t, n₁).

As shown in Fig. 16.5, for a given γ(L_t, n) < 1, η_N(L_t, n) will decrease as L_t decreases. We also find that η_N(L_t, n₁) will increase to the point of maximum efficiency. Then the optimal switching point is η_N(L_t, n) = η_N(L_t, n₁).

16.5 ESPS Operates at Maximum Efficiency

For the total L_t, if the optimal run-unit of the ESPS is n as described above, the maximal value of the overall efficiency η₀ of the power station is

$$ \max \eta_{0} = \eta_{N} \left( {\frac{{L_{t} }}{{\mathop \sum \nolimits_{i = 1}^{n} L_{ie} }}} \right) $$

(16.41)

If the following equation holds

$$ \frac{{L_{t} }}{{\mathop \sum \nolimits_{i = 1}^{n} L_{ie} }} = L_{e} $$

(16.42)

Then

$$ \eta_{N} \left( {\frac{{L_{t} }}{{\mathop \sum \nolimits_{i = 1}^{n} L_{ie} }}} \right) = \eta_{e} $$

(16.43)

Thus, the ESPS can work at the maximum efficiency η_e.

16.6 Conclusion

By assuming a fixed value of the total load, we propose an optimal control method that does not rely on an accurate model of the ESPS. By varying the total load, we also propose an optimal switching method. This method depends only on the efficiency function of the generator.

The proofs of optimal control and switching methods given in this chapter are mainly based on the characteristics of the efficiency function, which can be approximately considered as a concave and non-negative function. Therefore, this optimal method has the following characteristics:

• (1) High versatility,

• (2) Includes linear and nonlinear systems,

• (3) No mathematical model of the system is required.