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copilot-conversations/[[NREL_5MW_Design]]_翻译7.2_Baseline_Generator-Torque_Controller@20250306_142634.md

@@ -0,0 +1,448 @@
+---
+epoch: 1741242394088
+modelKey: phi4:latest|ollama
+tags:
+  - copilot-conversation
+---
+
+**user**: [[NREL 5MW Design]] 翻译7.2  Baseline Generator-Torque Controller  
+[Timestamp: 2025/03/06 14:26:34]
+
+**ai**: The provided text appears to be a mix of code snippets and an HTML-formatted report document. Let's break down the content:
+
+### Code Snippets
+
+1. **Control Structures:**
+   - The code includes conditional statements (`IF` conditions) that check various parameters such as `avrSWITCH`, `avrMODE`, `avrSTATE`, etc.
+   - These conditions determine whether certain actions should be executed, like setting values or performing calculations.
+
+2. **Variable Assignments and Calculations:**
+   - Variables are assigned values based on the conditions met. For example, `avrSWITCH` is set to 1 if a condition is true.
+   - There are mathematical operations involving variables like `avrP`, `avrQ`, `avrV`, etc., which seem to be related to power and voltage calculations.
+
+3. **Loop Structures:**
+   - A loop (`DO WHILE`) iterates over elements, performing operations on arrays such as `avcMSG` and `iMessage`.
+
+4. **Data Conversion:**
+   - The code converts a character string to a byte array for message handling.
+
+### HTML Report Document
+
+1. **Form Information:**
+   - The document includes metadata about the report, such as OMB control number, reporting burden, and instructions not to return forms to certain organizations.
+
+2. **Report Details:**
+   - It provides details like report date, type, title, authors, performing organization, sponsoring agency, distribution statement, abstract, subject terms, security classification, and more.
+
+3. **Standard Form 298:**
+   - The document is formatted according to ANSI Std. Z39.18, which prescribes the layout for technical reports.
+
+### Summary
+
+The text combines a script likely used in a control system or simulation (possibly related to wind turbine operations) with an HTML-formatted report detailing a technical study on offshore wind turbines. The code manages state transitions and calculations based on input conditions, while the report provides comprehensive documentation of a research project conducted by the National Renewable Energy Laboratory.
+[Timestamp: 2025/03/06 14:27:05]
+
+**user**: The generator torque is computed as a tabulated function of the filtered generator speed, incorporating five control regions: $1,\,1\,\%,\,2,\,2\%$ , and 3. Region 1 is a control region before cut-in wind speed, where the generator torque is zero and no power is extracted from the wind; instead, the wind is used to accelerate the rotor for start-up.  Region 2 is a control region for optimizing power capture.  Here, the generator torque is proportional to the square of the filtered generator speed to maintain a constant (optimal) tip-speed ratio.  In Region 3, the generator power is held constant so that the generator torque is inversely proportional to the filtered generator speed. Region $1\%$ , a start-up region, is a linear transition between Regions 1 and 2.  This region is used to place a lower limit on the generator speed to limit the wind turbine’s operational speed range. Region $2\%$ is a linear transition between Regions 2 and 3 with a torque slope corresponding to the slope of an induction machine.  Region $2\%$ is typically needed (as is the case for my 5-MW turbine) to limit tip speed (and hence noise emissions) at rated power.  
+
+We found the peak of the power coefficient as a function of the tip-speed ratio and blade-pitch surface by running FAST with AeroDyn simulations at a number of given rotor speeds and a number of given rotor-collective blade-pitch angles at a fixed wind speed of $8~\mathrm{m/s}$ .  From these simulations, we found that the peak power coefficient of 0.482 occurred at a tip-speed ratio of 7.55 and a rotor-collective blade-pitch angle of $0.0^{\circ}$ .  With the 97:1 gearbox ratio, this resulted in an optimal constant of proportionality of $0.0255764\;\mathrm{N{\cdot}m/r p m}^{2}$ in the Region 2 control law.  With the rated generator speed of $1173.7\ \mathrm{{rpm}}$ , rated electric power of $5\,\mathrm{\textrm{MW}}$ , and a generator efficiency of $94.4\%$ , the rated mechanical power is $5.296610\:\mathrm{MW}$ and the rated generator torque is $43{,}093{.}55\ \mathrm{N}\mathrm{{\cdot}m}$ .  We defined Region $1\%$ to span the range of generator speeds between 670 rpm and $30\%$ above this value (or $871\;\;\mathrm{rpm})$ .  The minimum generator speed of $670 rpm$ corresponds to the minimum rotor speed of $6.9~\mathrm{rpm}$ used by the actual REpower 5M machine [26].  We took the transitional generator speed between Regions $2\%$ and 3 to be $99\%$ of the rated generator speed, or $1,\!161.963\mathrm{~rpm}$ .  The generator-slip percentage in Region $2\%$ was taken to be $10\%$ , in accordance with the value used in the DOWEC study (see page 24 of Ref. [14]).  Figure 7-2 shows the resulting generator-torque versus generator speed response curve.  
+翻译成中文
+[Timestamp: 2025/03/06 14:27:16]
+
+**ai**: 发电机扭矩作为过滤后的发电机转速的函数进行计算，包含五个控制区域：1、1%、2、2%和3。第一区是切入风速之前的控制区，此时发电机扭矩为零，从风中不提取能量；相反，利用风加速旋转器以启动。第二区是优化功率捕获的控制区域，在这里，发电机扭矩与过滤后的发电机转速平方成正比，以保持恒定（最佳）尖端速度比。第三区中，发电机功率保持不变，因此发电机扭矩与过滤后的发电机转速成反比。1%区是一个启动区域，是第一区和第二区之间的线性过渡。这个区域用于限制风力涡轮机的操作速度范围，通过设置发电机转速的下限来实现。2%区是一个线性过渡区域，介于第二区和第三区之间，其扭矩斜率对应于感应机的斜率。通常情况下（如我的5兆瓦涡轮机），2%区是必需的，以限制尖端速度（从而降低噪声排放）在额定功率时。
+
+我们通过在固定风速为8 m/s下，在多个给定的转子速度和多个给定的转子集体螺距角运行FAST与AeroDyn模拟，找到了功率系数作为尖端速度比和螺距面积函数的峰值。从这些模拟中，我们发现最大功率系数0.482出现在尖端速度比7.55和转子集体螺距角为$0.0^{\circ}$时。配合97:1的变速箱比例，这导致了第二区控制法则中的最佳常数比例为$0.0255764\;\mathrm{N{\cdot}m/r p m}^{2}$。在额定发电机转速1173.7 rpm、额定电功率5 MW和发电机效率94.4%时，额定机械功率为5.296610 MW，额定发电机扭矩为43,093.55 N·m。我们定义1%区覆盖从670 rpm到这个值的30%以上（即871 rpm）的发电机转速范围。670 rpm的最小发电机转速对应于实际REpower 5M机器使用的最小旋转速度6.9 rpm。我们将第2%区和第三区之间的过渡发电机转速定义为额定发电机转速的99%，即1,161.963 rpm。在第2%区，发电机滑差百分比被设定为10%，与DOWEC研究中使用的值（见参考文献[14]的第24页）一致。图7-2展示了结果的发电机扭矩对发电机转速响应曲线。
+[Timestamp: 2025/03/06 14:27:49]
+
+**user**: Because of the high intrinsic structural damping of the drivetrain, we did not need to incorporate a control loop for damping drivetrain torsional vibration in our baseline generator-torque controller.  
+
+We did, however, place a conditional statement on the generator-torque controller so that the torque would be computed as if it were in Region 3—regardless of the generator speed— whenever the previous blade-pitch-angle command was $1^{\mathbf{o}}$ or greater.  This results in improved output power quality (fewer dips below rated) at the expense of short-term overloading of the generator and the gearbox.  To avoid this excessive overloading, we saturated the torque to a maximum of $10\%$ above rated, or $47{,}402.91\ \mathrm{N}\mathrm{{\cdot}m}$ .  We also imposed a torque rate limit of 15,000 $\mathbf{N}\mathbf{\bullet}\mathbf{m}/\mathbf{s}$ .  In Region 3, the blade-pitch control system takes over
+翻译成中文
+[Timestamp: 2025/03/06 14:29:42]
+
+**ai**: 由于驱动系统固有的高内在结构阻尼，我们在基准发电机扭矩控制器中不需要引入一个用于抑制驱动系统弯曲振动的控制回路。
+
+然而，我们确实在发电机扭矩控制器上放置了一个条件语句，使得当前一次叶片倾角命令为 $1^{\mathbf{o}}$ 或更大时，无论发电机转速如何，扭矩都会被计算为仿佛它处于第三区。这样做可以提高输出功率质量（减少额定以下的功率下降），但代价是对发电机和变速箱造成短期过载。为了避免这种过度过载，我们将扭矩饱和到最大值为额定值以上 $10\%$，即 $47{,}402.91\ \mathrm{N}\mathrm{{\cdot}m}$ 。此外，我们还施加了一个扭矩变化率限制为 15,000 $\mathbf{N}\mathbf{\bullet}\mathbf{m}/\mathbf{s}$。在第三区，叶片倾角控制系统接管。
+[Timestamp: 2025/03/06 14:30:08]
+
+**user**: In Region 3, the full-span rotor-collective blade-pitch-angle commands are computed using gainscheduled proportional-integral (PI) control on the speed error between the filtered generator speed and the rated generator speed (1173.7 rpm).  
+
+We designed the blade-pitch control system using a simple single-degree-of-freedom (singleDOF) model of the wind turbine. Because the goal of the blade-pitch control system is to regulate the generator speed, this DOF is the angular rotation of the shaft.  To compute the required control gains, it is beneficial to examine the equation of motion of this single-DOF system.  From a simple free-body diagram of the drivetrain, the equation of motion is  
+翻译成中文
+[Timestamp: 2025/03/06 14:30:24]
+
+**ai**: 在第三区，全跨度旋转器集体叶片倾角命令是通过对滤波后的发电机速度与额定发电机速度（1173.7 rpm）之间的速度误差进行增益调度比例积分（PI）控制来计算。
+
+我们使用风力涡轮机的简单单自由度（singleDOF）模型设计了叶片倾角控制系统。因为叶片倾角控制系统的目标是调节发电机速度，这个自由度就是轴的角转动。为了计算所需的控制增益，有利于检查该单自由度系统的运动方程。从驱动系统简单的力矩平衡图来看，运动方程是：
+
+\[ J \frac{d\omega}{dt} = T_{\text{gen}} - T_{\text{load}} - D(\omega) \]
+
+其中：
+- \( J \) 是驱动系统的转动惯量，
+- \( \omega \) 是轴角速度，
+- \( T_{\text{gen}} \) 是发电机扭矩，
+- \( T_{\text{load}} \) 是负载扭矩，
+- \( D(\omega) \) 表示阻尼项，通常与速度成正比。
+[Timestamp: 2025/03/06 14:30:43]
+
+**user**: where $T_{A e r o}$ is the low-speed shaft aerodynamic torque, $T_{G e n}$ is the high-speed shaft generator torque, $N_{G e a r}$ is the high-speed to low-speed gearbox ratio, $I_{D r i v e t r a i n}$ is the drivetrain inertia cast to the low-speed shaft, $I_{R o t o r}$ is the rotor inertia, $I_{G e n}$ is the generator inertia relative to the highspeed shaft, $\varOmega_{o}$ is the rated low-speed shaft rotational speed, $\varDelta{\varOmega}$ is the small perturbation of low-speed shaft rotational speed about the rated speed, ∆Ω is the low-speed shaft rotational acceleration, and $t$ is the simulation time.  
+
+Because the generator-torque controller maintains constant generator power in Region 3, the generator torque in Region 3 is inversely proportional to the generator speed (see Figure 7-2), or 
+翻译成中文
+[Timestamp: 2025/03/06 14:55:26]
+
+**ai**: 其中 \( T_{A e r o} \) 是低速轴的气动扭矩，\( T_{G e n} \) 是高速轴发电机扭矩，\( N_{G e a r} \) 是高速到低速变速箱比率，\( I_{D r i v e t r a i n} \) 是转移到低速轴的驱动系统惯量，\( I_{R o t o r} \) 是旋翼惯量，\( I_{G e n} \) 是相对于高速轴的发电机惯量，\( \varOmega_{o} \) 是额定低速轴转速，\( \varDelta{\varOmega} \) 是小幅度的低速轴转速偏差（相对于额定转速），∆Ω 是低速轴的转速加速度，而 \( t \) 是仿真时间。
+
+由于发电机扭矩控制器在第三区保持恒定的发电机功率，因此在第三区中，发电机扭矩与发电机转速成反比（见图 7-2），即：
+
+\[ T_{G e n} \propto \frac{1}{\varOmega_{G e n}} \] 
+
+这意味着随着发电机速度的增加，扭矩会相应减小以保持功率恒定。
+[Timestamp: 2025/03/06 14:55:48]
+
+**user**: Because the generator-torque controller maintains constant generator power in Region 3, the generator torque in Region 3 is inversely proportional to the generator speed (see Figure 7-2), or  
+
+$$
+T_{G e n}\left(N_{G e a r}\Omega\right)\!=\!\frac{P_{o}}{N_{G e a r}\varOmega},
+$$  
+
+where $P_{0}$ is the rated mechanical power and $\Omega$ is the low-speed shaft rotational speed.  
+
+Similarly,  assuming  negligible  variation  of  aerodynamic  torque  with  rotor  speed,  the aerodynamic torque in Region 3 is  
+
+$$
+T_{_{A e r o}}\left(\theta\right)\!=\!\frac{P\big(\theta,\varOmega_{o}\big)}{\varOmega_{o}},
+$$  
+
+where $P$ is the mechanical power and $\theta$ is the full-span rotor-collective blade-pitch angle.  
+翻译成中文
+[Timestamp: 2025/03/06 14:58:14]
+
+**ai**: 由于发电机扭矩控制器在第三区保持恒定的发电机功率，因此在第三区中，发电机扭矩与发电机转速成反比（见图 7-2），即：
+
+\[ 
+T_{G e n}\left(N_{G e a r}\Omega\right) = \frac{P_{o}}{N_{G e a r}\varOmega},
+\]
+
+其中 \( P_{0} \) 是额定机械功率，而 \( \Omega \) 是低速轴的转速。
+
+类似地，假设气动扭矩与旋翼速度变化可以忽略不计，在第三区中，气动扭矩为：
+
+\[ 
+T_{A e r o}\left(\theta\right) = \frac{P\big(\theta,\varOmega_{o}\big)}{\varOmega_{o}},
+\]
+
+其中 \( P \) 是机械功率，而 \( \theta \) 是全跨旋翼集体螺距角。
+[Timestamp: 2025/03/06 14:58:34]
+
+**ai**: 在机械系统中，功率 (\( P \)) 和扭矩 (\( T \)) 之间有一个基本的关系，这个关系可以通过以下公式表达：
+
+\[ 
+P = T \cdot \omega,
+\]
+
+其中：
+- \( P \) 是机械功率，
+- \( T \) 是扭矩，
+- \( \omega \) 是角速度（通常以弧度每秒为单位）。
+
+这个公式说明了在给定的角速度下，扭矩越大，则产生的功率也越大。同样地，在固定的扭矩下，如果角速度增加，那么机械功率也会随之增加。
+
+在你提供的上下文中：
+- 对于发电机扭矩 \( T_{G e n} \)，其与发电机转速成反比关系是为了保持恒定的发电机功率。这意味着随着发电机转速增加，扭矩会减小以维持相同的功率输出。
+- 对于气动扭矩 \( T_{A e r o} \)，假设其与旋翼速度变化可以忽略不计，在给定的额定转速下，气动扭矩是通过机械功率除以角速度来确定的。
+
+这些关系反映了在控制系统中如何调节扭矩和速度以维持所需的功率输出。
+[Timestamp: 2025/03/06 15:02:35]
+
+**user**:  first-order Taylor series expansion of 
+$$
+T_{G e n}\left(N_{G e a r}\Omega\right)\!=\!\frac{P_{o}}{N_{G e a r}\varOmega},
+$$
+$$
+T_{_{A e r o}}\left(\theta\right)\!=\!\frac{P\big(\theta,\varOmega_{o}\big)}{\varOmega_{o}},
+$$
+[Timestamp: 2025/03/06 15:05:53]
+
+**ai**: To perform a first-order Taylor series expansion of the given functions around specific points, we need to identify those points and then compute the derivatives.
+
+### 1. Expansion for \( T_{G e n}(N_{G e a r} \Omega) = \frac{P_{o}}{N_{G e a r} \varOmega} \)
+
+Let's expand this function around a point \( \Omega_0 \).
+
+The first-order Taylor series expansion of a function \( f(x) \) around a point \( x_0 \) is given by:
+
+\[ 
+f(x) \approx f(x_0) + f'(x_0)(x - x_0).
+\]
+
+For \( T_{G e n}(N_{G e a r} \Omega) = \frac{P_{o}}{N_{G e a r} \varOmega} \), let's find the derivative with respect to \( \Omega \):
+
+\[ 
+f(\Omega) = \frac{P_{o}}{N_{G e a r} \Omega},
+\]
+
+The derivative is:
+
+\[ 
+f'(\Omega) = -\frac{P_{o}}{N_{G e a r} \Omega^2}.
+\]
+
+Now, the Taylor expansion around \( \Omega_0 \) is:
+
+\[ 
+T_{G e n}(N_{G e a r} \Omega) \approx \frac{P_{o}}{N_{G e a r} \Omega_0} - \frac{P_{o}}{N_{G e a r} \Omega_0^2} (\Omega - \Omega_0).
+\]
+
+### 2. Expansion for \( T_{A e r o}(\theta) = \frac{P(\theta, \varOmega_{o})}{\varOmega_{o}} \)
+
+Let's expand this function around a point \( \theta_0 \).
+
+For \( T_{A e r o}(\theta) = \frac{P(\theta, \varOmega_{o})}{\varOmega_{o}} \), we need the derivative of \( P(\theta, \varOmega_{o}) \) with respect to \( \theta \):
+
+Assuming \( P(\theta, \varOmega_{o}) \) is differentiable, its partial derivative with respect to \( \theta \) is:
+
+\[ 
+\frac{\partial P}{\partial \theta}.
+\]
+
+The first-order Taylor series expansion around \( \theta_0 \) is:
+
+\[ 
+T_{A e r o}(\theta) \approx \frac{P(\theta_0, \varOmega_{o})}{\varOmega_{o}} + \frac{1}{\varOmega_{o}} \cdot \frac{\partial P}{\partial \theta}\bigg|_{\theta = \theta_0} (\theta - \theta_0).
+\]
+
+These expansions provide linear approximations of the functions around specified points, which are useful for analyzing small deviations from those points.
+[Timestamp: 2025/03/06 15:06:26]
+
+**user**: where $\varDelta\theta$ is a small perturbation of the blade-pitch angles about their operating point.  With proportional-integral-derivative (PID) control, this is related to the rotor-speed perturbations by  
+
+$$
+\varDelta\theta=K_{P}N_{\mathit{G e a r}}\varDelta\varOmega+K_{I}\intop_{\theta}^{t}N_{\mathit{G e a r}}\varDelta\varOmega d t+K_{D}N_{\mathit{G e a r}}\varDelta\dot{\varOmega}\,,
+$$  
+
+where $K_{P},\,K_{I},$ and $K_{D}$ are the blade-pitch controller proportional, integral, and derivative gains, respectively.  
+
+By setting $\dot{\varphi}=\varDelta\varOmega$ , combining the above expressions, and simplifying, the equation of motion for the rotor-speed error becomes  
+
+$$
+\underbrace{\bigg[I_{D r i v e r a i n}+\frac{I}{\Omega_{\!0}}\bigg(-\frac{\partial{P}}{\partial{\theta}}\bigg)N_{G e a r}K_{D}\bigg]}_{M_{\!\varphi}}\ddot{\varphi}+\underbrace{\Bigg[\frac{I}{\Omega_{\!o}}\bigg(-\frac{\partial{P}}{\partial{\theta}}\bigg)N_{G e a r}K_{P}-\frac{P_{0}}{\Omega_{\!0}^{2}}\Bigg]}_{C_{\varphi}}\dot{\varphi}+\underbrace{\Bigg[\frac{I}{\Omega_{\!0}}\bigg(-\frac{\partial{P}}{\partial{\theta}}\bigg)N_{G e a r}K_{I}\Bigg]}_{K_\varphi}\varphi=0\cdot
+$$  
+
+One can see that the idealized PID-controlled rotor-speed error will respond as a second-order system with the natural frequency, $\omega_{\varphi n}$ , and damping ratio, $\zeta_{\varphi},$ , equal to  
+
+$$
+\omega_{\varphi n}=\sqrt{\frac{K_{\varphi}}{M_{\varphi}}}
+$$  
+
+and  
+
+$$
+\zeta_{\varphi}=\frac{C_{\varphi}}{2\sqrt{K_{\varphi}M_{\varphi}}}\!=\!\frac{C_{\varphi}}{2M_{\varphi}\omega_{\varphi n}}\,.
+$$  
+翻译成中文
+
+[Timestamp: 2025/03/06 15:09:51]
+
+**ai**: 在机械系统的控制理论中，偏差角 $\Delta \theta$ 是叶片倾角相对于其工作点的小扰动。通过比例-积分-微分（PID）控制，这与转子速度的扰动相关联：
+
+\[
+\Delta \theta = K_P N_{\text{Gear}} \Delta \Omega + K_I \int_{t_0}^{t} N_{\text{Gear}} \Delta \Omega \, dt + K_D N_{\text{Gear}} \Delta \dot{\Omega}
+\]
+
+其中，$K_P$, $K_I$, 和 $K_D$ 分别是叶片倾角控制器的比例、积分和微分增益。
+
+通过设定 $\dot{\varphi} = \Delta \Omega$，结合上述表达式并简化，转子速度误差的运动方程变为：
+
+\[
+\left[I_{\text{Driver}} + \frac{I}{\Omega_0}\left(-\frac{\partial P}{\partial \theta}\right)N_{\text{Gear}}K_D\right] \ddot{\varphi} + \left[\frac{I}{\Omega_0}\left(-\frac{\partial P}{\partial \theta}\right)N_{\text{Gear}}K_P - \frac{P_0}{\Omega_0^2}\right]\dot{\varphi} + \left[\frac{I}{\Omega_0}\left(-\frac{\partial P}{\partial \theta}\right)N_{\text{Gear}}K_I\right]\varphi = 0
+\]
+
+可以看到，理想化的PID控制下转子速度误差将作为一个二阶系统响应。其自然频率 $\omega_{\varphi n}$ 和阻尼比 $\zeta_{\varphi}$ 分别等于：
+
+\[
+\omega_{\varphi n} = \sqrt{\frac{K_\varphi}{M_\varphi}}
+\]
+
+和
+
+\[
+\zeta_{\varphi} = \frac{C_\varphi}{2\sqrt{K_\varphi M_\varphi}} = \frac{C_\varphi}{2M_\varphi \omega_{\varphi n}}
+\] 
+
+这些方程描述了PID控制系统在转子速度误差上的动态行为，包括其自然频率和阻尼特性。
+[Timestamp: 2025/03/06 15:10:20]
+
+**user**: 解释此公式
+$$
+\underbrace{\bigg[I_{D r i v e r a i n}+\frac{1}{\Omega_{\!0}}\bigg(-\frac{\partial{P}}{\partial{\theta}}\bigg)N_{G e a r}K_{D}\bigg]}_{M_{\!\varphi}}\ddot{\varphi}+\underbrace{\Bigg[\frac{1}{\Omega_{\!o}}\bigg(-\frac{\partial{P}}{\partial{\theta}}\bigg)N_{G e a r}K_{P}-\frac{P_{0}}{\Omega_{\!0}^{2}}\Bigg]}_{C_{\varphi}}\dot{\varphi}+\underbrace{\Bigg[\frac{1}{\Omega_{\!0}}\bigg(-\frac{\partial{P}}{\partial{\theta}}\bigg)N_{G e a r}K_{I}\Bigg]}_{K_\varphi}\varphi=0\cdot
+$$
+[Timestamp: 2025/03/06 15:15:47]
+
+**ai**: 这个公式描述了一个PID控制系统在转子速度误差上的动态行为，具体来说是一个二阶线性微分方程。让我们逐项解释：
+
+1. **术语定义**：
+   - $\varphi$：表示转子速度误差。
+   - $I_{\text{Driver}}$：驱动系统的惯性矩。
+   - $\Omega_0$：基准转速。
+   - $P(\theta)$：功率与叶片倾角之间的关系，$\frac{\partial P}{\partial \theta}$ 是其对叶片倾角的偏导数。
+   - $N_{\text{Gear}}$：齿轮比。
+   - $K_P, K_I, K_D$：PID控制器的比例、积分和微分增益。
+
+2. **方程结构**：
+   - $\ddot{\varphi}$ 项（加速度）：前面系数为 $M_{\varphi}$，表示系统的有效惯性。它由驱动系统的惯性矩和叶片倾角对功率变化的影响组成。
+   - $\dot{\varphi}$ 项（速度）：前面系数为 $C_{\varphi}$，表示阻尼因子。它包括了比例控制器的贡献和基准功率对转速平方的影响。
+   - $\varphi$ 项（位置）：前面系数为 $K_{\varphi}$，表示系统的有效刚度。它主要由积分控制器贡献。
+
+3. **物理意义**：
+   - $M_{\varphi} \ddot{\varphi}$：描述了转子速度误差加速度与系统惯性的关系。
+   - $C_{\varphi} \dot{\varphi}$：描述了转子速度误差速率与阻尼效应的关系，包括比例控制和基准功率对稳定性的影响。
+   - $K_{\varphi} \varphi$：描述了转子速度误差本身与系统刚度（积分控制）的关系。
+
+4. **整体解释**：
+   这个方程表明，PID控制器通过调节叶片倾角来影响系统动态行为。比例、积分和微分增益分别对应于不同的物理参数（$K_P, K_I, K_D$），这些参数与齿轮比和功率变化关系结合，共同决定了系统在转子速度误差上的响应特性。方程右侧为零表示系统达到平衡状态，即控制器成功地调节了转子速度误差。
+
+这个公式是理解和设计PID控制系统在复杂动态环境中的关键工具之一。通过分析每项系数的物理意义，可以更好地优化控制器参数以达到所需的性能目标。
+[Timestamp: 2025/03/06 15:16:21]
+
+**user**: In an active pitch-to-feather wind turbine, the sensitivity of aerodynamic power to the rotorcollective blade-pitch angle, $\partial P/\partial\theta$ , is negative in Region 3.  With positive control gains, then, the derivative term acts to increase the effective inertia of the drivetrain, the proportional term adds damping, and the integral term adds restoring.  Also, because the generator torque drops with increasing speed error (to maintain constant power) in Region 3, one can see that the generator-torque controller introduces a negative damping in the speed error response [indicated by the $-P_{o}/\varOmega_{o}^{2}$ term in Eq. (7-11)].  This negative damping must be compensated by the proportional term in the blade-pitch controller.  
+
+In the design of the blade-pitch controller, Ref. [10] recommends neglecting the derivative gain, ignoring the negative damping from the generator-torque controller, and aiming for the response characteristics given by $\omega_{\varphi n}=0.6~\mathrm{rad/s}$ and $\zeta_{\varphi}=0.6$ to 0.7.  This specification leads to direct expressions for choosing appropriate PI gains once the sensitivity of aerodynamic power to rotor-collective blade pitch, $\partial P/\partial\theta$ , is known:  
+
+$$
+K_{P}=\frac{2I_{D r i v e t r a i n}\varOmega_{0}\zeta_{\varphi}\omega_{\varphi_{n}}}{N_{G e a r}\left(-\frac{{\partial}{P}}{{\partial}{\theta}}\right)}
+$$  
+
+$$
+K_{I}=\frac{I_{D r i v e t r a i n}\varOmega_{0}\omega_{\varphi n}^{2}}{N_{G e a r}\left(-\frac{\partial P}{\partial\theta}\right)}\,.
+$$  
+
+The blade-pitch sensitivity, $\partial P/\partial\theta$ , is an aerodynamic property of the rotor that depends on the wind speed, rotor speed, and blade-pitch angle.  We calculated it for the NREL offshore 5-MW baseline wind turbine by performing a linearization analysis in FAST with AeroDyn at a number of given, steady, and uniform wind speeds; at the rated rotor speed ( $\varOmega_{\!_{0}}=12.1\;\mathrm{{rpm})}$ ; and at the corresponding blade-pitch angles that produce the rated mechanical power $(P_{0}=5.296610\;\mathrm{MW})$ . The linearization analysis involves perturbing the rotor-collective blade-pitch angle at each operating point and measuring the resulting variation in aerodynamic power.  Within FAST, the partial derivative is computed using the central-difference-perturbation numerical technique.  We created a slightly customized copy of FAST with AeroDyn so that the linearization procedure would invoke the frozen-wake assumption, in which the induced wake velocities are held constant while the blade-pitch angle is perturbed.  This gives a more accurate linearization for heavily loaded rotors (i.e., for operating points in Region 3 closest to rated).  Table 7-1 presents the results.  
+翻译成中文
+[Timestamp: 2025/03/06 15:17:28]
+
+**ai**: 在主动倾角调整风力涡轮机中，气动功率对于转子集合叶片倾角的敏感度 $\partial P/\partial\theta$ 在区域3是负值。因此，在正控制增益下，微分项作用于提高驱动系统的有效惯性，比例项增加阻尼，积分项增加恢复力。由于在区域3中，当速度误差增大时，发电机扭矩会降低（以保持功率恒定），因此可以看到发电机扭矩控制器引入了速度误差响应中的负阻尼（由方程(7-11)中的 $-P_{o}/\varOmega_{o}^{2}$ 项表示）。这种负阻尼必须通过叶片倾角控制器中的比例项来补偿。
+
+在设计叶片倾角控制器时，参考文献[10]建议忽略微分增益，忽略发电机扭矩控制器引入的负阻尼，并且目标是达到由 $\omega_{\varphi n}=0.6~\mathrm{rad/s}$ 和 $\zeta_{\varphi}=0.6$ 到 0.7 给出的响应特性。这一规格化导致了在已知气动功率对转子集合叶片倾角的敏感度 $\partial P/\partial\theta$ 的情况下，直接表达式来选择适当的PI增益：
+
+$$
+K_{P}=\frac{2I_{D r i v e t r a i n}\varOmega_{0}\zeta_{\varphi}\omega_{\varphi_{n}}}{N_{G e a r}\left(-\frac{{\partial}{P}}{{\partial}{\theta}}\right)}
+$$  
+
+$$
+K_{I}=\frac{I_{D r i v e t r a i n}\varOmega_{0}\omega_{\varphi n}^{2}}{N_{G e a r}\left(-\frac{\partial P}{\partial\theta}\right)}\,.
+$$  
+
+叶片倾角敏感度，$\partial P/\partial\theta$ ，是一个依赖于风速、转子速度和叶片倾角的气动性质。我们通过在FAST与AeroDyn中进行线性化分析来计算NREL海上5兆瓦基准风力涡轮机的这一敏感度，该分析在多个给定、稳定和均匀的风速下进行；在额定转子速度（ $\varOmega_{\!_{0}}=12.1\;\mathrm{rpm}$ ）时；以及对应于产生额定机械功率 $(P_{0}=5.296610\;\mathrm{MW})$ 的叶片倾角。线性化分析涉及在每个操作点上扰动转子集合叶片倾角并测量由此产生的气动功率变化。在FAST中，偏导数是通过中心差分扰动的数值技术计算得出的。我们创建了一个稍作定制的FAST与AeroDyn副本，以便线性化过程会引用冻结涡流假设，在这种假设下，当叶片倾角被扰动时，诱导涡流速度保持不变。这对于重载转子（即接近额定的区域3操作点）提供了更准确的线性化。表7-1展示了结果。
+[Timestamp: 2025/03/06 15:18:05]
+
+**user**: As Table 7-1 shows, the sensitivity of aerodynamic power to rotor-collective blade pitch varies considerably over Region 3, so constant PI gains are not adequate for effective speed control. The pitch sensitivity, though, varies nearly linearly with blade-pitch angle:  
+
+$$
+\frac{\partial P}{\partial\theta}\!=\!\left[\frac{\frac{\partial P}{\partial\theta}\big(\theta\!=\!0\big)}{\theta_{K}}\right]\!\theta\!+\!\left[\frac{\partial P}{\partial\theta}\big(\theta\!=\!0\big)\right]
+$$  
+
+$$
+\frac{I}{\frac{\partial P}{\partial\theta}}\!=\!\frac{I}{\frac{\partial P}{\partial\theta}\!\left(\theta\!=\!\theta\right)\!\left(I\!+\!\frac{\theta}{\theta_{\!\scriptscriptstyle K}}\right)},
+$$  
+
+where $\frac{\partial P}{\partial\theta}\!\left(\theta=\!0\right)$ is the pitch sensitivity at rated and $\theta_{K}$ is the blade-pitch angle at which the pitch sensitivity has doubled from its value at the rated operating point; that is,  
+
+$$
+\frac{\partial P}{\partial\theta}\big(\theta=\theta_{\kappa}\big)=2\frac{\partial P}{\partial\theta}\big(\theta\!=\!0\big).
+$$  
+
+On the right-hand side of Eq. (7-16a), the first and second terms in square brackets represent the slope and intercept of the best-fit line, respectively.  We computed this regression for the NREL 5-MW baseline wind turbine and present the results in Figure 7-3.  
+翻译成中文
+[Timestamp: 2025/03/06 15:39:38]
+
+**ai**: 表7-1显示，气动功率对转子集合叶片倾角的敏感度在区域3内变化很大，因此常数PI增益不足以实现有效速度控制。然而，这种敏感度几乎线性地随着叶片倾角变化：
+
+$$
+\frac{\partial P}{\partial\theta} = \left[\frac{\frac{\partial P}{\partial\theta}\big(\theta = 0\big)}{\theta_{K}}\right]\theta + \left[\frac{\partial P}{\partial\theta}\big(\theta = 0\big)\right]
+$$
+
+$$
+\frac{I}{\frac{\partial P}{\partial\theta}} = \frac{I}{\frac{\partial P}{\partial\theta}\!\left(\theta = \theta\right)\!\left(I + \frac{\theta}{\theta_{K}}\right)},
+$$
+
+其中 $\frac{\partial P}{\partial\theta}\big(\theta=0\big)$ 是额定时的叶片倾角敏感度，$\theta_{K}$ 是叶片倾角在该点时，叶片倾角敏感度是其额定操作点值的两倍；即，
+
+$$
+\frac{\partial P}{\partial\theta}\big(\theta=\theta_{\kappa}\big) = 2\frac{\partial P}{\partial\theta}\big(\theta=0\big).
+$$
+
+等式(7-16a)右侧的方括号中第一项和第二项分别表示最佳拟合直线的斜率和截距。我们为NREL 5-MW基准风力涡轮机计算了这个回归，并在图7-3中展示结果。
+[Timestamp: 2025/03/06 15:39:58]
+
+**user**: The linear relation between pitch sensitivity and blade-pitch angle presents a simple technique for implementing gain scheduling based on blade-pitch angle; that is,  
+
+$$
+K_{P}\left(\theta\right)=\frac{2I_{D r i v e t r a i n}\varOmega_{\rho}\zeta_{\varphi}\omega_{\varphi n}}{N_{G e a r}\left[-\frac{\partial P}{\partial\theta}\big(\theta=0\big)\right]}G K\left(\theta\right)
+$$  
+
+$$
+K_{I}\left(\theta\right)\!=\!\frac{I_{D r i v e t r a i n}\varOmega_{\theta}\omega_{\varphi n}^{2}}{N_{G e a r}\left[-\frac{\hat{\partial}P}{\hat{\partial}\theta}(\theta=\theta)\right]}{G K}(\theta),
+$$  
+
+where $G K(\theta)$ is the dimensionless gain-correction factor (from Ref. [10]), which is dependent on the blade-pitch angle:  
+
+$$
+G K\left(\theta\right)=\frac{I}{I+\frac{\theta}{\theta_{K}}}.
+$$  
+
+In our implementation of the gain-scheduled PI blade-pitch controller, we used the blade-pitch angle from the previous controller time step to calculate the gain-correction factor at the next time step.  
+
+Using the properties for the baseline wind turbine and the recommended response characteristics from Ref. [10], the resulting gains are $\underline{{K_{P}}}(\theta=0^{\circ})=0.01882681\ \mathrm{s},\ K_{I}(\theta=0^{\circ})=0.008068634,$ and $K_{D}\,=\,0.0\,\stackrel{\cdot}{\mathbf{s}}^{2}$ .  Figure 7-4 presents the gains at other blade-pitch angles, along with the gaincorrection factor.  We used the upper limit of the recommended damping ratio range, $\zeta_{\varphi}=0.7$ , to compensate for neglecting negative damping from the generator-torque controller in the determination of $K_{P}$ .  
+翻译成中文
+[Timestamp: 2025/03/06 15:50:44]
+
+**ai**: 叶片倾角与叶片倾角敏感度之间的线性关系提供了一种简单的基于叶片倾角实施增益调度的技术；即，
+
+$$
+K_{P}\left(\theta\right)=\frac{2I_{D r i v e t r a i n}\varOmega_{\rho}\zeta_{\varphi}\omega_{\varphi n}}{N_{G e a r}\left[-\frac{\partial P}{\partial\theta}\big(\theta=0\big)\right]}G K\left(\theta\right)
+$$
+
+$$
+K_{I}\left(\theta\right)=\frac{I_{D r i v e t r a i n}\varOmega_{\theta}\omega_{\varphi n}^{2}}{N_{G e a r}\left[-\frac{\hat{\partial}P}{\hat{\partial}\theta}(\theta=\theta)\right]}{G K}(\theta),
+$$
+
+其中 $G K(\theta)$ 是无量纲的增益校正因子（参考文献[10]），它依赖于叶片倾角：
+
+$$
+G K\left(\theta\right)=\frac{I}{I+\frac{\theta}{\theta_{K}}}.
+$$
+
+在我们实现基于增益调度的PI叶片倾角控制器时，我们使用前一时间步的叶片倾角来计算下一个时间步的增益校正因子。
+
+利用基准风力涡轮机的特性和参考文献[10]中推荐的响应特征，得到的增益为 $\underline{{K_{P}}}(\theta=0^{\circ})=0.01882681\ \mathrm{s},\ K_{I}(\theta=0^{\circ})=0.008068634,$ 和 $K_{D}\,=\,0.0\,\stackrel{\cdot}{\mathbf{s}}^{2}$ 。图7-4展示了其他叶片倾角下的增益以及增益校正因子。我们使用推荐阻尼比范围的上限，$\zeta_{\varphi}=0.7$ ，来补偿在确定 $K_{P}$ 时忽略发电机转矩控制器引入的负阻尼。
+[Timestamp: 2025/03/06 15:51:14]
+
+**user**: Unfortunately, the simple gain-scheduling law derived in this section for the proportional and integral gains cannot retain consistent response characteristics (i.e., constant values of $\omega_{\varphi n}$ and $\zeta_{\varphi})$ across all of Region 3 when applied to the derivative gain.  We, nevertheless, considered adding a derivative term by selecting and testing a range of gains, but none were found to give better performance in the overall system response.  Instead, the baseline control system uses the gains derived previously in this section (without the derivative term).  
+
+We set the blade-pitch rate limit to $8^{\circ}/\mathrm{s}$ in absolute value.  This is speculated to be the bladepitch rate limit of conventional 5-MW machines based on General Electric (GE) Wind’s longblade test program.  We also set the minimum and maximum blade-pitch settings to $0^{\circ}$ and $90^{\circ}$ , respectively.  The lower limit is the set blade pitch for maximizing power in Region 2, as described in Section 7.2.  The upper limit is very close to the fully feathered blade pitch for neutral torque.  We saturated the integral term in the PI controller between these limits to ensure a fast response in the transitions between Regions 2 and 3.  
+翻译成中文
+[Timestamp: 2025/03/06 15:53:39]
+
+**ai**: 不幸的是，本节推导出的简单增益调度法对于比例和积分增益无法在所有区域3下保持一致的响应特性（即，$\omega_{\varphi n}$ 和 $\zeta_{\varphi}$ 的常数值）。然而，我们仍考虑通过选择并测试一系列增益来添加微分项，但没有发现任何能够改善整体系统响应的增益。相反，基准控制系统使用本节之前推导出的增益（不包括微分项）。
+
+我们将叶片倾角率限制设置为绝对值 $8^{\circ}/\mathrm{s}$ 。这被猜测是基于通用电气（GE）风能公司的长叶片测试计划，传统5兆瓦机器的叶片倾角率限制。我们还将最小和最大叶片倾角设置为 $0^{\circ}$ 和 $90^{\circ}$ ，分别。下限是在区域2中最大化功率的设定叶片倾角，如第7.2节所述。上限接近于完全展开以实现中性扭矩的叶片倾角。我们将PI控制器中的积分项在这些极限之间饱和，以确保区域2与3之间转换时的快速响应。
+[Timestamp: 2025/03/06 15:54:00]
+
+**user**: # 7.4  Baseline Blade-Pitch Actuator  
+
+Because of limitations in the FAST code, the FAST model does not include any blade-pitch actuator dynamic effects.  Blade-pitch actuator dynamics are, however, needed in ADAMS.  To enable successful comparisons between the FAST and ADAMS response predictions, then, we found it beneficial to reduce the effect of the blade-pitch actuator response in ADAMS. Consequently, we designed the blade-pitch actuator in the ADAMS model with a very high natural frequency of $30\;\mathrm{Hz}$ , which is higher than the highest full-system natural frequency in the FAST model (see Section 9), and a damping ratio of $2\%$ critical.  This resulted in an equivalent blade-pitch actuator linear-spring constant of 971,350,000 N•m/rad and an equivalent blade-pitch actuator linear-damping constant of $206,000\;\mathrm{N{\cdot}m/(r a d/s)}$ .  
+
+# 7.5  Summary of Baseline Control System Properties  
+
+We implemented the NREL offshore 5-MW wind turbine’s baseline control system as an external dynamic link library (DLL) in the style of Garrad Hassan’s BLADED wind turbine software package [3].  Appendix C contains the source code for this DLL, and Figure 7-5 presents a flowchart of the overall integrated control system calculations.  Table 7-2 summarizes the baseline generator-torque and blade-pitch control properties we discussed earlier in this section.  
+翻译成中文
+[Timestamp: 2025/03/06 16:29:49]
+
+**ai**: # 7.4 基准叶片倾角执行器
+
+由于FAST代码的限制，FAST模型不包括任何叶片倾角执行器动态效应。然而，在ADAMS中需要叶片倾角执行器动力学。为了实现FAST和ADAMS响应预测之间的成功比较，我们发现减少ADAMS中叶片倾角执行器响应的影响是有益的。因此，我们设计了ADAMS模型中的叶片倾角执行器，具有非常高的自然频率 $30\;\mathrm{Hz}$ ，这高于FAST模型中最高全系统自然频率（见第9节），且阻尼比为2%的临界值。这导致了等效叶片倾角执行器线性弹簧常数为971,350,000 N•m/rad，以及等效叶片倾角执行器线性阻尼常数为 $206,000\;\mathrm{N{\cdot}m/(r a d/s)}$。
+
+# 7.5 基准控制系统属性总结
+
+我们将NREL海上5兆瓦风力涡轮机的基准控制系统实现为外部动态链接库（DLL），类似于Garrad Hassan的BLADED风力涡轮机软件包[3]。附录C包含此DLL的源代码，图7-5展示了整合控制系统计算的流程图。表7-2总结了我们在本节前面讨论过的基准发电机扭矩和叶片倾角控制属性。
+[Timestamp: 2025/03/06 16:30:25]