Move AC fundamentals into bilingual primers

README.md

@@ -39,13 +39,14 @@ Market-neutral educational material — no venture-specific analysis.
 | 04 | [04_market_mechanics_companion_en.md](primers/grid/04_market_mechanics_companion_en.md) | [zh](primers/grid/04_market_mechanics_companion_zh.md) | Market mechanics companion — clearing, pricing, settlement. 市场机制手册——出清、定价、结算。 |
 | 05 | [05_trading_under_uncertainty_en.md](primers/grid/05_trading_under_uncertainty_en.md) | [zh](primers/grid/05_trading_under_uncertainty_zh.md) | Trading under uncertainty. 不确定性下的交易。 |
 
-**Topical primers 专题读物** (EN only):
+**Topical primers 专题读物**:
 
-| Doc 文档 | File | Contents 内容 |
-|---|---|---|
-| SCUC & SCED | [scuc_sced_en.md](primers/scuc_sced_en.md) | The two optimization problems that clear US wholesale markets; N-1 security, LMP as dual variables. 出清美国批发市场的两大优化问题。 |
-| Newsvendor 报童模型 | [newsvendor_en.md](primers/newsvendor_en.md) | The newsvendor problem applied to electricity bidding — underpins the bid loop in `pjm/design_models_*`. 报童模型在电力竞价中的应用。 |
-| Stochastic optimization 随机优化 | [stochastic_optimization_en.md](primers/stochastic_optimization_en.md) | Stochastic optimization in power systems — scenarios, recourse, hydro, offering under uncertainty. 电力系统中的随机优化。 |
+| Doc 文档 | EN | 中文 | Contents 内容 |
+|---|---|---|---|
+| Power grid & AC fundamentals 电网与交流电基础 | [power_grid_ac_fundamentals_en.md](primers/power_grid_ac_fundamentals_en.md) | [power_grid_ac_fundamentals_zh.md](primers/power_grid_ac_fundamentals_zh.md) | Grid structure, AC, impedance, inductors, capacitors, and Kirchhoff’s laws. 电网结构、交流电、阻抗、电感、电容与基尔霍夫定律。 |
+| SCUC & SCED | [scuc_sced_en.md](primers/scuc_sced_en.md) | EN only | The two optimization problems that clear US wholesale markets; N-1 security, LMP as dual variables. 出清美国批发市场的两大优化问题。 |
+| Newsvendor 报童模型 | [newsvendor_en.md](primers/newsvendor_en.md) | EN only | The newsvendor problem applied to electricity bidding — underpins the bid loop in `pjm/design_models_*`. 报童模型在电力竞价中的应用。 |
+| Stochastic optimization 随机优化 | [stochastic_optimization_en.md](primers/stochastic_optimization_en.md) | EN only | Stochastic optimization in power systems — scenarios, recourse, hydro, offering under uncertainty. 电力系统中的随机优化。 |
 
 ### `us_market/` — US market reference 美国市场综述
 

primers/power_grid_ac_fundamentals_en.md

@@ -0,0 +1,863 @@
+# Power Grid and AC Fundamentals
+
+## Overview
+
+This primer starts with the basic structure of the electrical grid and then builds toward alternating current, impedance, inductors, capacitors, and Kirchhoff’s laws.
+
+---
+
+## 1. What Is an Electrical Power Grid?
+
+An electrical power grid is a large interconnected system that moves electrical energy from power plants and other sources to homes, businesses, and factories.
+
+The grid has four main parts:
+
+1. **Generation**
+2. **Transmission**
+3. **Substations and transformers**
+4. **Distribution**
+
+### 1.1 Generation
+
+Electricity is produced by sources such as:
+
+- Natural-gas, coal, and nuclear plants
+- Hydroelectric stations
+- Wind turbines
+- Solar farms
+- Batteries and other energy-storage systems
+
+Traditional generators usually produce alternating current, or **AC**. Solar panels and batteries naturally produce direct current, or **DC**, so power-electronic inverters are used to connect them to the AC grid.
+
+### 1.2 Transmission
+
+Transmission lines carry large amounts of electrical power over long distances.
+
+Power is approximately:
+
+\[
+P = VI
+\]
+
+where:
+
+- \(P\) is power
+- \(V\) is voltage
+- \(I\) is current
+
+For the same amount of power, raising the voltage allows the current to be reduced.
+
+Transmission losses are approximately:
+
+\[
+P_{\text{loss}} = I^2R
+\]
+
+Because losses depend on the square of current, high-voltage transmission greatly reduces heating losses.
+
+### 1.3 Substations and Transformers
+
+Substations connect different parts of the grid. They contain equipment that can:
+
+- Raise or lower voltage
+- Connect or disconnect lines
+- Measure voltage and current
+- Protect equipment during faults
+- Control power flow
+
+Transformers raise voltage for long-distance transmission and lower it again near customers.
+
+### 1.4 Distribution
+
+Distribution networks carry electricity from local substations to customers.
+
+Additional transformers reduce the voltage to levels used by homes, offices, and industrial equipment.
+
+---
+
+## 2. Important Grid Quantities
+
+### 2.1 Voltage
+
+Voltage is the electrical potential difference between two points. It is the electrical “push” that drives current.
+
+In an AC grid, voltage has both:
+
+- A magnitude
+- A phase angle
+
+Different grid buses may have slightly different voltage magnitudes and phase angles.
+
+### 2.2 Current
+
+Current is the movement of electric charge through a conductor.
+
+In the grid, line current depends on:
+
+- Connected generation and loads
+- Voltage differences
+- Voltage phase-angle differences
+- Line impedance
+- Network topology
+
+### 2.3 Real, Reactive, and Apparent Power
+
+In AC systems, engineers distinguish between three types of power.
+
+#### Real power \(P\)
+
+Real power performs useful work and is measured in watts.
+
+Examples include:
+
+- Producing heat
+- Turning a motor
+- Producing light
+
+#### Reactive power \(Q\)
+
+Reactive power supports electric and magnetic fields. It is measured in vars.
+
+Reactive power moves back and forth between sources and reactive components such as inductors and capacitors.
+
+#### Apparent power \(S\)
+
+Apparent power is measured in volt-amperes.
+
+The three are related by:
+
+\[
+S^2 = P^2 + Q^2
+\]
+
+### 2.4 Frequency
+
+Frequency is the number of AC cycles per second.
+
+\[
+1\text{ Hz} = 1\text{ cycle per second}
+\]
+
+Typical grid frequencies are:
+
+- 60 Hz in the United States and some other countries
+- 50 Hz in many other parts of the world
+
+Frequency reflects the balance between generation and demand:
+
+- If demand exceeds generation, frequency tends to fall.
+- If generation exceeds demand, frequency tends to rise.
+
+---
+
+## 3. Buses, Branches, and Loads
+
+Power engineers often represent the grid as a network.
+
+### Bus
+
+A **bus** is a connection point where generators, loads, transformers, or transmission lines meet.
+
+### Branch
+
+A **branch** is a transmission line or transformer connecting two buses.
+
+### Load
+
+A **load** is equipment that consumes electrical power, such as:
+
+- Motors
+- Lighting
+- Heating systems
+- Computers and electronics
+
+A simplified network is:
+
+\[
+\text{Generator}
+\rightarrow
+\text{Bus 1}
+\rightarrow
+\text{Transmission line}
+\rightarrow
+\text{Bus 2}
+\rightarrow
+\text{Load}
+\]
+
+Real power grids contain many parallel paths. Electricity divides among those paths according to the network’s electrical properties rather than following one assigned route.
+
+---
+
+## 4. Direct Current and Alternating Current
+
+### 4.1 Direct Current
+
+With **direct current**, or DC, charge flows mainly in one direction.
+
+A battery is a common DC source.
+
+### 4.2 Alternating Current
+
+With **alternating current**, or AC, voltage and current repeatedly reverse direction.
+
+An ideal sinusoidal voltage is:
+
+\[
+v(t) = V_{\text{peak}}\sin(\omega t)
+\]
+
+The voltage:
+
+1. Rises positive
+2. Falls to zero
+3. Becomes negative
+4. Returns to zero
+5. Repeats
+
+Negative voltage does not mean “less than no electricity.” It means that the polarity between two reference points has reversed.
+
+### 4.3 How AC Is Generated
+
+A generator converts mechanical rotation into electrical energy.
+
+In a simplified generator:
+
+1. A magnetic field rotates relative to coils of wire.
+2. The magnetic flux through the coils changes.
+3. A voltage is induced.
+4. As the magnetic orientation reverses, the voltage reverses.
+
+This produces AC.
+
+Generators may be driven by:
+
+- Steam turbines
+- Gas turbines
+- Water turbines
+- Wind turbines
+- Engines
+
+### 4.4 RMS Voltage
+
+Because AC voltage changes continuously, engineers often describe it using its root-mean-square value.
+
+For a sine wave:
+
+\[
+V_{\text{RMS}} = \frac{V_{\text{peak}}}{\sqrt{2}}
+\]
+
+A 120 V household supply is approximately 120 V RMS, with a peak near 170 V.
+
+RMS voltage is useful because it produces the same heating effect in a resistor as an equal DC voltage.
+
+---
+
+## 5. How Electrical Energy Travels in AC Systems
+
+A wire already contains mobile electrons before it is energized.
+
+The power plant does not send a specific group of electrons all the way to a customer. Instead:
+
+1. The generator establishes an electromagnetic field.
+2. That field propagates along the circuit.
+3. Local electrons in the conductors begin moving.
+4. Energy is transferred to the load.
+
+In AC, electrons usually oscillate back and forth around approximately the same average position.
+
+Two different speeds are involved:
+
+- **Electron drift speed:** relatively slow
+- **Electromagnetic propagation speed:** very fast, often a significant fraction of the speed of light
+
+The electromagnetic energy flow is described by the Poynting vector:
+
+\[
+\mathbf{S} = \mathbf{E} \times \mathbf{H}
+\]
+
+where:
+
+- \(\mathbf{E}\) is the electric field
+- \(\mathbf{H}\) is the magnetic field
+- \(\mathbf{S}\) describes the direction and rate of energy flow
+
+The conductors guide the fields that carry energy toward the load.
+
+---
+
+## 6. Phase in AC Circuits
+
+Two AC waveforms are **in phase** when their peaks and zero crossings occur at the same times.
+
+They are **out of phase** when one waveform leads or lags the other.
+
+### Pure resistor
+
+For an ideal resistor:
+
+\[
+v(t) = i(t)R
+\]
+
+Voltage and current are in phase.
+
+### Pure inductor
+
+For an ideal inductor:
+
+- Voltage leads current by \(90^\circ\)
+- Current lags voltage by \(90^\circ\)
+
+### Pure capacitor
+
+For an ideal capacitor:
+
+- Current leads voltage by \(90^\circ\)
+- Voltage lags current by \(90^\circ\)
+
+---
+
+## 7. Resistance, Reactance, and Impedance
+
+### 7.1 Resistance
+
+Resistance opposes current and converts electrical energy into heat.
+
+\[
+P_{\text{loss}} = I^2R
+\]
+
+Resistance is measured in ohms.
+
+### 7.2 Reactance
+
+Reactance is opposition to AC caused by energy storage in electric or magnetic fields.
+
+It comes from:
+
+- Inductance
+- Capacitance
+
+Unlike ideal resistance, ideal reactance does not permanently consume average real power. It stores energy temporarily and returns it to the circuit.
+
+### 7.3 Impedance
+
+Impedance is the total opposition to AC.
+
+\[
+Z = R + jX
+\]
+
+where:
+
+- \(Z\) is impedance
+- \(R\) is resistance
+- \(X\) is reactance
+- \(j\) represents a \(90^\circ\) phase shift
+
+For AC circuits, Ohm’s law becomes:
+
+\[
+\underline{I} = \frac{\underline{V}}{Z}
+\]
+
+The underlines indicate phasor quantities.
+
+### Example
+
+If a line has:
+
+\[
+Z = 2 + j8\ \Omega
+\]
+
+then:
+
+- Resistance is \(2\ \Omega\)
+- Inductive reactance is \(8\ \Omega\)
+
+Its magnitude is:
+
+\[
+|Z| = \sqrt{2^2 + 8^2} \approx 8.25\ \Omega
+\]
+
+The line is mostly inductive because its reactance is much larger than its resistance.
+
+---
+
+## 8. Inductors
+
+An inductor is usually a coil of wire that stores energy in a magnetic field.
+
+Its key behavior is:
+
+> An inductor resists changes in current.
+
+### 8.1 How an Inductor Works
+
+When current flows through a wire, it creates a magnetic field.
+
+Winding the wire into a coil strengthens and concentrates that field.
+
+If the current changes, the magnetic field changes. That changing field induces a voltage that opposes the original change. This is Lenz’s law.
+
+The basic relationship is:
+
+\[
+v_L = L\frac{di}{dt}
+\]
+
+where:
+
+- \(v_L\) is inductor voltage
+- \(L\) is inductance in henries
+- \(di/dt\) is the rate of change of current
+
+### 8.2 DC Behavior
+
+When a DC source is first connected, the inductor opposes the sudden rise in current.
+
+In a series RL circuit:
+
+\[
+i(t) = \frac{V}{R}\left(1-e^{-tR/L}\right)
+\]
+
+The time constant is:
+
+\[
+\tau = \frac{L}{R}
+\]
+
+After the current becomes steady:
+
+\[
+\frac{di}{dt}=0
+\]
+
+so:
+
+\[
+v_L = 0
+\]
+
+An ideal inductor then behaves like a short circuit for steady DC.
+
+### 8.3 Switching Off an Inductor
+
+When current is interrupted suddenly, the collapsing magnetic field produces a voltage that tries to keep current flowing.
+
+This can create:
+
+- Sparks
+- Arcing
+- High-voltage transients
+- Damage to switches or transistors
+
+Protective devices such as flyback diodes provide a safe path for the current.
+
+### 8.4 Stored Energy
+
+An inductor stores energy in its magnetic field:
+
+\[
+W_L = \frac{1}{2}LI^2
+\]
+
+### 8.5 Inductive Reactance
+
+For sinusoidal AC:
+
+\[
+X_L = 2\pi fL
+\]
+
+Inductive reactance increases when:
+
+- Frequency increases
+- Inductance increases
+
+The impedance of an ideal inductor is:
+
+\[
+Z_L = j\omega L
+\]
+
+### 8.6 Inductors in the Grid
+
+Grid equipment contains inductance even when there is no obvious coil.
+
+Examples include:
+
+- Transmission lines
+- Transformers
+- Motors
+- Generators
+- Reactors
+
+Power-system reactors are large inductors used to:
+
+- Limit fault current
+- Control voltage
+- Absorb reactive power
+- Reduce transients
+
+---
+
+## 9. Capacitors
+
+A capacitor stores energy in an electric field.
+
+Its key behavior is:
+
+> A capacitor resists changes in voltage.
+
+### 9.1 Construction
+
+A basic capacitor has two conductive plates separated by an insulating dielectric.
+
+\[
+\text{plate} \; | \; \text{dielectric} \; | \; \text{plate}
+\]
+
+The dielectric may be air, ceramic, plastic film, oxide, or another insulating material.
+
+### 9.2 Charge and Voltage
+
+The stored charge is:
+
+\[
+Q = CV
+\]
+
+where:
+
+- \(Q\) is charge
+- \(C\) is capacitance in farads
+- \(V\) is voltage
+
+For ideal parallel plates:
+
+\[
+C = \frac{\varepsilon A}{d}
+\]
+
+Capacitance increases when:
+
+- Plate area increases
+- Plate spacing decreases
+- Dielectric permittivity increases
+
+### 9.3 Current-Voltage Relationship
+
+The fundamental capacitor equation is:
+
+\[
+i = C\frac{dv}{dt}
+\]
+
+A capacitor carries current when its voltage is changing.
+
+A finite current can only change capacitor voltage at a finite rate, so capacitor voltage cannot jump instantly in an ideal circuit.
+
+### 9.4 DC Charging
+
+When an uncharged capacitor is connected to a DC source through a resistor:
+
+\[
+v_C(t) = V\left(1-e^{-t/RC}\right)
+\]
+
+and:
+
+\[
+i(t) = \frac{V}{R}e^{-t/RC}
+\]
+
+The time constant is:
+
+\[
+\tau = RC
+\]
+
+After the capacitor is fully charged, its voltage becomes constant, so:
+
+\[
+\frac{dv}{dt}=0
+\]
+
+and therefore:
+
+\[
+i=0
+\]
+
+An ideal capacitor behaves like an open circuit for steady DC.
+
+### 9.5 Stored Energy
+
+A capacitor stores energy in its electric field:
+
+\[
+W_C = \frac{1}{2}CV^2
+\]
+
+### 9.6 Capacitive Reactance
+
+For sinusoidal AC:
+
+\[
+X_C = \frac{1}{2\pi fC}
+\]
+
+Capacitive reactance decreases when:
+
+- Frequency increases
+- Capacitance increases
+
+The impedance of an ideal capacitor is:
+
+\[
+Z_C = \frac{1}{j\omega C} = -jX_C
+\]
+
+### 9.7 Capacitors in the Grid
+
+Grid capacitance appears in:
+
+- Transmission lines
+- Underground cables
+- Capacitor banks
+- Series compensation equipment
+- Shunt compensation equipment
+
+Capacitor banks are used to:
+
+- Support voltage
+- Supply reactive power locally
+- Improve power factor
+- Reduce line current
+- Reduce \(I^2R\) losses
+
+Series capacitors offset part of line inductive reactance:
+
+\[
+X_{\text{net}} = X_L - X_C
+\]
+
+Shunt capacitors are connected to a bus and help support bus voltage.
+
+---
+
+## 10. Inductors and Capacitors Compared
+
+| Property | Inductor | Capacitor |
+|---|---|---|
+| Stores energy in | Magnetic field | Electric field |
+| Resists changes in | Current | Voltage |
+| Basic equation | \(v=L\,di/dt\) | \(i=C\,dv/dt\) |
+| Steady DC behavior | Short circuit ideally | Open circuit ideally |
+| AC phase relationship | Current lags voltage | Current leads voltage |
+| Reactance as frequency rises | Increases | Decreases |
+| Ideal impedance | \(j\omega L\) | \(1/(j\omega C)\) |
+
+---
+
+## 11. Why Current Can Reverse While Power Still Flows to a Load
+
+For a pure resistor:
+
+\[
+v(t)=V_{\text{peak}}\sin(\omega t)
+\]
+
+\[
+i(t)=I_{\text{peak}}\sin(\omega t)
+\]
+
+Instantaneous power is:
+
+\[
+p(t)=v(t)i(t)
+\]
+
+Therefore:
+
+\[
+p(t)=V_{\text{peak}}I_{\text{peak}}\sin^2(\omega t)
+\]
+
+Because \(\sin^2(\omega t)\) is never negative, the resistor receives energy during both halves of the cycle.
+
+When voltage and current both reverse, their product remains positive.
+
+For ideal inductors and capacitors, power alternates between positive and negative because energy is repeatedly stored and returned.
+
+---
+
+## 12. Kirchhoff’s Laws
+
+Kirchhoff’s laws describe the basic conservation rules that electrical networks must obey.
+
+### 12.1 Kirchhoff’s Current Law
+
+At any node or bus:
+
+\[
+\sum I_{\text{entering}} = \sum I_{\text{leaving}}
+\]
+
+Current cannot accumulate at an ideal node.
+
+In power-system form, this becomes a real- and reactive-power balance at each bus.
+
+A simplified real-power balance is:
+
+\[
+P_{\text{generation}} - P_{\text{load}}
+=
+P_{\text{exported}} + P_{\text{losses}}
+\]
+
+A similar equation applies to reactive power.
+
+### 12.2 Kirchhoff’s Voltage Law
+
+Around any closed electrical loop:
+
+\[
+\sum V = 0
+\]
+
+The total voltage rises equal the total voltage drops.
+
+For a transmission line:
+
+\[
+\underline{V}_1 - \underline{V}_2 = \underline{I}Z
+\]
+
+where:
+
+- \(\underline{V}_1\) and \(\underline{V}_2\) are bus-voltage phasors
+- \(\underline{I}\) is line current
+- \(Z\) is line impedance
+
+### 12.3 Role in Grid Analysis
+
+Together, Kirchhoff’s laws allow engineers to calculate:
+
+- Bus voltages
+- Line currents
+- Real-power flows
+- Reactive-power flows
+- Losses
+- Fault currents
+
+They are fundamental to:
+
+- Load-flow studies
+- Short-circuit studies
+- Stability studies
+- Protection design
+
+---
+
+## 13. How Power Flows Through an AC Grid
+
+In an interconnected AC network, power divides among all available paths.
+
+The distribution of power depends on:
+
+- Line resistance
+- Line reactance
+- Bus-voltage magnitudes
+- Voltage phase angles
+- Network connections
+
+A lower-impedance path generally carries more current than a higher-impedance path.
+
+In high-voltage transmission systems, inductive reactance is often larger than resistance. As a result, real-power flow is strongly influenced by the phase-angle difference between buses.
+
+Grid operators influence power flow by:
+
+- Adjusting generator output
+- Switching transmission lines
+- Changing transformer taps
+- Connecting or disconnecting capacitor banks or reactors
+- Using power-electronic controllers
+
+---
+
+## 14. Core Mental Models
+
+### AC
+
+\[
+\text{rotating generator}
+\rightarrow
+\text{alternating voltage}
+\rightarrow
+\text{alternating current}
+\rightarrow
+\text{energy transfer}
+\]
+
+### Inductor
+
+\[
+\text{changing current}
+\rightarrow
+\text{changing magnetic field}
+\rightarrow
+\text{induced voltage}
+\rightarrow
+\text{opposition to current change}
+\]
+
+### Capacitor
+
+\[
+\text{changing voltage}
+\rightarrow
+\text{movement of charge}
+\rightarrow
+\text{changing electric field}
+\rightarrow
+\text{current}
+\]
+
+### Grid analysis
+
+\[
+\text{network model}
++
+\text{impedances}
++
+\text{Kirchhoff’s laws}
+\rightarrow
+\text{voltages, currents, and power flows}
+\]
+
+---
+
+## 15. Summary
+
+The electrical grid is an interconnected AC network that transports energy from generators to loads.
+
+The most important ideas are:
+
+- High voltage reduces transmission current and resistive losses.
+- AC voltage and current repeatedly reverse direction.
+- Electrical energy is carried by electromagnetic fields guided by conductors.
+- Resistance consumes real power.
+- Inductors store energy in magnetic fields and resist changes in current.
+- Capacitors store energy in electric fields and resist changes in voltage.
+- Impedance combines resistance and reactance.
+- Kirchhoff’s Current Law enforces balance at buses.
+- Kirchhoff’s Voltage Law enforces voltage consistency around loops.
+- These principles form the foundation of power-flow, fault, and stability analysis.

primers/power_grid_ac_fundamentals_zh.md

@@ -0,0 +1,863 @@
+# 电网与交流电基础
+
+## 概述
+
+本文从电网的基本结构讲起，逐步介绍交流电、阻抗、电感、电容与基尔霍夫定律。
+
+---
+
+## 1. 什么是电力系统？
+
+电力系统是一个大型互联网络，把发电厂及其他电源产生的电能输送到家庭、商业和工业用户。
+
+电网主要由四个部分组成：
+
+1. **发电**
+2. **输电**
+3. **变电站与变压器**
+4. **配电**
+
+### 1.1 发电
+
+电力来源包括：
+
+- 天然气、煤炭和核电厂
+- 水电站
+- 风力发电机组
+- 光伏电站
+- 电池及其他储能系统
+
+传统发电机通常直接产生交流电（ **AC** ）。光伏组件和电池天然产生直流电（ **DC** ），因此需要通过电力电子逆变器接入交流电网。
+
+### 1.2 输电
+
+输电线路用于远距离输送大规模电力。
+
+功率近似为：
+
+\[
+P = VI
+\]
+
+其中：
+
+- \(P\) 为功率
+- \(V\) 为电压
+- \(I\) 为电流
+
+输送相同功率时，提高电压可以降低电流。
+
+输电损耗近似为：
+
+\[
+P_{\text{loss}} = I^2R
+\]
+
+损耗与电流的平方成正比，因此高压输电能显著减少发热损耗。
+
+### 1.3 变电站与变压器
+
+变电站连接电网的不同部分，其中的设备可以：
+
+- 升高或降低电压
+- 接通或断开线路
+- 测量电压与电流
+- 在故障时保护设备
+- 控制潮流
+
+变压器在远距离输电前升高电压，在靠近用户处再降低电压。
+
+### 1.4 配电
+
+配电网把电力从当地变电站输送到用户。
+
+配电变压器进一步把电压降到家庭、办公场所和工业设备使用的等级。
+
+---
+
+## 2. 重要的电网物理量
+
+### 2.1 电压
+
+电压是两点之间的电势差，是驱动电流的电学“推力”。
+
+交流电网中的电压同时具有：
+
+- 幅值
+- 相角
+
+不同母线的电压幅值和相角可能略有不同。
+
+### 2.2 电流
+
+电流是电荷在导体中的运动。
+
+电网中的线路电流取决于：
+
+- 接入的发电与负荷
+- 电压差
+- 电压相角差
+- 线路阻抗
+- 网络拓扑
+
+### 2.3 有功、无功与视在功率
+
+交流系统区分三类功率。
+
+#### 有功功率 \(P\)
+
+有功功率用于完成实际工作，以瓦特为单位。
+
+例如：
+
+- 产生热量
+- 驱动电机
+- 发出光
+
+#### 无功功率 \(Q\)
+
+无功功率用于维持电场和磁场，以乏（var）为单位。
+
+无功功率在电源与电感、电容等无功元件之间往返交换。
+
+#### 视在功率 \(S\)
+
+视在功率以伏安为单位。
+
+三者关系为：
+
+\[
+S^2 = P^2 + Q^2
+\]
+
+### 2.4 频率
+
+频率是交流电每秒完成的周期数。
+
+\[
+1\text{ Hz} = 1\text{ 个周期/秒}
+\]
+
+常见电网频率为：
+
+- 美国及部分国家为 60 Hz
+- 世界许多其他地区为 50 Hz
+
+频率反映发电与用电之间的平衡：
+
+- 用电大于发电时，频率往往下降。
+- 发电大于用电时，频率往往上升。
+
+---
+
+## 3. 母线、支路与负荷
+
+电力工程师通常把电网表示为一个网络。
+
+### 母线
+
+**母线** 是发电机、负荷、变压器或输电线路汇接的节点。
+
+### 支路
+
+**支路** 是连接两条母线的输电线路或变压器。
+
+### 负荷
+
+**负荷** 是消耗电力的设备，例如：
+
+- 电机
+- 照明
+- 供热系统
+- 计算机与电子设备
+
+一个简化网络为：
+
+\[
+\text{发电机}
+\rightarrow
+\text{母线 1}
+\rightarrow
+\text{输电线路}
+\rightarrow
+\text{母线 2}
+\rightarrow
+\text{负荷}
+\]
+
+真实电网包含许多并行路径。电力依据网络的电气特性在各条路径间分配，而不是沿某一条指定路线输送。
+
+---
+
+## 4. 直流电与交流电
+
+### 4.1 直流电
+
+在 **直流电** （DC）中，电荷主要沿一个方向流动。
+
+电池是常见的直流电源。
+
+### 4.2 交流电
+
+在 **交流电** （AC）中，电压和电流的方向周期性反转。
+
+理想正弦电压为：
+
+\[
+v(t) = V_{\text{peak}}\sin(\omega t)
+\]
+
+电压会：
+
+1. 上升到正值
+2. 降至零
+3. 变为负值
+4. 回到零
+5. 重复循环
+
+负电压并不意味着“比没有电还少”，而是两个参考点之间的极性发生了反转。
+
+### 4.3 交流电如何产生
+
+发电机把机械旋转转换为电能。
+
+在简化的发电机中：
+
+1. 磁场相对线圈旋转。
+2. 穿过线圈的磁通量发生变化。
+3. 线圈中产生感应电压。
+4. 磁场方向反转时，电压方向也反转。
+
+由此产生交流电。
+
+发电机的原动机可以是：
+
+- 蒸汽轮机
+- 燃气轮机
+- 水轮机
+- 风力机
+- 发动机
+
+### 4.4 电压有效值
+
+交流电压持续变化，因此工程上常用均方根值，即有效值描述它。
+
+对于正弦波：
+
+\[
+V_{\text{RMS}} = \frac{V_{\text{peak}}}{\sqrt{2}}
+\]
+
+标称 120 V 的家庭电源约为 120 V 有效值，峰值接近 170 V。
+
+有效值的意义在于，它在电阻上产生的热效应与数值相同的直流电压相同。
+
+---
+
+## 5. 电能如何在交流系统中传输
+
+导线在通电前就已经含有可移动电子。
+
+发电厂并不是把某一群电子一路送到用户。实际过程是：
+
+1. 发电机建立电磁场。
+2. 电磁场沿电路传播。
+3. 导体内的局部电子开始运动。
+4. 能量被传递给负荷。
+
+在交流系统中，电子通常只是在大致相同的平均位置附近来回振荡。
+
+这里涉及两种不同的速度：
+
+- **电子漂移速度：** 相对较慢
+- **电磁传播速度：** 非常快，通常是光速的相当一部分
+
+电磁能流可用坡印廷矢量表示：
+
+\[
+\mathbf{S} = \mathbf{E} \times \mathbf{H}
+\]
+
+其中：
+
+- \(\mathbf{E}\) 为电场
+- \(\mathbf{H}\) 为磁场
+- \(\mathbf{S}\) 表示能量流动的方向与速率
+
+导体为把能量送向负荷的电磁场提供引导。
+
+---
+
+## 6. 交流电路中的相位
+
+两个交流波形的峰值和过零点同时出现时，称为 **同相** 。
+
+一个波形超前或滞后于另一个波形时，称为 **不同相** 。
+
+### 纯电阻
+
+对于理想电阻：
+
+\[
+v(t) = i(t)R
+\]
+
+电压与电流同相。
+
+### 纯电感
+
+对于理想电感：
+
+- 电压超前电流 \(90^\circ\)
+- 电流滞后电压 \(90^\circ\)
+
+### 纯电容
+
+对于理想电容：
+
+- 电流超前电压 \(90^\circ\)
+- 电压滞后电流 \(90^\circ\)
+
+---
+
+## 7. 电阻、电抗与阻抗
+
+### 7.1 电阻
+
+电阻阻碍电流，并把电能转化为热量。
+
+\[
+P_{\text{loss}} = I^2R
+\]
+
+电阻的单位是欧姆。
+
+### 7.2 电抗
+
+电抗是电场或磁场储能对交流电造成的阻碍。
+
+它来自：
+
+- 电感
+- 电容
+
+与理想电阻不同，理想电抗不永久消耗平均有功功率，而是暂时储存能量并将其返还电路。
+
+### 7.3 阻抗
+
+阻抗是对交流电的总阻碍。
+
+\[
+Z = R + jX
+\]
+
+其中：
+
+- \(Z\) 为阻抗
+- \(R\) 为电阻
+- \(X\) 为电抗
+- \(j\) 表示 \(90^\circ\) 的相移
+
+在交流电路中，欧姆定律写作：
+
+\[
+\underline{I} = \frac{\underline{V}}{Z}
+\]
+
+下划线表示相量。
+
+### 示例
+
+若某条线路的阻抗为：
+
+\[
+Z = 2 + j8\ \Omega
+\]
+
+则：
+
+- 电阻为 \(2\ \Omega\)
+- 感抗为 \(8\ \Omega\)
+
+阻抗的模为：
+
+\[
+|Z| = \sqrt{2^2 + 8^2} \approx 8.25\ \Omega
+\]
+
+由于电抗远大于电阻，该线路以感性为主。
+
+---
+
+## 8. 电感
+
+电感通常由线圈构成，在磁场中储存能量。
+
+其核心特性是：
+
+> 电感阻碍电流的变化。
+
+### 8.1 电感如何工作
+
+电流流过导线时会产生磁场。
+
+把导线绕成线圈，可以增强并集中磁场。
+
+电流变化会引起磁场变化，变化的磁场又会产生一个阻碍原有变化的感应电压，这就是楞次定律。
+
+基本关系为：
+
+\[
+v_L = L\frac{di}{dt}
+\]
+
+其中：
+
+- \(v_L\) 为电感电压
+- \(L\) 为电感量，单位为亨利
+- \(di/dt\) 为电流变化率
+
+### 8.2 直流特性
+
+刚接入直流电源时，电感会阻碍电流突然上升。
+
+在串联 RL 电路中：
+
+\[
+i(t) = \frac{V}{R}\left(1-e^{-tR/L}\right)
+\]
+
+时间常数为：
+
+\[
+\tau = \frac{L}{R}
+\]
+
+电流稳定后：
+
+\[
+\frac{di}{dt}=0
+\]
+
+因此：
+
+\[
+v_L = 0
+\]
+
+此时，理想电感对稳态直流相当于短路。
+
+### 8.3 电感断电
+
+电流突然中断时，正在消失的磁场会产生试图维持原有电流的电压。
+
+这可能造成：
+
+- 火花
+- 电弧
+- 高压暂态
+- 开关或晶体管损坏
+
+续流二极管等保护器件可为电流提供安全通路。
+
+### 8.4 储存的能量
+
+电感在磁场中储存能量：
+
+\[
+W_L = \frac{1}{2}LI^2
+\]
+
+### 8.5 感抗
+
+对于正弦交流电：
+
+\[
+X_L = 2\pi fL
+\]
+
+在以下情况下，感抗增大：
+
+- 频率升高
+- 电感量增大
+
+理想电感的阻抗为：
+
+\[
+Z_L = j\omega L
+\]
+
+### 8.6 电网中的电感
+
+即使没有明显的线圈，电网设备也具有电感。
+
+例如：
+
+- 输电线路
+- 变压器
+- 电机
+- 发电机
+- 电抗器
+
+电力系统中的电抗器是大型电感，用于：
+
+- 限制故障电流
+- 控制电压
+- 吸收无功功率
+- 减小暂态过程
+
+---
+
+## 9. 电容
+
+电容在电场中储存能量。
+
+其核心特性是：
+
+> 电容阻碍电压的变化。
+
+### 9.1 结构
+
+基本电容器由两块导体极板和其间的绝缘介质组成。
+
+\[
+\text{极板} \; | \; \text{介质} \; | \; \text{极板}
+\]
+
+介质可以是空气、陶瓷、塑料薄膜、氧化物或其他绝缘材料。
+
+### 9.2 电荷与电压
+
+储存的电荷为：
+
+\[
+Q = CV
+\]
+
+其中：
+
+- \(Q\) 为电荷
+- \(C\) 为电容量，单位为法拉
+- \(V\) 为电压
+
+对于理想平行板电容器：
+
+\[
+C = \frac{\varepsilon A}{d}
+\]
+
+在以下情况下，电容量增大：
+
+- 极板面积增大
+- 极板间距减小
+- 介质介电常数增大
+
+### 9.3 电流—电压关系
+
+电容的基本方程为：
+
+\[
+i = C\frac{dv}{dt}
+\]
+
+电容两端电压变化时，就会有电流流动。
+
+有限电流只能以有限速率改变电容电压，因此理想电路中的电容电压不能瞬间跳变。
+
+### 9.4 直流充电
+
+未充电的电容通过电阻接入直流电源时：
+
+\[
+v_C(t) = V\left(1-e^{-t/RC}\right)
+\]
+
+并且：
+
+\[
+i(t) = \frac{V}{R}e^{-t/RC}
+\]
+
+时间常数为：
+
+\[
+\tau = RC
+\]
+
+电容充满后，电压保持恒定，因此：
+
+\[
+\frac{dv}{dt}=0
+\]
+
+从而：
+
+\[
+i=0
+\]
+
+理想电容对稳态直流相当于开路。
+
+### 9.5 储存的能量
+
+电容在电场中储存能量：
+
+\[
+W_C = \frac{1}{2}CV^2
+\]
+
+### 9.6 容抗
+
+对于正弦交流电：
+
+\[
+X_C = \frac{1}{2\pi fC}
+\]
+
+在以下情况下，容抗减小：
+
+- 频率升高
+- 电容量增大
+
+理想电容的阻抗为：
+
+\[
+Z_C = \frac{1}{j\omega C} = -jX_C
+\]
+
+### 9.7 电网中的电容
+
+电网中的电容存在于：
+
+- 输电线路
+- 地下电缆
+- 电容器组
+- 串联补偿设备
+- 并联补偿设备
+
+电容器组用于：
+
+- 支撑电压
+- 就地提供无功功率
+- 改善功率因数
+- 降低线路电流
+- 减少 \(I^2R\) 损耗
+
+串联电容器抵消一部分线路感抗：
+
+\[
+X_{\text{net}} = X_L - X_C
+\]
+
+并联电容器接在母线上，用于支撑母线电压。
+
+---
+
+## 10. 电感与电容对比
+
+| 特性 | 电感 | 电容 |
+|---|---|---|
+| 储能场所 | 磁场 | 电场 |
+| 阻碍其变化的量 | 电流 | 电压 |
+| 基本方程 | \(v=L\,di/dt\) | \(i=C\,dv/dt\) |
+| 稳态直流特性 | 理想情况下短路 | 理想情况下开路 |
+| 交流相位关系 | 电流滞后电压 | 电流超前电压 |
+| 频率升高时的电抗 | 增大 | 减小 |
+| 理想阻抗 | \(j\omega L\) | \(1/(j\omega C)\) |
+
+---
+
+## 11. 电流反向时，功率为何仍可流向负荷
+
+对于纯电阻：
+
+\[
+v(t)=V_{\text{peak}}\sin(\omega t)
+\]
+
+\[
+i(t)=I_{\text{peak}}\sin(\omega t)
+\]
+
+瞬时功率为：
+
+\[
+p(t)=v(t)i(t)
+\]
+
+因此：
+
+\[
+p(t)=V_{\text{peak}}I_{\text{peak}}\sin^2(\omega t)
+\]
+
+由于 \(\sin^2(\omega t)\) 永不为负，电阻在交流周期的两个半周都吸收能量。
+
+当电压和电流同时反向时，两者乘积仍为正。
+
+对于理想电感和电容，功率会在正值与负值之间交替，因为能量不断被储存并返还。
+
+---
+
+## 12. 基尔霍夫定律
+
+基尔霍夫定律描述电气网络必须遵守的基本守恒规则。
+
+### 12.1 基尔霍夫电流定律
+
+在任意节点或母线上：
+
+\[
+\sum I_{\text{流入}} = \sum I_{\text{流出}}
+\]
+
+电流不能在理想节点上积聚。
+
+在电力系统中，这体现为每条母线上的有功和无功平衡。
+
+简化的有功平衡为：
+
+\[
+P_{\text{发电}} - P_{\text{负荷}}
+=
+P_{\text{送出}} + P_{\text{损耗}}
+\]
+
+无功功率也有类似的平衡方程。
+
+### 12.2 基尔霍夫电压定律
+
+沿任意闭合回路：
+
+\[
+\sum V = 0
+\]
+
+所有电压升之和等于所有电压降之和。
+
+对于输电线路：
+
+\[
+\underline{V}_1 - \underline{V}_2 = \underline{I}Z
+\]
+
+其中：
+
+- \(\underline{V}_1\) 和 \(\underline{V}_2\) 为母线电压相量
+- \(\underline{I}\) 为线路电流
+- \(Z\) 为线路阻抗
+
+### 12.3 在电网分析中的作用
+
+结合使用基尔霍夫两条定律，工程师可以计算：
+
+- 母线电压
+- 线路电流
+- 有功潮流
+- 无功潮流
+- 损耗
+- 故障电流
+
+它们是以下分析的基础：
+
+- 潮流分析
+- 短路分析
+- 稳定性分析
+- 保护设计
+
+---
+
+## 13. 功率如何在交流电网中流动
+
+在互联交流网络中，功率会在所有可用路径之间分配。
+
+功率分布取决于：
+
+- 线路电阻
+- 线路电抗
+- 母线电压幅值
+- 电压相角
+- 网络连接关系
+
+低阻抗路径通常比高阻抗路径承载更多电流。
+
+高压输电系统的感抗通常大于电阻，因此有功潮流很大程度上受母线间相角差影响。
+
+电网调度机构通过以下方式影响潮流：
+
+- 调整发电机出力
+- 投切输电线路
+- 调整变压器分接头
+- 投切电容器组或电抗器
+- 使用电力电子控制器
+
+---
+
+## 14. 核心心智模型
+
+### 交流电
+
+\[
+\text{发电机旋转}
+\rightarrow
+\text{交变电压}
+\rightarrow
+\text{交变电流}
+\rightarrow
+\text{能量传输}
+\]
+
+### 电感
+
+\[
+\text{电流变化}
+\rightarrow
+\text{磁场变化}
+\rightarrow
+\text{产生感应电压}
+\rightarrow
+\text{阻碍电流变化}
+\]
+
+### 电容
+
+\[
+\text{电压变化}
+\rightarrow
+\text{电荷移动}
+\rightarrow
+\text{电场变化}
+\rightarrow
+\text{产生电流}
+\]
+
+### 电网分析
+
+\[
+\text{网络模型}
++
+\text{阻抗}
++
+\text{基尔霍夫定律}
+\rightarrow
+\text{电压、电流与潮流}
+\]
+
+---
+
+## 15. 总结
+
+电网是把电能从发电机输送到负荷的互联交流网络。
+
+最重要的概念包括：
+
+- 高电压可以降低输电电流和电阻损耗。
+- 交流电压与电流周期性反转方向。
+- 电能由导体引导的电磁场传输。
+- 电阻消耗有功功率。
+- 电感在磁场中储能，并阻碍电流变化。
+- 电容在电场中储能，并阻碍电压变化。
+- 阻抗由电阻与电抗共同组成。
+- 基尔霍夫电流定律约束母线处的平衡。
+- 基尔霍夫电压定律约束回路中的电压一致性。
+- 这些原理构成潮流、故障和稳定性分析的基础。