Use upright font for derivative operator

Author: dietmarw

OpenHPL/ElectroMech/Generators/SimpleGen.mo

@@ -23,7 +23,7 @@ model SimpleGen "Model of a simple generator with mechanical connectors"
 where ω<sub>a</sub> is angular velocity and J<sub>a</sub> is moment of inertia.</p>
 
 <p>From energy balance:</p>
-<p>$$ \\frac{dK_a}{dt} = \\dot{W}_s - \\dot{W}_{f,a} - \\dot{W}_g $$</p>
+<p>$$ \\frac{\\mathrm{d}K_a}{\\mathrm{d}t} = \\dot{W}_s - \\dot{W}_{f,a} - \\dot{W}_g $$</p>
 <p>where:</p>
 <ul>
 <li>Ẇ<sub>s</sub> is turbine shaft power</li>

OpenHPL/ElectroMech/Generators/SynchGen.mo

@@ -127,29 +127,29 @@ equation
 </ul>
 
 <h5>Phase Shift Angle Dynamics</h5>
-<p>$$ \\frac{d\\delta_e}{dt} = (\\omega - \\omega_s)\\frac{n_p}{2} $$</p>
+<p>$$ \\frac{\\mathrm{d}\\delta_e}{\\mathrm{d}t} = (\\omega - \\omega_s)\\frac{n_p}{2} $$</p>
 <p>where \\(n_p\\) is number of poles, \\(\\omega\\) and \\(\\omega_s\\) are generator and grid angular velocities.</p>
 
 <h5>Swing Equation</h5>
-<p>$$ \\frac{d\\omega}{dt}=\\frac{\\dot{W}_s-P_e}{J\\omega} $$</p>
+<p>$$ \\frac{\\mathrm{d}\\omega}{\\mathrm{d}t}=\\frac{\\dot{W}_s-P_e}{J\\omega} $$</p>
 
 <h5>Transient Operation</h5>
 <p>$$
 \\begin{array}{c}
-T_{qo}'\\frac{dE_d'}{dt} =-E_d' + (x_q' - x_q)I_q \\\\
-T_{do}'\\frac{dE_q'}{dt} = -E_q' + (x_d - x_d')I_d + E_f
+T_{qo}'\\frac{\\mathrm{d}E_d'}{\\mathrm{d}t} =-E_d' + (x_q' - x_q)I_q \\\\
+T_{do}'\\frac{\\mathrm{d}E_q'}{\\mathrm{d}t} = -E_q' + (x_d - x_d')I_d + E_f
 \\end{array}
 $$</p>
 <p>where \\(T_{do}'\\) and \\(T_{qo}'\\) are d-/q-axis transient open-circuit time constants.</p>
 
 <h5>Excitation System</h5>
 <p>Field voltage dynamics:</p>
-<p>$$ \\frac{dE_f}{dt} = \\frac{-E_f + K_E\\left(V_{tr}-V_t-V_{stab}\\right)}{T_E} $$</p>
+<p>$$ \\frac{\\mathrm{d}E_f}{\\mathrm{d}t} = \\frac{-E_f + K_E\\left(V_{tr}-V_t-V_{stab}\\right)}{T_E} $$</p>
 <p>where \\(K_E\\) is excitation system gain, \\(T_E\\) is excitation time constant, \\(V_{tr}\\) is voltage reference set point, 
 and \\(V_t = \\sqrt{\\left(E_d'-R_aI_d-x_q'I_q\\right)^2+\\left(E_q'-R_aI_q+x_d'I_d\\right)^2}\\) is terminal voltage.</p>
 
 <h5>Stabilization</h5>
-<p>$$ \\frac{dV_{stab}}{dt} = \\frac{-V_{stab} + K_F\\frac{dE_f}{dt}}{T_{FE}} $$</p>
+<p>$$ \\frac{\\mathrm{d}V_{stab}}{\\mathrm{d}t} = \\frac{-V_{stab} + K_F\\frac{\\mathrm{d}E_f}{\\mathrm{d}t}}{T_{FE}} $$</p>
 <p>where \\(K_F\\) is stabilizer gain and \\(T_{FE}\\) is stabilizer time constant.</p>
 
 <h5>Output Power</h5>

OpenHPL/Functions/KP07/package.mo

@@ -33,7 +33,7 @@ With the finite volume method, we divide the grid into small control volumes/cel
 conservation laws. The semi-discrete (time-dependent ODEs) central-upwind scheme can be written as:
 </p>
 <p>
-$$ \\frac{d}{dt}\\bar{U}_j(t) = -\\frac{H_{j+\\frac{1}{2}}(t) - H_{j-\\frac{1}{2}}(t)}{\\Delta x} + \\bar{S}_j(t) $$
+$$ \\frac{\\mathrm{d}}{\\mathrm{d}t}\\bar{U}_j(t) = -\\frac{H_{j+\\frac{1}{2}}(t) - H_{j-\\frac{1}{2}}(t)}{\\Delta x} + \\bar{S}_j(t) $$
 </p>
 <p>
 Here, Ū<sub>j</sub> are the cell centre average values, while H<sub>j±1/2</sub>(t) are the central 

OpenHPL/Waterway/Pipe.mo

@@ -86,10 +86,10 @@ small pressure variations.</p>
 <h5>Mass and Momentum Balance</h5>
 
 <p><strong>Mass Balance:</strong> For incompressible water, the mass in the filled pipe is constant:</p>
-<p>$$ \\frac{dm_\\mathrm{c}}{dt} = \\dot{m}_\\mathrm{c,in} - \\dot{m}_\\mathrm{c,out} = 0 $$</p>
+<p>$$ \\frac{\\mathrm{d}m_\\mathrm{c}}{\\mathrm{d}t} = \\dot{m}_\\mathrm{c,in} - \\dot{m}_\\mathrm{c,out} = 0 $$</p>
 
 <p><strong>Momentum Balance:</strong> The momentum balance is expressed as:</p>
-<p>$$ \\frac{dM_\\mathrm{c}}{dt} = \\dot{M}_\\mathrm{c,in} - \\dot{M}_\\mathrm{c,out} + F_\\mathrm{p,c} + F_\\mathrm{g,c} + F_\\mathrm{f,c} $$</p>
+<p>$$ \\frac{\\mathrm{d}M_\\mathrm{c}}{\\mathrm{d}t} = \\dot{M}_\\mathrm{c,in} - \\dot{M}_\\mathrm{c,out} + F_\\mathrm{p,c} + F_\\mathrm{g,c} + F_\\mathrm{f,c} $$</p>
 <p>where:</p>
 <ul>
 <li>M<sub>c</sub> = m<sub>c</sub> v<sub>c</sub> is the momentum</li>

OpenHPL/Waterway/Reservoir.mo

@@ -107,8 +107,8 @@ the reservoir outlet pressure is simply:</p>
 
 <p><strong>Detailed Model:</strong> For a detailed model with dynamics and inflow,
 the mass and momentum balances are:</p>
-<p>$$ H\\frac{d\\dot{m}}{dt} = \\frac{\\rho}{A}\\dot{V}^2 + A(p_\\mathrm{atm}-p_\\mathrm{o}) + \\rho gHA - F_\\mathrm{f,r} $$</p>
-<p>$$ \\frac{dm}{dt} = \\dot{m}_\\mathrm{i} - \\dot{m}_\\mathrm{o} $$</p>
+<p>$$ H\\frac{\\mathrm{d}\\dot{m}}{\\mathrm{d}t} = \\frac{\\rho}{A}\\dot{V}^2 + A(p_\\mathrm{atm}-p_\\mathrm{o}) + \\rho gHA - F_\\mathrm{f,r} $$</p>
+<p>$$ \\frac{\\mathrm{d}m}{\\mathrm{d}t} = \\dot{m}_\\mathrm{i} - \\dot{m}_\\mathrm{o} $$</p>
 <p>where ṁ is the reservoir mass flow rate, A is the cross-sectional area,
 p<sub>atm</sub> and p<sub>o</sub> are atmospheric and outlet pressures, and F<sub>f,r</sub> is
 the friction term (typically small for large reservoirs).</p>

OpenHPL/Waterway/RunOff_zones.mo

@@ -136,7 +136,7 @@ each elevation zone. Using the mass balance, the change in the dry snow storage
 as follows:
 </p>
 <p>
-$$ \\frac{dV_\\mathrm{s,d}}{dt}=\\dot{V}_\\mathrm{p,s}-\\dot{V}_\\mathrm{d2w} $$
+$$ \\frac{\\mathrm{d}V_\\mathrm{s,d}}{\\mathrm{d}t}=\\dot{V}_\\mathrm{p,s}-\\dot{V}_\\mathrm{d2w} $$
 </p>
 <p>
 Here, the flow of the precipitation in the form of snow is denoted as \\(\\dot{V}_\\mathrm{p,s}\\). This precipitation 
@@ -188,7 +188,7 @@ from the snow-free areas. The net runoff to the next segment (upper zone) is als
 the volume of the soil moisture storage, \\(V_\\mathrm{s,m}\\), is found as follows:
 </p>
 <p>
-$$ \\frac{dV_\\mathrm{s,m}}{dt}=\\dot{V}_\\mathrm{s2s}-\\dot{V}_\\mathrm{s2u}-\\alpha_\\mathrm{e}\\dot{V}_\\mathrm{s,e} $$
+$$ \\frac{\\mathrm{d}V_\\mathrm{s,m}}{\\mathrm{d}t}=\\dot{V}_\\mathrm{s2s}-\\dot{V}_\\mathrm{s2u}-\\alpha_\\mathrm{e}\\dot{V}_\\mathrm{s,e} $$
 </p>
 <p>
 Here, \\(\\dot{V}_\\mathrm{s2u}\\) is the net runoff to the next segment (the upper zone). \\(\\dot{V}_\\mathrm{s,e}\\) 
@@ -234,7 +234,7 @@ evapotranspiration from the lakes in the catchment area are also taken into acco
 The upper zone characterises components with quick runoff. The following mass balance is used for the upper zone description:
 </p>
 <p>
-$$ \\frac{dV_\\mathrm{u,w}}{dt}=\\dot{V}_\\mathrm{s2u}-\\dot{V}_\\mathrm{u2l}-\\dot{V}_\\mathrm{u2s}-\\dot{V}_\\mathrm{u2q} $$
+$$ \\frac{\\mathrm{d}V_\\mathrm{u,w}}{\\mathrm{d}t}=\\dot{V}_\\mathrm{s2u}-\\dot{V}_\\mathrm{u2l}-\\dot{V}_\\mathrm{u2s}-\\dot{V}_\\mathrm{u2q} $$
 </p>
 <p>
 Here, \\(V_\\mathrm{u,w}\\) is the water volume in the upper zone that depends on the saturation threshold, s<sub>T</sub>, 
@@ -266,7 +266,7 @@ The lower zone characterises the lake and the groundwater storages and defines t
 The following mass balance equation is used for the lower zone description:
 </p>
 <p>
-$$ \\frac{dV_\\mathrm{l,w}}{dt}=\\dot{V}_\\mathrm{u2l}+a_\\mathrm{L}\\dot{V}_\\mathrm{p}-\\dot{V}_\\mathrm{l2b}-a_\\mathrm{L}\\dot{V}_\\mathrm{e} $$
+$$ \\frac{\\mathrm{d}V_\\mathrm{l,w}}{\\mathrm{d}t}=\\dot{V}_\\mathrm{u2l}+a_\\mathrm{L}\\dot{V}_\\mathrm{p}-\\dot{V}_\\mathrm{l2b}-a_\\mathrm{L}\\dot{V}_\\mathrm{e} $$
 </p>
 <p>
 The water volume in the lower zone is denoted as \\(V_\\mathrm{l,w}\\). As mentioned previously, \\(\\dot{V}_\\mathrm{p}\\) 

OpenHPL/Waterway/SurgeTank.mo

@@ -159,8 +159,8 @@ connecting the conduit, surge volume, and penstock. Four different surge tank co
 <h5>Mass and Momentum Balances</h5>
 
 <p>All surge tank types are modeled using mass and momentum balance equations:</p>
-<p>$$ \\frac{dm}{dt} = \\dot{m}_\\mathrm{s,in} = \\rho \\dot{V} $$</p>
-<p>$$ \\frac{d(mv)}{dt} = \\dot{m}_\\mathrm{i}v_\\mathrm{i} + F_\\mathrm{p} + F_\\mathrm{g} + F_\\mathrm{f} $$</p>
+<p>$$ \\frac{\\mathrm{d}m}{\\mathrm{d}t} = \\dot{m}_\\mathrm{s,in} = \\rho \\dot{V} $$</p>
+<p>$$ \\frac{\\mathrm{d}(mv)}{\\mathrm{d}t} = \\dot{m}_\\mathrm{i}v_\\mathrm{i} + F_\\mathrm{p} + F_\\mathrm{g} + F_\\mathrm{f} $$</p>
 
 <h5>Water Mass and Geometry</h5>